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A supplementary lecture note for the book Approximation algorithmsby V. Vazirani

2007 4§<Æl

<ª $í 9 þj&hoHKZO

IntroductionSet Cover

Part I

Chapter 1, 2



NP-optimization problemsApproximation algorithmsLower bounding OPTWell-characterized problems

NP-optimization problems

&ñ_ 1.1

An NP-optimization, Π consists of the followings.

A set of valid instances, DΠ, recognizable in polynomial time. All thenumbers are rational. The size of an instance I ∈ DΠ, denoted by |I | isthe number of bits needed to write I in binary.

Each instance I ∈ DΠ has a nonempty set of feasible solutions, SΠ(I ).Every solution s ∈ SΠ(I ) is of length polynomially bounded in |I |.Also, there is polynomial time algorithm that, given I and s, decideswhether s ∈ SΠ(I ).

There is a polynomial time computable objective functionobjΠ : SΠ(I )→ Q+.

Π is specified to be either a maximization problem or a minimizationproblem.




When Π is a minimization problem, s ∈ SΠ(I ) is an optimalsolution for an instance I if objΠ(s) is the largest among thefeasible solutions of SΠ(I ).

OPTΠ(I ) or simply OPT will denote the largest objective value.




Approximation algorithms

Let Π be a minimization (maximization) problem, and δ : Z+ → Q+ afunction with δ ≥ 1 (δ ≤ 1, resp.). An algorithm A is said to be factor δapproximation algorithm, or δ-approximation of Π if, on each instance I ,A produces a feasible solution s ∈ SΠ(I ) such that

objΠ(s) ≤ δ(|I |)OPTΠ(I )

(objΠ(s) ≥ δ(|I |)OPTΠ(I ), resp.).

&ñ_ 1.2

(Vertex Cover) Given an undirected graph G = (V ,E ), and a costfunction c : V → Q+, find a minimum cost vertex cover, namelyU ⊆ V such that every edge has at least one endpoint in U.

Notation: c(v) = cv c(U) =∑

v∈U cv .

cv ¿º 19 M:, Cardinality Vertex Cover (CVC) ¦ ÂÒÉr.




Lower bounding OPT

NP-hard NP-þj&ho ëH]j_ OPT\¦ ½ H Érכ K\¦ ½ H õכ ðøÍtÐ NP-hards (=?)."f H·ú¦o1pu¦ n ½+É M:H OPT\¦ @/½+É Ãº eH °úכs ¹כ9. ½Órçß >íß0pxôÇ OPT_ ôÇ(lower bound) (þjèo ëH]j_ âÄº)¦Ð:x 6 xôÇ.sQôÇ Lower bounding¦ CVC \V\¦ :xK ¶ú(R Ð.

&ñ_ 1.3

(Matching) Given an undirected graph G = (V ,E), an edge set M ⊂ E iscalled a matching if no two edges of M shares an endpoint.

$í|9 1.4

|M| ≤ |C | ∀ matching M and cover C.




·ú¦o1pu 1.5

Find a maximal matching M of G and return the M-coveredvertices.

s ∈ S is maximal if no other element of S is “larger than” s andmaximum if s is “larger than” or equal to any other element.

&ño 1.6

Algorithm 1.5 is a 2-approximation of CVC.

Proof




Some questions

1. A better analysis of Algorithm 1.5 is possible?

2. A better algorithm with the same lower bounding scheme?

3. A better approximation?




Well-characterized problems Π

&ñ_ 1.7

&ñëH]j Π ∈ NP ⇐⇒ Π has a YES-certificate: Π_ ²ús ‘\V’9M:, s\¦ ½Órçß\ SX > H H >rF.

$í|9 1.8

P ⊆ NP.

&ñ_ 1.9

Π ∈ NP ⇐⇒ Πc ∈ co−NP.

$í|9 1.10

P ⊆ co−NP.




VC: GH |C | ≤ k vertex cover C\¦ tH?VCc : G_ H vertex cover CH |C | > k?

• VC NP\ 5Åq H Érכ ~1> ·ú Ãº e. YES-certificateÉr ~1> ëß[þt Ãº el M:ëHs. 7£¤, VCc

H co-NP\ 5ÅqôÇ. ÕªQ,VCH YES-certificateõH ²úo, /'îr NO-certificate (VCc_

YES-certificate)s >rF t ·ú§Ü¼ 9, z]jÐ >rF t ·ú§H¦~ÃÎ[þts¦ e.

ÆÒ8£¤ 1.11

VCc /∈ NP (equiv, VC /∈ co−NP).

ÕªQ , 6£§õ °ú Ér ÆÒ8£¤s %ir $íwn > )a.

ÆÒ8£¤ 1.12

NP 6= co−NP.




Widely accepted picture

$í|9 1.13

ÆÒ8£¤ 1.12¦ &ñ #Q*ôÇ NP-hard ëH]j No-certificate¦|9 Ãº \O.

(=t [O"î½+É (.כ




&ñ_ 1.14

&ñëH]j Π well-characterized. ⇔ Π ∈ NP ∩ co−NP.

·ú¡_ 7H_\ _ # Π well characterizedH Érכ ½Ó·ú¦o1pu¦ H y©ôÇ ‘~½Ó7£x’s )a.%i&hÜ¼Ð, sìrÕªAáÔ\"f_ n&!Q, þj@/ f±l(matching) ëH]j, Õªo¦ +þA>S\(LP)Ér well-characterization¦°úHH z\ H # ½Ó·ú¦o1pu¦ ÆÒ½Ùþ¡~ @/³ð&h

\V.




\V 1.15

Áº¾Ó ÕªAáÔ G sìrÕªAáÔs þj@/ f±lü< þjè vertexcover_ ß¼lH °ú . (Konig_ &ño)"f VCH sìr ÕªAáÔ\"fH well-characterized. z]jÐsìrÕªAáÔ\"f vertex coverëH]jH ½Ó·ú¦o1pus >rFôÇ.

ÃÐ¦ 1.16

sìrÕªAáÔ_ &ñs \OÜ¼ ¿º °úכs °ú t ·ú§Ér \V\¦ ~1> ëß[þt

Ãº e. \V\¦ [þt#Q, f.ËÃº rÐ. ¢ôÇ x'H ÕªAáÔH¢-af±l\¦ °úÜ¼"f þjè VC_ ß¼l ¢-af±l_ß¼l 5 Ð ß¼. sH x'H ÕªAáÔ n\¦ /BNÄ» t ·ú§H¿º>h_ Us 5 f.ËÃºrÐ\¦ °úl M:ëHX<, yy þjè 3>h_n ¹כ9l M:ëHs.




\V 1.17

f.ËÃº n&!Qü< þj@/ß¼l f±l (Tutte-Berge Formula)

\V 1.18

LP-duality

"f, sìrÕªAáÔ_ f±lü< °ú s þj@/ß¼l f±l, Õªo¦LPH well-characterized.ú§Ér min-max 'a>H LP-dualityÐ K$3½+É Ãº e. (Konig_ &ñoü< Tutte-Berge Formula ðøÍt.)+'\ Ðxtëß LP-dualityH ½Ó·ú¦o1pu ÷rëß m HKZO >hµ1Ï\ BÄº Ä»6 x .



Set CoverLayering

Greedy N±Ó§hæ

1. C ←− ∅;

2. while C 6= U do&³F © Ér îçHq6 x(α¦ )¦ °úH S_ |9½+Ë, say,S\¦ ×þ; S − C_ y "é¶è v[þt_ , p(v) = αÐ &ñ_;

C ←− C ∪ S ;

3. ×þ)a |9½+Ë[þt¦ Ø¦§4.



Set CoverLayering

Tight example

U = 1, 2, . . . , n, c(1, 2, . . . , n) = 1 + ε,c(1) = 1

n , c(2) = 1n−1 , . . ., c(n) = 1.

Essentially the best possible approximation!



Set CoverLayering

Layering

n ×æu w : V → Q+ #Q" ©Ãº c > 0 >rF #w(v) = cdeg(v)\¦ ëß7á¤ degree-weighted¦ ÂÒØÔ.

l:r&ño 2.4

w degree-weighteds w(V ) ≤ 2OPT.

¤ ÃZ: U\¦ þj&h&!Q¦ . &!Qsl M:ëH\∑v∈U deg(v) ≥ |E |. "f w(U) ≥ c |E |. ÕªX<

w(V ) = c∑

v∈V deg(v) = 2c |E |. "f, w(V ) ≤ 2OPT.

e__ ×æu w\ @/K"f, 6£§õ °ú s þj@/ degree-weighted×æu\¦ &ñ_ : Ãº 0 n\¦ ¿º ]jôÇ.c = minw(v)/deg(v)\¦ >íß ¦ DhÐîr ×æut(v) = cdeg(v)\¦ &ñ_ôÇ. Õªo¦ ïß#×æuw ′(v) = w(v)− t(v)\¦ >íßôÇ.



Set CoverLayering

6£§õ °ú Ér ·ú¦o1pu¦ &h6 xôÇ.

G0 = GÐ Z~H. G0Ð ÂÒ' Ãº 0 n |9½+Ë D0\¦ ¿º

]j ¦ w\ @/ # þj@/ degree-weighted ×æu\¦ >íßôÇ.s M: 0_ ïß#×æu\¦ °úH n|9½+Ë¦ W0¦ .

G1¦ n |9½+Ë V − (W0 ∪D0)Ð Ä»)a ÕªAáÔ¦ . 0A_õ&ñ¦ ïß#×æu\¦ ×æuÐ 6 x H G1\ ìøÍ4¤ôÇ.

s õ&ñÉr H n_ Ãº 0s ÷&H Gk Òqt$í|c M: t

ìøÍ4¤ôÇ. C = W0 ∪ · · · ∪Wk−1¦ KÐ Òqt$íôÇ.



Set CoverLayering

t0, t1, . . . , tk−1¦ G0,G1, . . . , tk−1_ þj@/ degree-weighted ×æu¦ .



Set CoverLayering

&ño 2.5

0A_ ·ú¦o1puÉr e__ ×æu\ @/ # 2-HKZOs.

¤ ÃZ: Äº C n&!Qe¦ Ðs. ëß m,V − C = D0 ∪ · · · ∪ DkH zÐ ÂÒ', #Q" iü< j\ @/ #,u ∈ Di , v ∈ DjH _ps. i ≤ j¦ . ÕªQ (u, v)Gi_ ñs#Q 9 sH u Gi\"f Ãº 0sH \כ íHs.

Hu\¦ 7£x"î l 0AK C ∗\¦ þj&h n&!Q¦ . ëß

v ∈ C ∩Wjs w(v) =∑

i≤j ti (v) )a. "f

w(C ) =∑k−1

i=0 ti (C ∩ V (Gi )). ôÇ¼# v ∈ Djs

w(v) ≥∑

i<j ti (v) =∑

i≤j ti (v) (tj(v) = cdegGj(v) = 0). "f

e__ n |9½+Ë S\ @/ # w(S) ≥∑k−1

i=0 ti (S ∩ V (Gi )).



Set CoverLayering

Äº y i\ @/K, C ∗ ∩ V (Gi )H Gi_ n&!Q )a. "f·ú¡_ l:r&ño\ _ #

ti (C ∩ V (Gi )) ≤ ti (V (Gi )) ≤ 2ti (C∗ ∩ V (Gi )).

w(C ) =∑k−1

i=0 ti (C ∩ V (Gi )) ≤ 2∑k−1

i=0 ti (C∗ ∩ V (Gi )) ≤ 2w(C ∗).

Tight example

Kn,n with unit vertex weights.


Steiner Tree and TSP

Part II

Chapter 3


Steiner Tree and TSPMulticutõ &ñÃº¾¡§3lqâì2£§ëH]j: Áº_ âÄºMST\¦ 6 xôÇ StT ·ú¦o1puBjàÔaË: TSP

Multicutõ &ñÃº¾¡§3lqâì2£§ëH]j: Áº_ âÄº

&ñ_ 3.1

RcçÃÍÏ â«l/ÈÁ(MStT) G ¢-a ñ (complete) ÕªAáÔs¦ ñ_q6 xs 6£§õ °ú s yÂÒ1pxd¦ ëß7á¤ H Û¼s-Áº ëH]j\¦ BjàÔaË:

Û¼s- ÁºëH]j¦ :

cij + cjk ≥ cik , ∀ i , j , k ∈ V .

&ño 3.2

e__ Û¼s-Áº ëH]j\¦, H>Ãº Ä»t÷&2¤ "f, BjàÔaË:Û¼s- Áº ëH]jÐ ½Ó8Ö Ãº e.

¤ ÃZ: 0A_ &ñ_ü< °ú s Û¼s-Áº ëH]j e¦ M:, °ú Ér n|9½+Ë¦ ¢-a ñ ÕªAáÔ G ′\"f Û¼s-Áº ëH]j\¦ &ñ_ôÇ. SH 1lx9 > Z~H.q6 x¦ 6£§õ °ú s &ñ_ôÇ.

c ′ij = G_ i − j þjéßâÐ q6 x.



ñq6 x c ′s yÂÒ1pxd¦ ëß7á¤ H Érכ © . ¢ôÇ G ′_ ñq6 xÉr G_ ñq6 xÐ ß¼t ·ú§. "f, G ′_ þj&hKHG_ þj&hKÐ q6 xs ß¼t ·ú§. G ′_ þj&hK_ ñ\ @/6£x H þjéßâÐ_ ñ\¦ ¿º í<Ê H G_ ÂÒìrÕªAáÔH Õª q6 xs °ú . ëß rÐ\¦ í<Ê rÐ Òqtlt ·ú§¦ M:t ñ\¦ ]j # G_ þj&h 0pxK\¦ %3¦ Ãº e.



MST\¦ 6 xôÇ StT ·ú¦o1pu

tFKÂÒ' G yÂÒ1pxd¦ ëß7á¤ôÇ¦ &ñ .

MST eð5 StT N±Ó§hæ

SÐ Ä»)a ÂÒìrÕªAáÔ\"f þjè©Áº(MST)\¦ ½ôÇ.

&ño 3.3

0A ·ú¦o1pu K_ q6 xÉr þj&h q6 x_ 2C\¦ Åt ·ú§H.

¤ ÃZ: q6 x OPT\¦ þj&hÛ¼s- Áº\¦ Òqty . Õªo¦ ñ\¦ ¿º CÐ4¤ôÇ. sXO> %3Ér ÕªAáÔ_ 9Q rÐ\¦ ½ . (\V\¦ [þt#Q 6£§õ°ú s U·sÄºÃÐÒoÜ¼Ð ½½+É Ãº e.) Õªo¦ ’Short-cutting’¦ 6 x #,S_ n[þt_ Kx9 âÐ\¦ ½ôÇ.



yÂÒ1pxd\ _ # Kx9 âÐ_ UsH OPT_ ¿ºC\¦ Åt ·ú§H. Kx9 âÐH S_ n\¦ í<Ê H gË>Áº\ s. "f &ño_ 7£x"îs =åQèß.

tight example



BjàÔaË: TSP

&ñ_ 3.4

TSP]: ¢-a ñ ÕªAáÔ G , q6£§ ñ q6 x c.·]: þjèq6 x ÈÒ#Q, 7£¤, H n\¦ &ñSXy ôÇ í<Ê H rÐ.

ñq6 xs yÂÒ1pxd¦ ëß7á¤ BjàÔaË: TSP¦ .

&ño 3.5

#Q" ½Órçß >íß0px <ÊÃº α(n)\ @/K"f TSP_ α(n)-HHÔ¦0px .

¤ ÃZ: 6£§õ °ú s ÕªAáÔ G = (V ,E) 0A_ HCÐ ÂÒ' n|9½+Ë V_ ¢-a ñÕªAáÔ G ′\ &ñ_)a TSPÐ_ ½Ó8¦ ÒqtyKÐ. G ′_ ñ e_ q6 xceH,

ce =

1, e ∈ Enα(n), e /∈ E .

ëß TSP_ α(n)-H 0px , s\¦ 6 x # ½Órçß\ &ñëH]j

HC\¦ Û¦ Ãº e6£§¦ ·ú Ãº e. <ª $í 9 þj&hoHKZO


RcçÃÍÏ TSP· Dø5 2-FD9

1. G_ MST\¦ ½ôÇ.2. MST_ ñ[þt¦ s×æ4¤ôÇ.3. 9Q rÐ\¦ ½ôÇ.4. 9Q rÐ\¦ âÄ» 9, n[þts %6£§ H íH"f@/Ð ‘short-cutting’¦:x # ÈÒ#Q\¦ ëßH.

Tight example



>h)a 32-H

1. G_ MST\¦ ½ôÇ.2. MST_ f.ËÃº Ãº n[þtÐ Ä»)a G_ ÂÒìrÕªAáÔ\"f þjèq6 x¢-af±l M¦ ½ #, s ñ[þt¦ MST\ ÆÒôÇ.3. 0A ÕªAáÔH H n Ãº ÃºsÙ¼Ð >rF H 9Q rÐ\¦½ôÇ.4. 9Q rÐ\¦ âÄ» 9 n[þts %6£§ H íH"f@/Ð ’short-cutting’¦6 x # ÈÒ#Q\¦ ëßH.

l:r&ño 3.6

U ⊆ V , |U|H Ãº, Õªo¦ Ms UÐ Ä»)a ÂÒìrÕªAáÔ_ þjèq6 x¢-af±l¦ . ÕªQ c(M) ≤ 1

2OPT.

¤ ÃZ: þj&hÈÒ#Q τ\¦ Òqty . Õªo¦ τ ′¦ ÈÒ#Q τ\¦ âÄ» 9 U\

short-cutting¦ &h6 x 9 ëßH ÈÒ#Q¦ . ÕªQ c(τ ′) ≤ c(τ). ÕªX<,

τH U_ ¿º>h_ ¢-af±l_ ½+Ë|9½+Ës )a. "f s×æ, q6 xs Ér

Érכ12OPT \¦ Åt 3lwôÇ. "f þjèq6 x ¢-af±l ÕªQ .



0A\"f ½ôÇ 9Q âÐ_ q6 xÉr

≤ c(MST ) + c(M) ≤ OPT +1

2OPT =

3

2OPT.

tight example

ÆÒ8£¤ 3.7

BjàÔaË: TSP_ 43-H 0px .


Multiway Cut and k-cut

Part III

Chapter 4


Multiway Cut and k-cut k-'pV,]Xéß, k-]XéßëH]j

k-'pV,]Xéß, k-]XéßëH]j

&ñ_ 4.1

âí5¢>(cut) )a Áº¾ÓÕªAáÔ G = (V ,E)_ V\¦ ìr½+É(partition) He__ ¿º |9½+Ë, U ∪ (V \ U)\ 5geH ñ_ |9½+Ë¦ ]Xéß(cut)s¦ôÇ. ëß s ∈ U, t ∈ V \ Us s − t ]Xéßs¦ ôÇ.

0Aü< °ú s ]XéßÜ¼Ð ìro ¦ z·Ér n[þt¦ 'pV,s¦ ÂÒØÔ.

&ñ_ 4.2

k-'aiÛyâí5(k-terminal cut £¿ÐM multyway cut)%KV]: Áº¾Ó ÕªAáÔ G = (V ,E), q6£§ ñ q6 x c, 'pV, |9½+Ës1, s2, . . . , sk ⊆ V .·]: ]j H 'pV,[þts "fÐ ìro÷&H ñ |9½+Ë, 7£¤,'pV,]Xéß(terminal cut) ×æ\"f þjèq6 x¦ °úH .כ



• k ≥ 3s NP-hard.

&ñ_ 4.3

k-âí5(k-cut)%KV]: Áº¾Ó ÕªAáÔ G = (V ,E ), q6£§ ñ q6 x c .·]: ]j H G k >h_ ¹èÐכ ìro÷&H ñ |9½+Ë, 7£¤,k-]Xéß(terminal cut) ×æ\"f þjèq6 x¦ °úH .כ

• k ¦&ñ)a ©Ãºs ½Órçß\ Û¦ Ãº e6£§. ÕªXOt ·ú§Ü¼NP-hard.



k-'aiÛyâí5¢> N±Ó§hæ

1. y i = 1, 2, , . . . , k\ @/K 'pV, si\¦ Ér H'pV,ÐÂÒ' ¦wnrvH þjèq6 x ]Xéß Ci\¦ ½ôÇ.2. Ci[þt ×æ\"f © q6 xs H ,כ say, Ck\¦ ]jü@rv¦ Qt

]Xéß_ ½+Ë|9½+Ë C ≡⋃k−1

i=1 Ci\¦ KÐ ôÇ.

• si\¦ ¦wnrvH þjè]XéßÉr, S \ si\¦ _ nÐ »¡¤ôÇÕªAáÔ\"f, »¡¤ nü< si\¦ ìro H þjèq6 x ]Xéß."f þj@/âì2£§ ·ú¦o1pu¦ 6 x ½Órçß\ ½½+É Ãº

e.



&ño 4.4

k-'pV,]Xéß ·ú¦o1puÉr (2− 2k)-HK\¦ Ð©ôÇ.

¤ ÃZ: þj&hK\¦ A¦ . H ñq6 xs q6£§sÙ¼Ð, AH G\¦, si\¦ ôÇ >h

m í<Ê H &ñSXy k>h_ ¹èÐכ ìroôÇ¦ &ñ½+É Ãº e. s M:, 'pV,si\¦ í<Ê H >¹èüכ Ér ¹è[þtכ s\ 5geH A_ ñ[þt¦ Ai¦ æ¼.A_ y ñH ¿º >h_ \¹èכ 5geÜ¼Ù¼Ð

A =k⋃

i=1

Ai ,

k∑i=1

c(Ai ) = 2c(A).

c(Ci ) ≤ c(Ai )sÙ¼Ð C 2-HH ¦כ ~1> ·ú Ãº etëß, FK 8 Érìr$3s 0px :

c(C) ≤∑k

i=1 c(Ci )− c(Ck) ≤∑k

i=1 c(Ai )− c(Ck)

≤ 2c(A)− 2kc(A).

t ÂÒ1px ñH c(Ck) #Q" c(Ai )Ð ß¼l M:ëHs.



tight example

·ú¦o1puÜ¼Ð %3Ér K_ q6 xÉr

(2− ε)(k − 1) = (2− ε)(1− 1

k

)k = (2− ε)

(1− 1

k

)OPT ∀ε > 0.



&ñ_ 4.5

§¿h-Ññ ÈÁ ñq6 x c\¦ ÈÁ/×ÕªAáÔ G = (V ,E ) e¦M:, n|9½+Ë Vü< ñ×æu w\¦ tH Áº T 6£§_$í|9¦ ëß7á¤½+É M: ¦o-Êê Áº¦ ôÇ.i) H e ∈ E\ @/K T\"f ñ e\¦ ]j ªìr÷&Hn|9½+Ë Uü< V \ UÐ &ñ÷&H G_ ]Xéß_ q6 xs we

)a: c(U;V \ U) = we .ii) H n © u, v ∈ V\ @/K, þjèq6 x u − v ]Xéß_ °úכsGü< T\"f °ú .

• þj@/âì2£§ ·ú¦o1pu¦ n − 1 &h6 x ¦o-Êê ÁºH %3¦ Ãº e(Exer. 4.3-4.6)!• sH

(n2

)>h n ©¦ ìro H þjè]Xéß[þt¦ T_ n − 1>h

_ ]XéßÜ¼Ð ¿º èq Ãº eH _ps.



k-âí5¢> N±Ó§hæ

1. G_ G-H Áº T\¦ ½ôÇ.2. T_ ñ ×æ\"f ×æu © Ér k − 1>h_ ñ\¦ YJ,G\"f s\ @/6£x H ]Xéß_ ½+Ë|9½+Ë C\¦ ½ôÇ.,

&ño 4.6

0A_ ·ú¦o1puÉr (2− 2k )-HK\¦ Ð©ôÇ.

¤ ÃZ: Äº C z]jÐ k>h s©_ ¹èÐכ G\¦ ìroôÇH¦כ Ð. (kÐ ß¼ k>h ÷&2¤ C\"f ñ\¦ NS )a.)T\"f k − 1_ ñ\¦ ]j , &ñSXy k>h_ >¹èüכ s\¹è[þtכ @/6£x H k>h_ n|9½+Ës ÒqtlHX<, s n|9½+Ë[þtÉr ¿º G\"f "fÐ ìro)aH ¦כ ·ú Ãº e."f k>h s©_ ¹èÐכ G ìro)a.



k-]Xéß ëH]j_ þj&hK\¦ A¦ . A\¦ ]j½+É M: ÒqtlH k>h_ _¹èכn|9½+Ë¦ V1, V2, . . . ,Vk , Õªo¦ Vi\¦ Ér n[þtõ ìro H ]Xéß¦

Ai¦ . ÕªQk∑

i=1

c(Ai ) = 2c(A).

s ×æ\"f ñq6 xs © H ¦כ Ak¦ .

ëß T\"f ×æu yy c(Ai ) i = 1, 2, . . . , k − 1Ð ß¼t ·ú§Ér k − 1>h_

ñ\¦ ¹1Ô¦ Ãº e 7£x"îÉr =åQ> )a. s\¦ 0AK T\"f V1, V2, . . . ,Vk_

y n|9½+Ë¦ _ nÐ »¡¤ôÇ. s M: n|9½+Ë s\"f zeH

ñ_ |9½+Ë¦ B¦ . ëß sXO> %3Ér ÕªAáÔ Áº m Áº

|c M:t ñ\¦ ]jôÇ. zÉr k − 1>h_ ñ|9½+Ë¦ B ′s¦ . ëß s

Áº\"f (u, v) ∈ B ′s¦ (u, v) Vi\ @/6£x H n\ ²ú9e¦ . Õª

Q wuvH þjè u − v ]Xéß_ q6 xs¦ AiH u − v ]Xéß ×æ sl M:

ëH\ wuv ≤ c(Ai ) $íwnôÇ.



CH T_ ñ×æ\"f q6 xs © Ér k − 1>h_ ñ_ q6 xsl M:ëH\,

c(C) ≤∑

e∈B′ we ≤∑k−1

i=1 c(Ai )

≤(1− 1

k

) ∑ki=1 c(Ai ) = 2

(1− 1

k

)c(A).

tight example


k-CenterFeedback Vertex Set

Part IV

Chapter 5, 6



k-G'p'ëH]j

• k-G'p'ëH]jH BjàÔaË:s âÄº, #Q" H>Ãº α(n) Ô¦0px.

• BjàÔaË: k-G'p'ëH]jH #Q" ε > 0\ @/K"f (2− ε)-H Ô¦0px .

• Hochbaumõ ShmoysH 2-HKZO >hµ1Ï.



k-G'p'ëH]j

BjàÔaË: k-G'p'ëH]j_ Ér 'a&h

ñ\¦ q6 x 7£x íH"fÐ &ñ\PôÇ: ce1 ≤ ce2 ≤ · · · ≤ cem .Ei ≡ e1, e2, . . . , ei. Gi ≡ (V ,Ei )Ð &ñ_ôÇ. k-G'p'ëH]jHß¼l k\¦ Åt ·ú§Ér dominating set¦ Gi×æ\"f þjè °úכ

i∗\¦ ×þ H ëH]jü< 1lxus. ÕªQ cei∗ þj&h3lq&h<ÊÃº

°úכs )a.• þjè ß¼l dominating set¦ ½ H ëH]jH NP-hards.

&ñ_ 5.3

V¹כÜ«8è«(square graph) Áº¾ÓÕªAáÔ G = (V ,E )_]jYLÕªAáÔ G 2

H °ú Ér n|9½+Ë Vü<, G\"f Us 2s _âÐÐ ÷&H n çß_ ñ |9½+ËÜ¼Ð ÅÒ#QtH ÕªAáÔÐ

&ñ_ôÇ.



k-G'p'ëH]j

l:r&ño 5.4

G 2_ îß&ñ|9½+Ë(stable set, independent set), I_ ß¼lH G_dominating set D_ ß¼l Ð °ú .

¤ ÃZ: G_ n|9½+ËÉr |D|>h_ Z>Ð &!Q)a. G_ Z>Ér G 2_

9þtaË:s. ôÇ>h_ 9þtaË:\"f þj@/ ôÇ>h_ îß&ñ|9½+Ë_ n ×þ|c Ãº e. "f, |I | ≤ |D|.



k-G'p'ëH]j

&ño 5.6

0A_ ·ú¦o1puÉr 2-HK\¦ Ð©ôÇ.

¤ ÃZ: Äº MjH maximal îß&ñ|9½+ËÜ¼Ð+, G 2j "f G_ dominating sets

)a. s M:, yÂÒ1pxd\ _K G\"f G'p' Mj_ n[þts 6 x H ñH

G 2j ñ q6 x 2C\¦ Åt ·ú§H. "f, l:r&ño 5.5 \ _K &ño $íwn.

Tight example



k-G'p'ëH]j

&ño 5.7

P 6= NP, #Q*ôÇ ε > 0\ @/K"f BjàÔaË: k-G'p'_(2− ε)-HKZOÉr Ô¦0px .

¤ ÃZ: G = (V ,E )\ |D| ≤ k dominating set D >rF HH &ñëH]j\¦ Ð. G\ ñ\¦ ' # ¢-a ñ ÕªAáÔÐ ëß[þt. "é¶A ñ\H 1, DhÐ ')a ñ\H 2_ q6 x¦ ïr. Õªo¦s ÕªAáÔ\"f &ñ_)a BjàÔaË: k-G'p' ëH]j\ (2− ε)-HKZO¦&h6 xôÇ.

²ús “\V” ëH]jH 3lq&h<ÊÃº þj@/ (2− ε) K\¦ HKZOs ½½+É ,¦sכ “m” ëH]jH 3lq&h<ÊÃº þjè 2 K\¦½½+É .sכ



k-G'p'ëH]j

&ñ_ 5.8

g ä»n RcçÃÍÏ k-¤d'a%KV9§4: yÂÒ1pxd¦ ëß7á¤ H ñq6 x cuv , uv ∈ Eõ n×æu w : V → Q+\¦ Áº¾Ó ¢-a ñÕªAáÔ G = (V ,E ),©Ãº k ∈ Q+.þj&hK: ×æu_ ½+Ës k\¦ Åt ·ú§H V_ ÂÒìr|9½+Ë S ×æ\"f,V_ y n\"f S_ þjè q6 x sÖ© nt_ q6 x_þj@/°úכs þjè ÷&H :כminS⊆V :w(S)≤k maxv∈V minu∈S cvu.

wdom(H) ≡ n ×æu\¦ ÕªAáÔ H_ dominating set_×æu ×æ\"f þjè°úכ.



k-G'p'ëH]j

l:r&ño 5.9

ÕªAáÔ H\"f n u_ þjè ×æu sÖ© n\¦ s(u), I\¦H2_ îß&ñ|9½+Ës¦ ½+É M:,

w(s(u) : u ∈ I) ≤ wdom(H).

¤ ÃZ: D\¦ H_ þjè×æu dominating sets¦ . ÕªQH_ H n\¦ &!Q H D_ y n\¦ ×ædÜ¼Ð H "fÐn §|9½+Ës \OH Z>s |D| >h >rF 9 s[þtÉr H2\"fH

¿º 9þtaË:s )a. "f IH y Z>\"f ôÇ>h s©_ n\¦ |9Ãº \O. Õªo¦ I_ n[þtÉr #Q" Z>\ 5Åq ¦ "f Õª Z>_ ×æd n\¦ sÖ©nÐ °úH. "f,w(s(u) : u ∈ I) ≤ wdom(H).

si (u) ≡ Gi\"f n u_ þjè ×æu sÖ©n



k-G'p'ëH]j

g ä»n RcçÃÍÏ k-¤d'a N±Ó§hæ

1. G 21 ,G

22 , . . . ,G

2m\¦ ½$íôÇ.

2. G 2i _ maximal îß&ñ|9½+Ë Mi\¦ ½ôÇ.

3. Si = si (u) : u ∈ Mi\¦ ½ôÇ.4. w(Si ) ≤ k þjè i\¦ j¦ , Sj\¦ KÐ ôÇ.

&ño 5.10

3-H

¤ ÃZ: n ×æu \OH âÄºü< °ú s, l:r&ño 5.9\ _ #,cej ≤ OPTe¦ ·ú Ãº e.

H n vH Mj_ #Q" n uü< Gj\"f ¿º >h s _ ñ\¦ âÐÐ

)a. ¢ôÇ u ∈ Mjü< sj(u)H &ñ_\ , Gj_ ôÇ>h_ ñÐ )a.H G_ nü< Sj_ #Q" nH, "f, Gj_ ñ\¦ [j>h s âÐÐ

)a. yÂÒ1pxd\ _K Õª âÐ_ UsH ≤ 3cej ≤ 3OPT )a.



k-G'p'ëH]j

Tight example



x×¼Ñþ n|9½+Ë

x×¼Ñþ n|9½+Ë

&ñ_ 6.1

qÞ«9c gTÒ¼

9§4: n ×æu w : V → Q+\¦ Áº¾Ó ÕªAáÔ

G = (V ,E ).þj&hK: ]j G\¦ ÁºrÐ(acyclic) ÕªAáÔÐ ëß×¼H n|9½+Ë ×æ\"f ×æu_ ½+Ës þjè ÷&H .כ

• ]j G\¦ ÁºrÐ(acyclic) ÕªAáÔÐ ëßHH Érכ G_H rÐü< §ôÇH õכ °ú .

GF[2] ≡ (field)Ð"f_ 0, 1,ÃÐ¦: (field) = 8¾ºl8(commutative division ring)

0 + 0 = 0, 0 + 1 = 1 + 0 = 1, 1 + 1 = 0,

0 · 0 = 0 · 1 = 1 · 0 = 0, 1 · 1 = 1



x×¼Ñþ n|9½+Ë

G_ ñ_ >hÃº ëßpu_ "é¶¦ 7' /BNçß 0, 1|E |\"f,G_ éßíHrÐ C_ :£¤$í 7' χC

[þtÐ Òqt$í)a(span) ÂÒìr/BNçß,7£¤, rÐ/BNçß_ "é¶¦ cyclomatic number, cyc(G )¦ ôÇ. rÐ/BNçß_ "é¶è[þtÉr H n Ãº 2 ÂÒìrÕªAáÔ ÷&H כ¦ ·ú Ãº e.

&ño 6.2

cyc(G ) = |E | − |V |+ κ(G ).

Äº G_ rÐ/BNçßÉr y _¹èכ rÐ/BNçß_ ½+Ë(directsum)s )a. "f, G )a ÕªAáÔ¦ &ñ ¦ &ño\¦7£x"î )a: cyc(G ) = |E | − |V |+ 1.

T\¦ G_ gË>Áº¦ . Tµ1Ú_ y ñ e T_ ñü< ëß×¼H rÐ[þtÉr ¿º +þA1lqwn :£¤$í 7'\¦ ëßH. "f,cyc(G ) ≥ |E | − (|V | − 1).



x×¼Ñþ n|9½+Ë

ôÇ¼# T_ y ñ\ _K &ñ_÷&H ]Xéß (S ;Sc)_ :£¤$í 7'H,H rÐ_ :£¤$í 7'ü< "fÐ Ãºfs. 7£¤ ?/&hs 0s )a. = , ]Xéßõ rÐH ½Ó© Ãº >h_ ñ[þt¦ /BNÄ» l M:ëHs.

sQôÇ ]XéßÉr |V | − 1>h >rF HX< ¿º +þA1lqwns. 7£¤,rÐ /BNçß_ #/BNçß_ "é¶Ér þjè |V | − 1s ÷& 9, "f rÐ/BNçß_ "é¶Ér |E | − (|V | − 1)¦ Åt ·ú§H.



x×¼Ñþ n|9½+Ë

)a ÕªAáÔ G\"f n v\¦ ]j½+É M:, cyclomatic number_yè\¦ δG (v)¦ . &ño 6.2\¦ Gü< G \ v\ &h6 x ,cyc(G ) = |E | − |V |+ 1, Õªo¦cyc(G \ v) = |E | − degG (v)− |V |+ 1 + κ(G \ v)\¦ %3H. sÐÂÒ', )a ÕªAáÔ G\"fH

δG (v) = degG (v)− κ(G \ v). (6.1)

l:r&ño 6.3

H G_ ÂÒìr ÕªAáÔs δH(v) ≤ δG (v).

¤ ÃZ: v\¦ ]j½+É M:, %ò¾Ó¦ ~ÃÎH ÂÒìrÉr s\¦ í<Ê H ¹èëßכs. "f, Gü< H ÷&%3¦ &ñ . (6.1)\ _ #6£§¦ 7£x"î H õכ °ú .



x×¼Ñþ n|9½+Ë

degH(v)− κ(H \ v) ≤ degG (v)− κ(G \ v).

ªAá¤_ ½Ó[þt¦ q§ . Äº degH(v) ≤ degG (v)s. H\"fv\¦ ]j # ÒqtlH ¦¹è[þtכ c1, c2, . . ., ck¦ . G\ëßeH ñ ×æ\"f v\ ²ú9 et ·ú§Ér ñH s ¦¹è[þtכ r&>hÃº\¦ yè rvHX< ëß ¹¡§s )a. "f, ëß G \ v ¹כè\¦ 8 t¦ e, sH ]j÷&H v\ ²ú9 eÜ¼ 9 G\ëßeH ñ M:ëH\ ÆÒ ÷&H .¹è[þtsכ ÕªQ, sQôÇ ¹èכ ôÇ>h © degG (v) degH(v)Ð þjè 1 ß¼. "f ÂÒ1px ñ $íwn.



x×¼Ñþ n|9½+Ë

ëß9 F = v1, v2, . . . , vf x×¼Ñþn|9½+Ës F\¦ ]j cyclomatic numberH 0s ÷&Ù¼Ð, G0 ≡ G ,Gi ≡ G \ v1, . . . , vi(i = 1, 2, · · · )¦ ,

cyc(G ) =f∑

i=1

δGi−1(vi ).

l:r&ño 6.3\ _ #,

cyc(G ) ≤∑e∈F

δG (v). (6.2)



x×¼Ñþ n|9½+Ë

l:r&ño 6.4

F minimal x×¼Ñþ n|9½+Ës∑

v∈F δG (v) ≤ 2cyc(G).

¤ ÃZ: %ir G ÷&%3¦ &ñ½+É Ãº e. F = v1, v2, . . . , vf Õªo¦F\¦ ]j k>h_ ¹èכ Òqt|¦ . s×æ G\"f F_ f ôÇ>h_ n\ ñÐ ÷&H ¹è[þtsכ t>h¦ ,

f∑i=1

κ(G − vi ) = f + t. (6.3)



x×¼Ñþ n|9½+Ë

&ño 6.2\ _ #∑f

i=1 δG (vi ) =∑f

i=1(degG (vi )− κ(G − vi )),cyc(G ) = |E | − |V |+ 1sÙ¼Ð, ëß 6£§¦ Ðs 7£x"îÉr =åQèß:

f∑i=1

(degG (vi )− κ(G − vi )) ≤ 2(|E | − |V |).

sÉrכ (6.3)\ _ # 6£§õ 1lxus:

f∑i=1

degG (vi ) ≤ 2(|E | − |V |) + f + t.

f>h_ n\¦ ]j # %3Ér k>h_ y ¹èHכ Áº )a, "f ¹èכ ?/ÂÒ_ ñ_ Ãº\¦ ½+Ë

|V | − f − k. (6.4)



x×¼Ñþ n|9½+Ë

]Xéß (F ;V \ F )_ ñ_ >hÃº_ ôÇ¦ ½KÐ. FminimalsÙ¼Ð y viH F_ Ér n\¦ í<Ê t ·ú§H rÐ\í<Ê÷&#Q ôÇ. "f (0A_ ÕªaË>õ °ú s) y viH ¿º >h s©

_ ñÐ ÷&H ¦\¹èכ þjè t¦ e#Q ôÇ. yvi\ @/ # s ¿º >h_ ñ ×æ \¦ ]j ¿º f>h_ ñ]j)a. nü< ¹èכ ôÇ ©¦ ×æ4¤ H ñ\¦ ôÇ >hm ]j %iÜ¼Ù¼Ð, sXO> f>h\¦ ]jôÇ Êê\, t>h_ ¹è[þtÉrכ #y F_ n ×æ\ þjè ôÇ>hü< ñÐ ÷&#Q 9, k − t>h_ ¹è[þtÉrכ þjè ¿º >hü< ñÐ ÷&#Q ôÇ. "f(F ;V \ F )_ ñ_ >hÃºH þjè

f + t + 2(k − t) = f + 2k − t. (6.5)

"f, (6.4)õ (6.5)\ _ #∑fi=1 degG (vi ) ≤ 2|E | − 2(|V | − f − k)− (f + 2k − t)

= 2(|E | − |V |) + f + t.



x×¼Ñþ n|9½+Ë

&ñ_ 6.5

#Q" ©Ãº c > 0\ @/K y n v_ ×æu wc = cδG (v)\¦ ëß7á¤ w\¦cyclomatics¦ ÂÒÉr.

(6.2)\ _ # w cyclomatics ccyc(G) ≤ OPT s. ¢ôÇ l:r&ño6.4\ _ # w cyclomatics¦ F minimal x×¼Ñþn|9½+Ësw(F ) ≤ 2ccyc(G)s:

ccyc(G) ≤ OPT ≤ w(F ) ≤ 2ccyc(G).

"f, ×æu w cyclomatics e__ minimal x×¼Ñþ|9½+Ë F\ @/K

2£§&ño 6.6

w(F ) ≤ 2OPT.



x×¼Ñþ n|9½+Ë

s¦כ 9ìøÍ&h ×æu\ &h6 x l 0A # 6£§õ °ú Ér ‘layering’¦ Òqty .

c ← minv∈V

wv

δG (v)

; tv ← cδG (v); w ′

v ← wv − tv .

¢ôÇ ïß#×æu w ′v ªÃº n|9½+Ë¦ V ′(( V ), G ′

¦ V ′Ü¼Ð Ä»)a ÂÒ

ìrÕªAáÔ¦ .

sQôÇ õ&ñ¦ G @/ G ′\, ÁºrÐ ÕªAáÔ |c M:t, ìøÍ4¤ôÇ:G = G0 ) G1 ) · · · ) Gk

V = V0 ) V1 ) · · · ) Vk .

¢ôÇ, t i (1 = 0, 1, . . .)H Gi\"f &ñ_)a cyclomatic ×æu, w i (1 = 0, 1, . . .)HGi_ ïß#×æu¦ : w 0 ≡ w , w 1 ≡ w − t0, . . .. Õªo¦ ¼#_©tk = w k¦ Z~. ÕªQ, ∑

i :v∈Vi

t iv = wv . (6.6)



x×¼Ñþ n|9½+Ë

&ño 6.8

2-H.

¤ ÃZ: F ∗\¦ þj&hK¦ . ÕªQ F ∗ ∩ViH Gi_ x×¼Ñþ|9½+Ë

s)a. Õªo¦ t iUc 7ø5 Gi_ x×¼Ñþ |9½+Ë_ þjè×æu\¦

OPTi¦ ,

OPT = w(F ∗) =k∑

i=0

t i (F ∗ ∩ Vi ) ≥k∑

i=0

OPTi .

¢ôÇ F0 %ir 6£§õ °ú s ìrK½+É Ãº e:

w(F0) =k∑

i=0

t i (F0 ∩ Vi ) =k∑

i=0

t i (Fi ).



x×¼Ñþ n|9½+Ë

l:r&ño 6.7\ _ #, FiH Gi_ minimal x×¼Ñþ|9½+Ës. Õªo¦ H 0 ≤ i ≤ k − 1\ @/K ti cyclomatic ×æusÙ¼Ðti (Fi ) ≤ 2OPTi . "f,

w(F0) ≤ 2k∑

i=0

OPTi ≤ 2OPT.

Tight example



x×¼Ñþ n|9½+Ë

minimal x×¼Ñþ |9½+Ë¦ ½ H ~½ÓZO

V_ÂÒìr|9½+Ë UÐÄ»)aÕªAáÔ G [U]\¦Òqty . U ← ∅Ü¼Ðr #, G [U]\ rÐ µ1ÏÒqt½+É M: t V_ n\¦ 'ôÇ.ÕªQ F := V \ UH minimal x×¼Ñþ |9½+Ës )a.

s ~½ÓZOÉr l:r&ño 6.7\"f Fü< F ′¦ ½ HX< Û¼XO>

&h6 x½+É Ãº e: 0A_ ~½ÓZOÜ¼Ð H\"f F\¦ ½ôÇ Êê\ s\¦ G\SX©r~ M:H V \ V ′_ n ×æ\"f rÐ µ1ÏÒqt÷&t ·ú§¦M:t 'ôÇ. s ×æ '÷&t ·ú§Ér n[þt_ |9½+Ës F ′s)a.

ÃÐ¦ 6.9

Ä»¾ÓÕªAáÔ\"fH O(log |V | log log |V |)-H 0px(Seymour[95], Even et al[95]).


FPTASStrong NP-hardness & FPTAS

Part V

Chapter 8-10



¢-a½ÓHKZOçH(FPTAS)


&ñ_ 7.1

FD9Bß!B þj&hoëH]j Π_ 0pxK x_ 3lq&h<ÊÃº\¦ fΠ(x)¦ . 6£§õ °ú Ér ·ú¦o1pu A\¦ HKZOçH s¦ ÂÒÉr:e__ ëH]j \V Iü< ε > 0¦ 9§4 6£§¦ ëß7á¤ H 0pxKx\¦ Ø¦§4ôÇ.

Π þj@/oëH]js fΠ(x) ≥ (1− ε)OPT,þjèoëH]js fΠ(x) ≤ (1 + ε)OPT.

A_ Ãº'rçßs ¦&ñ)a ε > 0\ @/ # ½Óds½ÓrçßHKZOçH(PTAS, polynomial time approximationscheme)s¦ ôÇ. 8, 1

εõ Π 9§4ß¼l_ ½Ód s¢-a½ÓrçßHKZOçH (FPTAS, fully polynomial timeapproximation scheme)s¦ ôÇ.




ëH]j 7.2

:#×%KV ÂÒx wj ∈ Z+, ò6 x pj ∈ Z+\¦ n>h_ ¾¡§3lq,j ∈ N = 1, 2, . . . , n¦ ÂÒx W ∈ Z+\¦ Cz©\ ò6 x_

½+Ës þj@/ ÷&2¤ ×þ # V,H.

:#×%KV· Dø5 ËÂ/× N±Ó§hæ

〈I 〉 ≡ ëH]j \V I_ s 9§4ß¼l|I | ≡ ëH]j \V I_ unary 9§4ß¼l

&ñ_ 7.3

ëH]j Π_ H ëH]j \V I\¦, #Q" ½Ó<ÊÃº p\ @/ #p(|I |)-rçß\ ÉÒH KZO¦ Ä»½Órçß KZOs¦ ÂÒÉr.




·ú¦o1pu 7.4

ËÂ/×N±Ó§hæ I: z(W ′, l) =¾¡§3lq 1, 2, . . . , l(l ≤ n)Ð ÂÒx W ′(≤W )¦ Cz©\"f %3¦ Ãº eH þj@/ ò6 x. (W ′ ≥ w1s z(W ′, 1) = p1,0 ≤W ′ < w1s z(W ′, 1) = 0Ü¼Ð ílo. W ′ < 0s z(W ′, l) = −∞Ð&ñ_.

z(W ′, l) = maxz(W ′ − wl , l − 1) + pl , z(W ′, l − 1). (7.7)

·ú¦o1pu 7.5

ËÂ/×N±Ó§hæ II: w(Z , l) =¾¡§3lq 1, 2, . . . , l(l ≤ n)Ð ò6 x Z\¦ %3l0AK ¹ôÇכ9 Cz©_ þjè ÂÒx (⇒ w(0, l) = 0, l = 1, 2, . . . , n.) Z < 0sw(Z , l) =∞Ð &ñ_.

w(Z , l) = minw(Z − pl , l − 1) + wl ,w(Z , l − 1). (7.8)

OPT = w(Z , n) ≤ B þj@/ Z .O(n2 max pj) rçß ·ú¦o1pu.




:#×%KV· Dø5 FPTAS

sn#Q: X<s'_ &ñSX\¦ ±úÆÒ#Q (±úÉr o Ãº\¦ 9&ñÂÒìr Áºr)9§4ß¼l\¦ nõ 1

ε(ε )_ ½ÓdÜ¼Ð ôÇ.

P ≡ maxpj, pj = b pj

εP/ncÐ ëH]j\¦ H # Ä»½Ó·ú¦o1pu II\¦

&h6 x # ½ôÇ þj&hK(¾¡§3lq|9½+Ë) S\¦ HKÐ 6 x.

$í|9 7.6

p(S) ≥ (1− ε)OPT.

¤ ÃZ: Äº H |9½+Ë S ⊆ N\ @/K p(S) ≥∑

i∈S(pj

εP/n−1) ≡ 1

εP/np(S)−|S |.

"f, (εP/n)p(S) ≥ p(S)− (εP/n)|S | ≥ p(S)− εP ≥ p(S)− εOPT."f, "é¶A ëH]j_ þj&hK\¦ S∗¦ (εP/n)p(S∗) ≥ (1− ε)OPT.

ÕªX<, p(S) ≥ (εP/n)p(S) ≥ (εP/n)p(S∗)."f, p(S) ≥ (1− ε)OPT.

0A_ ·ú¦o1pu_ Ãº'rçßÉr O(n2b PεP/nc) = O(n2b n

εc) = O( n3

ε). "f

Cz©ëH]j\¦ 0AôÇ FPTAS )a.<ª $í 9 þj&hoHKZO


Strong NP-hardness & FPTAS


&ñ_ 8.1

NP_ H ëH]j, 9§4X<s' unaryÐ ³ð&³)a ΠÐ½Ó8s 0px , Π\¦ strongly NP-hard¦ ôÇ.

&ñ_ 8.2

ëH]j Π_ H ëH]j \V I, #Q" ½Ó<ÊÃº p >rF #,|I | ≤ p(〈I 〉)\¦ ëß7á¤ Π\¦ qÃºuëH]j (no numberproblem)s¦ ôÇ. ÕªXOt ·ú§Ü¼ ÃºuëH]j (numberproblem)¦ ôÇ.

Πp = I ∈ Π : |I | ≤ p(〈I 〉).




$í|9 8.3

ëß Π NP-completes¦ qÃºuëH]j Ä»½Órçß·ú¦o1puÉr P 6= NP ôÇ Ô¦0px .

&ñ_ 8.4

ëß #Q" ½Ó<ÊÃº p\ @/K Πp NP-complete(NP-hard)sΠ\¦ strongly NP-complete (strongly NP-hard)¦ ôÇ.


Euclidean TSP

Part VI

Chapter 11


Euclidean TSP fo(Euclidean) TSP

fo(Euclidean) TSP

d-"é¶ /BNçß_ ¿º &h x , y s\ 6£§õ °ú Ér fo\¦ Òqty

:√∑d

i=1(xi − yi )2.

&ñ_ 9.1

ÒÏ¤n>h(Euclidean) TSP: d-"é¶ fo /BNçß(Euclideanspace)\ n>h_ &hs e¦ M:, s[þt¦ í<Ê H fo_ ½+Ës© Ér íHr\¦ ½ôÇ.

ÄºoH d = 2 âÄº, 7£¤ î 0A_ ëH]j\¦ &ñôÇ. (9ìøÍ&hâÄºÐ ~1> SX©½+É Ãº e.)



Äº, H &h[þt¦ í<Ê H © Ér &ñy+þA_ Us\¦ Ls¦ , LÉr 4n2, Õªo¦ &ñy+þA_ çßs 1 0A\H n>h_ &h[þts 0Au ¦ e¦ &ñ½+É Ãº e (nÉr 2_[þv ]jYLs¦ &ñ.):L = 4n2, L = 2k , k = 2 + 2 log2 n.

¢ôÇ mÉr ½çß [kε , 2kε ]\ 5Åq H 2_ [þv ]jYLÃº¦ .

⇒ m = O( log nε ).

l:rìr½+É

L× L→ 4 L2 ×

L2 (level 1) → · · · → 4i L

2i × L2i (level i) → · · ·

sQôÇ l:rìr½+ÉÉr H n d¦ W1>hm tH Áº

TÐ ³ðr½+É Ãº e. sM: Áº_ y nH Óüt:r ôÇ >h_&ñy+þA¦ _pôÇ.



í_O

íHr &ñy+þA¦ [þt¦ ±úM:H ½Ó© í_O¦ âÄ»ôÇ.



&ñ_ 9.2

τ n>h_ &hõ #Q" í_O ÂÒìr|9n?Ü¼Ð sÀÒ#Q íHr(tour)s 9, í_O¦ ]jü@ôÇ Qt ÂÒìr\"fH ñ_ §µ1ÏÒqt t ·ú§¦ M:, l:rìr½+É\ úH íHr¦ ÂÒØÔ.l:rìr½+É\ úH íHr τ #Q" í_O ¿º¦ íõ #âÄ» t ·ú§¦ M:, l:rìr½+É\ úH § ]jôÇ)a íHr¦ÂÒØÔ.

l:r&ño 9.3

τ\¦ l:rìr½+É\ úH íHr¦ , Us τ\¦ Åt ·ú§H§ ]jôÇ)a íHr\¦ ½½+É Ãº e.

¤ ÃZ: ª¼#\"f Short-Cutting.



l:r&ño 9.4

l:r ìr½+É\ úH ]jôÇ)a § íHr ×æ\ þj&h íHr\¦

2O(m) = nO( 1ε) rçß îß\ ½½+É Ãº e.

¤ ÃZ â«Xn: 1lx&h>S\ZO\ _ # Áº T_ y &ñy+þA_ H “valid visit”_ q6 x_ ³ð\¦ Ä»tK çß. T_ U·sk = O(log n)sÙ¼Ð &ñy+þA_ >hÃºH n_ ½Ó<ÊÃº (>íßK¦.(כ

τ\¦ l:r ìr½+É\ úH ]jôÇ)a § íHr ×æ\ þj&h íHr¦ . τ Áº T_ #Q" &ñy+þA¦ [þt¦ H 8úxSÃºH 8m¦Åt ·ú§H. τ ×æ\"f S\ 5Åq H ÂÒìrÉr, S_ W1>h_ _ í_O[þt\"f r ¦ =åQH þj@/ 4m>h_ âÐÐ sÀÒ#Q .sXO> W1_ í_O[þtÉr °ú Ér âÐ_ rõ =åQs Hd\

¦ tÖ¦ Ãº eHX<, 6 x)a í_O_ |9½+Ë Õª f±l\¦ validvisitss¦ ÂÒÉr.



y í_OÉr 0, 1, Õªo¦ 2 t 6 x|c Ãº eÜ¼Ù¼Ð 34m = nO( 1

ε)_ 0px$ís etëß s×æ\"f Ãº, 2r>h_ í_Os

[þt¦ ±úM: 6 x÷&#Q t0>tH âÄºëß Òqty )a. sH22r¦ Åt ·ú§Ü¼ 9 Õª ÃºH %ir nO( 1

ε)\¦ Åt ·ú§H. "f

S_ valid visit_ ÃºH nO( 1ε).

ôÇ >h_ valid visit_ þj&h q6 xÉr #QbG> ½½+É? S iP: YU6\_ n¦ . Õªo¦ e__ valid visit V\¦ Òqty .SH YU6\ i + 1\ W1>h_ &ñy+þAÜ¼Ð ìr½+É ÷& 9, s M:, W1 >h_ ?/ÂÒ fs þj@/ 4m>h_ ÆÒ í_O¦ °ú> )a.



W1 >h_ &ñy+þA_ H í_Os [þt¦ H &hÜ¼Ð 6 x÷&H â

Äº_ ÃºH %ir nO( 1ε). "f W1 >h y+þA_ H valid visit_

½+Ës ëß×¼H âÄº_ Ãº nO( 1ε)s 9 s[þt_ y y+þA\ K©

H valid visit_ þjè q6 xÉr sp ·ú¦ e.



W1 >h y+þA_ H valid visit_ ½+Ë ×æ\"f Vü< úH þt]כ ëß¦ ¦9ôÇ. Õª ×æ\"f q6 x_ ½+Ës þjè ÷&H sכ V_ þjèq6 xs )a.

ëß\ l:r&ño 9.4\"f ½ôÇ, l:r ìr½+É\ úH ]jôÇ)a §íHr ×æ_ þj&hK_ q6 xs ≤ (1 + ε)OPTs PTAS )a. ÕªQ, ÕªXOt ·ú§Ér \V\¦ ëß[þtÃº e.

sQôÇ &h¦ K l 0AK 6£§õ °ú Ér Áº0AÐ ×þ)a ìr½+É

¦ Òqty : l:rìr½+É¦ (a, b) (0 ≤ a, b < L)ëßpu s1lxôÇ(a, b)-s1lx ìr½+É¦ Òqty . 7£¤, l:rìr½+É\"f ýa³ð (x , y)\¦ tH Ãºfõ Ãºî¦ yy (a + x) mod Lü< (b + y) mod L\¦t2¤ s1lxôÇ ìr½+É¦ ÒqtyKÐ.



ÄºoH ëß a, b\¦ Áº0AÐ ×þ (a, b)-s1lx ìr½+É\ úH ]jôÇ)a §íHr ×æ_ þj&hK_ q6 xs ≤ (1 + ε)OPT |c SXÒ¦s 1

2s©s Hd¦ Ð9

.sכ

π\¦ þj&h íHr, N(π)\¦ π [þt¦ Ãºî ¢H ÃºfÜ¼Ð § H SÃº¦ . (&h¦ § H âÄº, ¿ºÜ¼Ð G'p.) ÕªQ, 6£§Ér ~1> 7£x"î½+É Ãº e.

l:r&ño 9.5

N(π) ≤ 2 ·OPT.



&ño 9.6

a, b\¦ [0, L)\"f Áº0AÐ ×þ π\¦ (a, b)-s1lx ìr½+É\ úÆÒ#Q, íHr_Us 7£x_ l@/ °úכÉr 2εOPT\¦ Åt ·ú§H.

¤ ÃZ: π\¦ (a, b)-s1lx ìr½+É\ úÆÒl 0AK"fH #Q" l¦ t±ú M:, í_O¦ tt ·ú§Ü¼ K© ìr¦ ¿º>hÐ ¾º#Q © Ér í_O¦ t2¤ K

ôÇ. sM:, sÐôÇ íHr q6 x 7£xH l_ í_O çß¦ Åt ·ú§H. lsi-P: YU6\_ s |c SXÒ¦Ér 2i

Ls 9, s M: í_O çßÉr L

2i ms )a. "f,

íHr q6 x 7£x_ l@/ °úכÉr ∑i

L

2im

2i

L=

k

m.

ÕªX<, kε≤ m ≤ 2k

εsÙ¼Ð s °úכÉr ε¦ Åt ·ú§H. "f l:r&ño 9.5\¦

6 x &ño 7£x"î)a. .



l:r&ño 9.7

ïáÔ ÂÒ1pxd(Markov Inequality): X q6£§ SXÒ¦Ãºs

Pr(X ≥ kE [X ]) ≤ 1

k.

¤ ÃZ: E [X ] =∫∞0 xf (x)dx ≥ c Pr[X ≥ c] ∀ c > 0. c ← kE [X ]Ð

@/9.

&ño 9.6\"f íHr7£x_ l@/°úכÉr E [X ] ≤ 2εOPT. "f, ïáÔ ÂÒ1pxd\"f k = 2Ð Z~Ü¼ 6£§õ °ú Ér z¦ %3H.

2£§&ño 9.8

Áº0AÐ ×þôÇ (a, b)-s1lx ìr½+É\ úH Us (1 + 4ε)OPT\¦Åt ·ú§H íHr >rF½+É SXÒ¦s þjè 1

2s )a.


Introduction to LP-DualitySet Cover via Dual Fitting

Rounding Applied to Set CoverSet Cover via the Primal-Dual Schema

Part VII

Chapter 12-15




LP-lìøÍ HKZO

Part II. LP-lìøÍ HKZO

Review

The LP-duality theorem

Min-max relations and LP-duality

Two LP-based algorithm design techniques

Rounding

Primal-dual schema

Also we can use LP-duality to analyze approximation algorithms,by dual fitting.




Dual fitting analysis


ëH]j 11.1

Set Cover (SC)]: Ä»ôÇ|9½+Ë U, |U| = n. U_ ÂÒìr|9½+Ë_ c+t7S = S1, . . . ,Sm, c : S −→ Q+.i¦\B: ½+Ë U ÷&H S_ þjèq6 x ÂÒìr c+t7.

min∑

S∈S cSxS

sub. to∑

S :e∈S xS ≥ 1 ∀e ∈ U

xS ∈ 0, 1 S ∈ S.





+þA>S\ ¢-aoH 6£§õ °ú s jþt Ãº e:

(fractional covering)

OPTp = min∑

S∈S cSxS (11.9)

sub. to∑

S :e∈S xS ≥ 1 ∀e ∈ U ←→ ye

xS ≥ 0

(fractional packing)

max∑

e∈U ye (11.10)

sub. to∑

e:e∈S ye ≤ cS ∀S ∈ Sye ≥ 0 ∀e ∈ U





Recall

&³Ft &!Q)a n[þt¦ C¦ ½+É M:,S_ |9½+Ë S_ îçHq6 x¦ 6£§õ °ú s &ñ_:

pe =c(S)

|S − C | , ∀e ∈ S − C .

Greedy N±Ó§hæ1. C ←− ∅;

2. while C 6= U do&³F © Ér îçHq6 x(α¦ )¦ °úH

S_ |9½+Ë, say, S\¦ ×þ;S − C_ y "é¶è e[þt_ , pe = αÐ &ñ_;

C ←− C ∪ S ;3. ×þ)a |9½+Ë[þt¦ Ø¦§4.





#l"f pe\¦ +þA¢-ao_ ©@/ëH]j(fractional packing)_ KÐ ¦9 , ½Ó©0pxK ÷&tH ·ú§H (HW: 13.2).

ÕªQ, 6£§õ °ú s &ñ_ 0pxK )a:

ye :=pe

Hn.

¤ ÃZ: S_ e__ |9½+Ë S k>h_ "é¶è\¦ ¦ . s[þt¦ Greedy ·ú¦o1puÜ¼Ð &!Q÷&H íH"fÐ e1, e2, . . ., ek¦ . ei &!Q÷&H ìøÍ4¤éß>\¦

ÒqtyKÐ. s íHçß þjèôÇ k − i + 1>h_ &!Q÷&t ·ú§Ér "é¶è >rFôÇ."f SH e\¦ þj@/ c(S)/(k − i + 1)_ îçHq6 xÜ¼Ð &!Q½+É Ãº e. ·ú¦o1puÉr © $§4 > e\¦ &!Q Ù¼Ð pe ≤ c(S)/(k − i + 1)."f,

yei ≤ 1Hn· c(S)

k−i+1

⇒∑k

i=1 yei ≤c(S)Hn·(

1k

+ 1k−1

+ · · ·+ 11

)= Hk

Hn· c(S) ≤ c(S).

"f, y ©@/ 0pxK ÷&Ù¼Ð,∑e∈U

pe = Hn(∑e∈U

ye) ≤ Hn ·OPT.




LP-rounding for Set Cover


f := S_ |9½+Ë[þt\ y "é¶è Ø¦&³ H SÃº ×æ þj@/ °úכ.

·ú¦o1pu 12.1

LP-rounding algorithm

1. +þA¢-aoëH]j_ þj&hK\¦ ½ôÇ;

2. xS ≥ 1f S\¦ ¿º ¦Ér.





&ño 12.2

·ú¦o1pu 12.1Ér H>Ãº f\¦ Ð©ôÇ.

¤ ÃZ: e__ e ∈ U\¦ Òqty . e\¦ í<Ê H |9½+ËÉr ú§f>h, "f þjè ôÇ>h_ |9½+Ë SH fractional cover_ K\"fxS ≥ 1

f ÷&ôÇ. "f, 0A_ ·ú¦o1pus ×þôÇ |9½+Ë S_|9½+Ë CH 0pxK ÷&#Q ôÇ.Õªo¦ xS ≥ 1

f âÄº\ëß 1s ÷&Ù¼Ð, fractional solution[þtÉrfC\¦ íõ # 7£x t ·ú§H. "f, Ø¦§4)a &ñÃºK_ 3lq&h<ÊÃº °úכÉr fractional cover_ 3lq&h<ÊÃº °úכ OPTp_ fC\¦ Åt·ú§H.





·ú¦o1pu 12.3

È· (Pþb N±Ó§hæ

1. +þA¢-aoëH]j_ þj&hK x\¦ ½ôÇ;

2. y |9½+Ës xS_ SXÒ¦Ð ×þ÷&2¤ Áº0AÐ

y |9½+Ë¦ ×þ # C\¦ ½ôÇ.(·ú¡õ z»s ¦ SXÒ¦s yy xS , 1− xS

1lx¦ ~4R ·ú¡s |9½+Ë S\¦ K\ í<Êr.)

3. 0A_ õ&ñ¦ c log n ìøÍ4¤ #Õª ½+Ë|9½+Ë C′¦ KÐ ôÇ.(c_ °úכÉr A\ [O"î.)

E[c(C)] =∑S∈S

cS · Pr[S ×þHd] =∑S∈S

cSxS = OPTp





U+ ÇÐ xjS¿ a CUc +B &!aÁGö È·ÖR?

S_ k>h_ |9½+Ës "é¶è a\¦ í<ÊôÇ¦ . Õª þj&hK_°úכ[þt¦ x1, x2, . . ., xks¦ . ÕªQ x1 + x2 + · · ·+ xk ≥ 1.s |9½+Ë ×æ þjèôÇ ×þ|c SXÒ¦Ér

1− (1− x1)(1− x2) · · · (1− xk) ≥ 1− (1− 1

k)k ≥ 1− 1

e.

·ú¦o1pu\"f 6£§s ëß7á¤÷&2¤ c\¦ úH:(1

e

)c log n

≤ 1

4n.





ÕªQ,

Pr[a C′Ü¼Ð &!Q÷&t ·ú§6£§] ≤(

1

e

)c log n

≤ 1

4n.

"f C′s 0pxK u SXÒ¦Ér ≤ n · 14n ≤

14 .

ôÇ¼#, E[c(C′)] ≤ c log nOPTpsÙ¼Ð, ïáÔ ÂÒ1pxd¦ &h6 x½+ÉÃº e: X q6£§ SXÒ¦Ãºs, e__ ªÃº t\ @/K

Pr[X ≥ t] ≤ E[X ]t .

Pr[c(C′) ≥ 4c log n ·OPTp] ≤ Pr[c(C′) ≥ 4E[c(C′]] ≤ 1

4.

"f C′s 0pxKs¦ Õª q6 xs 4c log n ·OPTp\¦ Åt ·ú§¦

SXÒ¦Ér12 s©s )a.





12 -&ñÃº$í¦ 6 xôÇ n&!Q 2-H ·ú¦o1pu.

min∑

v∈V cvxv (12.11)

sub. to xu + xv ≥ 1 ∀(u, v) ∈ E

xv ∈ 0, 1, v ∈ V .

min∑

v∈V cvxv (12.12)

sub. to xu + xv ≥ 1 ∀(u, v) ∈ E

xv ≥ 0, v ∈ V .





l:r&ño 12.4

(12.12)Ér 12-&ñÃº$í¦ °úH. 7£¤, Õª =Gt&h[þtÉr ¿º Õª "é¶è_ °úכs 0, 1

2,

¢H 1s )a.

¤ ÃZ: "é¶è_ °úכs 0, 12, ¢H 1s °úכ¦ °úH 0pxK xH Ér ¿º 0pxK

_ ¦2¤½+ËÜ¼Ð ³ðr÷&H ¦כ Ðs.

V+ =

v :

1

2< xv < 1

, V− =

v : 0 < xv <

1

2

.

ε > 0\ @/ #, 6£§õ °ú Ér K\¦ &ñ_ :

yv =

xv + ε, v ∈ V+

xv − ε, v ∈ V−xv , Qt.

, zv =

xv − ε, v ∈ V+

xv + ε, v ∈ V−xv , Qt.

#l"f, y 6= zs 9 yü< z q6£§ |õ (12.12)_ 0pxK ÷&2¤ ε¦ Øæìry > ú¦ Ãº e.ÕªX<, x = 1

2(y + z).

sQôÇ 12-&ñÃº$í¦ 6 x 2-HKZO¦ ~1> ëß[þt Ãº e. =t [O"î

# Ð.




©@/$í¦ 6 xôÇ Set Cover ·ú¦o1pu


A = (aij) ∈ Qm×n

min cT x (13.13)

Ax ≥ b

x ≥ 0.

max bT y (13.14)

AT y ≤ c

y ≥ 0.





H-"é¶-©Ð#Ä»|

α ≥ 1H j = 1, . . . , n\ @/K, xj = 0s,cj

α ≤∑m

i=1 aijyi ≤ cj .(¢H, 1αcT ≤ yTA ≤ cT .)

H-©@/-©Ð#Ä»|

β ≥ 1H i = 1, . . . ,m\ @/K, yi = 0s,bi ≤

∑nj=1 aijxj ≤ βbi .(¢H, b ≤ Ax ≤ βb.)





l:r&ño 13.1

(x , y) H-©Ð#Ä»|¦ ëß7á¤ H "é¶-©@/ 0pxK ©s(bT y ≤)cT x ≤ αβbT ys $íwnôÇ.

¤ ÃZ:cT x ≤ αyTAx ≤ αβyTb.

Set Cover\¦ 0AôÇ "é¶-©@/ ·ú¦o1puα = 1, β = f .

H-"é¶-©Ð#Ä»|∀S , xS > 0⇒

∑e∈S ye = cS .

H-©@/-©Ð#Ä»|∀e, ye > 0⇒

∑S :e∈S xS ≤ f .





·ú¦o1pu 13.2

Set Cover xjS-3×7 FD9 N±Ó§hæ1. x ← 0, y ← 0;

2. H "é¶è &!Q|c M: t 6£§¦ ìøÍ4¤ôÇ:f &!Q÷&t ·ú§Ér e_ "é¶è e\¦ ×þôÇ;1px ñÐ ëß7á¤÷&H ©@/ ]jds ÒqtU M: t

ye\¦ 7£xr;1px ñÐ ëß7á¤÷&H ]jd_ |9½+Ë[þt¦ ¿º &!Q\ V,¦

s\ xS\¦ \OX<sàÔôÇ;s M: Dh\v> &!Q÷&H "é¶è[þt¦ \OX<sàÔ ôÇ.

3. |9½+Ë&!Q x\¦ Ø¦§4ôÇ.





&ño 13.3

·ú¦o1pu 13.2Ér f -HK\¦ Ð©ôÇ.


Maximum Satisfiability

Part VIII

Chapter 16


Maximum Satisfiability MAX-SAT¦ 0AôÇ +þA>S\ îrç

MAX-SAT¦ 0AôÇ +þA>S\ îrç

&ñ_ 14.1

Max-Sat]: n>h_ ÂÒÖ¦Ãº x = (x1, x2, . . ., xn)_ o'XO[þt_disjunctionÜ¼Ð sÀÒ#Q ]X(clause) c[þt_ |9½+Ë C_conjunctive normal form <ÊÃº,

f =∧

c∈Cc

c .

y ]X c ∈ C_ ×æu wc .i¦\B: f_ ëß7á¤÷&H ]X[þt_ ×æu_ ½+Ës þj@/ ÷&H x_o°úכ.

H ]X_ o'XO_ >hÃº k>h s s MAX-kSATs¦ ÂÒØÔ.



]X_ o'XOs ú§¦ M:

·ú¦o1pu 14.2

Max-Sat N±Ó§hæy Ãº 1lqwn&hÜ¼Ð 1

2 SXÒ¦Ð True ÷&2¤ o°úכ¦Òqt$íôÇ.

K\"f ]X c_ ×æu °úכ¦ Wc¦ , c_ o'XO_ >hÃº\¦k¦ E [Wc ] = (1− 1

2k )wc Hd¦ ~1> ·úÃº e.

(1− 12k ) ≥ 1

2sÙ¼Ð ×æu_ l@/°úכÉr

E [W ] =∑c∈C

E [Wc ] ≥1

2

∑c∈C

wc ≥1

2OPT.

H ]X\"f k ≥ 2s E [W ] ≥ 34OPT Hd¦ ·ú Ãº e.



Derandomization

·ú¦o1pu 14.2\¦ &ñ&h ·ú¦o1puÜ¼Ð derandomize # Ð. 7£¤, 1lx9ôÇHu\¦, SXÒ¦&hs m, &ñ&hÜ¼Ð Ð©½+É Ãº eH ~½ÓZO¦ Ð.

E [W |x1 = a1, . . . , xi = ai ]

= E [W |x1 = a1, . . . , xi = ai , xi+1 = True] · 12

+E [W |x1 = a1, . . . , xi = ai , xi+1 = False] · 12

"f, E [W |x1 = a1, . . . , xi = ai , xi+1 = True],¢H E [W |x1 = a1, . . . , xi = ai , xi+1 = False]H E [W |x1 = a1, . . . , xi = ai ]Ð&ôÇ.

"f sQôÇ 'a>\¦ xi[þt_ e_Ð &ñôÇ íH"f\ (' íH"f¦

&ñ ) &h6 x H xi[þts o°úכ¦ °ú> ÷&H íHçß, E [W ]ü< °ú

H 3lq&h<ÊÃº\¦ K ìøÍ×¼r "f &ñ&hÜ¼Ð ½Kt> )a.



H i\ @/K l@/°úכ E [W |x1 = a1, . . . , xi = ai ]\¦ ½Órçß\>íß½+É Ãº e, sH ½Órçß ·ú¦o1pus )a. o°úכs1lqwn&hÜ¼Ð ÅÒ#QtH âÄº sH ½Órçß\ >íß½+É Ãº e.(= ?)

Notice: s "é¶oH l@/°úכ¦ ½Órçß\ >íß½+É Ãº e Ãºo°úכs 1lqwn&hÜ¼Ð ÅÒ#Qtt ·ú§H âÄº\ &h6 x)a:

E [W |x1 = a1, . . . , xi = ai ]

= E [W |x1 = a1, . . . , xi = ai , xi+1 = True]×Pr[xi+1 = True|x1 = a1, . . . , xi = ai ]

+E [W |x1 = a1, . . . , xi = ai , xi+1 = False]×Pr[xi+1 = False|x1 = a1, . . . , xi = ai ].



]X_ o'XOs &h¦ M:

S+c : ]X c\ "é¶+þAÜ¼Ð H Ãº |9½+Ë

S−c : ]X c\ ÂÒ&ñÜ¼Ð H Ãº |9½+Ë

yi =

1, xj = True0, xj = False.

max∑

c∈C wczc (14.15)

sub. to∑

i∈S+c

yi +∑

i∈S−c(1− yi ) ≥ zc ∀c ∈ C

zc ∈ 0, 1, ∀c ∈ C,yi ∈ 0, 1, ∀i .

max∑

c∈C wczc (14.16)

sub. to∑

i∈S+c

yi +∑

i∈S−c(1− yi ) ≥ zc ∀c ∈ C

0 ≤ zc ≤ 1, ∀c ∈ C,0 ≤ yi ≤ 1, ∀i .



·ú¦o1pu 14.3

Max-Sat LP (Pþb N±Ó§hæ+þA>S\ 14.16\¦ Û¦#Q þj&hK (y∗, z∗)\¦ ½ôÇ.y Ãº xi SXÒ¦ y∗i Ð True ÷&2¤ 1lqwn&hÜ¼Ð o°úכ¦&ñôÇ.

LP îrç ·ú¦o1puÜ¼Ð %3Ér o°úכs ëß7á¤ H ]X_ ×æu½+Ë¦ W , Õª ×æ ]X c\ K© H ÂÒìr¦ Wc¦ .



l:r&ño 14.4

c k>h_ o'XOÐ ëß[þt#Q&, E [Wc ] ≥(1−

(1− 1

k

)k)

wcz∗c .

¤ ÃZ: c_ Ãº[þt¦ ¼#_© x1, . . . , xk¦ .ÕªQ c ëß7á¤|c SXÒ¦Ér

1−∏

i∈S+c(1− yi )

∏i∈S−c

yi ≥ 1−(

1k

(∑i∈S+

c(1− yi ) +

∑i∈S−c

yi

))k

=

1−(1− 1

k

(∑i∈S+

cyi +

∑i∈S−c

(1− yi )))k

≥ 1− (1− z∗ck

)k .

#l"f g(z) := 1− (1− zk)kH 3lq(concave)<ÊÃºs. "f ½çß [0, 1]\"f

h(z) = g(1)z + g(0) =(1−

(1− 1

k

)k)

zÐ ß¼. "f · · · .

1−(1− 1

k

)kH k\ @/ôÇ yè<ÊÃºs. "f, H ]X_ o'XO_ Ãº

k\¦ Åt ·ú§H, E [W ] ≥(1−

(1− 1

k

)k) ∑

c∈C wcz∗c =(

1−(1− 1

k

)k)

OPTLP ≥(1−

(1− 1

k

)k)

OPT ≥(1− 1

e

)OPT.

·ú¦o1pu 14.2ü< ðøÍtÐ ·ú¦o1pu 14.3¦ derandomize½+É Ãº e.



34-FD9 N±Ó§hæ

6£§õ °ú Ér ·ú¦o1pu¦ ÒqtyKÐ.

1lx¦ ~. ·ú¡s ·ú¦o1pu 14.2\¦ +' ·ú¦o1pu 14.3\¦ Ãº'ôÇ.

l:r&ño 14.5

E [Wc ] ≥ 34wcz

∗c .

¤ ÃZ: E [Wc |·ú¡] = (1− 2−k)wc ≥ (1− 2−k)wcz∗c ,

E [Wc |+'] =(1−

(1− 1

k

)k)

wcz∗c . "f,

E [Wc ] ≥ 12

((1− 2−k) +

(1−

(1− 1

k

)k))

wcz∗c ≥ 3

4wcz∗c .



·ú¦o1pu 14.6

(&ñ&h 34 -H ·ú¦o1pu) 1. ·ú¦o1pu 14.2¦ derandomizeôÇ.

2. ·ú¦o1pu 14.3\¦ derandomizeôÇ.3. ¿º >h_ K ×æ Ér ¦כ Ø¦§4ôÇ.

l:r&ño 14.7

·ú¦o1pu 14.6Ér MAX-SAT_ 3/4-HK\¦ &ñ&hÜ¼ÐÐ©ôÇ.



·ú¡\"f E [W ] ≥ 34OPTLP , "f, ·ú¦o1pus ]j/BN H &ñÃºK_ integrality

gapÉr 34s©s. z]jÐ 6£§õ °ú Ér tight examples >rFôÇ.

\V 14.8

f = (x1 ∨ x2) ∧ (x1 ∨ x2) ∧ (x1 ∨ x2) ∧ (x1 ∨ x2). H ]X_ ×æu\¦ 1s¦ yi = 1

2, zc = 1s +þA¢-ao_ þj&hK Hd¦ ·úÃº e. "f

OPTLP = 4. ÕªQ, OPT = 3.

\V 14.9

f = (x1 ∨ x2) ∧ (x1 ∨ x2) ∧ (x1 ∨ x2). ]X_ ×æu\¦ yy 1, 1, 2 + εs¦ yi = 1

2, zc = 1s +þA¢-ao_ þj&hK Hd¦ ·úÃº e. "f, ¿º

randomized ·ú¦o1pu ¿º H Ãº\¦ 12_ SXÒ¦Ð TrueÐ t&ñôÇ.

derandomization éß>\"f x1¦ © $ |Ü¼Ð úÜ¼,E [W |x1 = True] = 3 + ε

2, E [W |x1 = False] = 3 + ε. "f x1Ér FalseÐ

t&ñôÇ. ÕªQ 3lq&h<ÊÃºH 3 + ε. ÕªQ, x1\¦ TrueÐ t&ñ 4 + ε_×æu\¦ %3¦ Ãº e.


Scheduling on Unrelated Parallel Machines

Part IX

Chapter 17


Scheduling on Unrelated Parallel Machines 1lqwn #î§>=áÔÐ[j"f Û¼H×¦aA

1lqwn #î§>=áÔÐ[j"f Û¼H×¦aA

ëH]j 15.1

â ÄZÝ~è«×Tz" â«Xä·ÿb

]: \O |9½+Ë J, áÔÐ[j"f |9½+Ë M, ½¹כ \Orçß pij ∈ Z+,i ∈ M, j ∈ J.i¦\B: ‘makespan’ 7£¤ t \Os =åQH íHçßt_rçß¦ þjèo H áÔÐ[j"f-\O Û¼H×¦.

pij H i ∈ M\ @/K °ú Ér âÄº, þjè makespan ëH]j¦ ÂÒØÔ 9, PTAS >rF. y áÔÐ[j"f_ 5Åq si e#Q, \Orçßspj/siÐ &ñ÷&H âÄº\ PTAS >rF (17.5).



Parametric pruning£· £ ø5 ¤n>ÌfCµìª

min tsub. to

∑i∈M xij = 1, ∀j ∈ J∑

j∈J pijxij ≤ t, ∀i ∈ M

xij ∈ 0, 1, ∀i ∈ M, ∀j ∈ J.

+þA¢-aoëH]j_ integrality gapÉr ÁºôÇ :

\V 15.2

m>h_ áÔÐ[j"fü< H áÔÐ[j"f\"f m_ Ãº'rçßs oHôÇ>h_ \O_ âÄº, þj&h 3lq&h<ÊÃº °úכÉr mstëß +þA¢-aoþj&hK 3lq&h<ÊÃº °úכÉr 1.



sXO>, 3lq&h<ÊÃºÐ H \Orçß¦ Ãº ª_ °úכ¦ °úH FGéß&h K\¦ ~½Ót l 0AK 6£§õ °ú s parametricpruning¦ #î' H +þA>S\¢-ao\¦ Òqty : y T ∈ Z+°úכ\

@/K, ST = (i , j) : pij ≤ TÐ &ñ_ . Õªo¦ LP(T )\¦ 6£§õ °ú Ér +þA>S\ 0px$íëH]jÐ &ñ_ :∑

i :(i ,j)∈STxij = 1 ∀j ∈ J (15.17)∑

j :(i ,j)∈STpijxij ≤ T , ∀i ∈ M

xij ≥ 0, ∀(i , j) ∈ ST .

sìrÃÐÒoÜ¼Ð LP(T ) 0pxK\¦ °úH þjè T\¦ ½½+É Ãº e.s\¦ T ∗¦ OPT_ ôÇs )a.

l:r&ño 15.3

LP(T )_ =Gt&hKH þj@/ m + n>h_ ªÃº "é¶è\¦ °úH.



2£§&ño 15.4

LP(T )_ =Gt&hKH þjè n −m>h_ \O\ @/K &ñÃº°úכ 7£¤ 1¦ °úH.

¤ ÃZ: =Gt&h K\"f &ñÃº°úכ¦ °úH \O, 7£¤ ôÇ>h_ áÔÐ[j"f\ Cu)a \O_ Ãº\¦ α, #Q >h_ áÔÐ[j"f\ ìrÃº°úכÜ¼Ð ¾º#Q \O_ Ãº\¦ β¦ . ÕªQ, ªÃº\¦ °úH "é¶èH þjèôÇ α+ 2βstëß ·ú¡_ l:r&ño 15.3\_ # ªÃº "é¶è_ >hÃºH þj@/ m + ns. "f α+ 2β ≤ m + n. ¢ôÇα+ β = n. ÕªQÙ¼Ð α ≥ n −m.

LP(T )_ =Gt&hK, x\ @/K 6£§õ °ú Ér n|9½+Ë J ∪M\¦ °úH sìrÕªA

áÔ G = (J ∪M,E)\¦ &ñ_ôÇ. (j , i) ∈ E ⇔ xij 6= 0. F ⊆ J\¦ #Q >h_ áÔÐ

[j"f\ ìrÃº°úכÜ¼Ð ¾º#Q \O_ |9½+Ës¦ . Õªo¦ H\¦ n |9½+Ë

F ∪M\ _ # Ä»)a G_ ÂÒìrÕªAáÔ¦ . Óüt:r H_ y ñ (j , i)H

0 < xij < 1\¦ ëß7á¤ôÇ.



H\H F_ n[þt¦ ¿º &!Q H f±l >rFôÇH ¦כ7£x"î½+É Ãº e(l:r&ño 15.6).

ªSm\ B+ Å]

n|9½+Ë V\¦ t 9, ñ_ >hÃº |V | s )a ÕªAáÔ\¦Ä»Áº(pseudo-tree)¦ ÂÒØÔ. ¢ôÇ ¹è[þtsכ ¿ºÄ»Áº ÕªAáÔ\¦ Ä»Õü(pseudo-forest)s¦ .



l:r&ño 15.5

LP(T )_ =Gt&hKÐ &ñ_)a sìrÕªAáÔ G = (J ∪M,E )HÄ»Õüs )a.

¤ ÃZ: G_ e__ ¦\¹èכ Gc¦ . LP(T )\¦ Gc_

\Oõ áÔÐ[j"f\ ²DGôÇ # %3Ér ëH]j\¦ LPc(T )¦ . Õªo¦ x×æ Gc_ \Oõ áÔÐ[j"f\ @/6£x÷&H ÂÒìr¦ xc¦ . QtH xc¦ . Gc G_ Ér ÂÒìrõ ÷&t ·ú§Ér ¹כèsl M:ëH\ xcH LPc(T )_ =Gt&hs )a. (ëß m,xcH LPc(T )_ "fÐ Ér ¿º &h_ ¦2¤½+ËÜ¼Ð ³ð&³÷&HX<,Gc 1lqwn)a ¹èslכ M:ëH\ y &hÉr xc\¦ ½+Ë

LP(T )_ 0pxK )a. sXO> %3Ér ¿º 0pxK\¦ ¦2¤½+Ë x ÷& 9 sH x =Gt&hsH \כ íHs )a.) "f,l:r&ño 15.3\¦ &h6 x GcH Ä»Áº )a.



l:r&ño 15.6

HH H \O¦ &!Q H f±l\¦ °úH.

¤ ÃZ: x\"f, f ôÇ >h_ áÔÐ[j"f\ 1Ð ½+É©÷&H \O[þtõÕª 1_ °úכ\ @/6£x÷&H ñ[þt¦ ]j # %3Ér ÕªAáÔ ÐH )a. 7£¤ HH G\"f °ú Ér ÌÃº_ nü< ñ ]j÷&#Q %3Ér ÕªAáÔsÙ¼Ð %ir Ä»Õüs )a. ¢ôÇ HH y \Os þjè2>h_ ñ\¦ ²ú¦ e(A ÕªaË> ÃÐ).



"f H_ H 9(leaf)Ér áÔÐ[j"f )a. y 9¦ sÖ©ôÇ\Oõ f±¦ s ¿º n\¦ H\"f ]jôÇ. s\¦ H ¹כè\ ìøÍ4¤ &h6 xôÇ. s M:, B éß> H 9Ér áÔÐ[j"f)a. (=?)

ëß s õ&ñ\"f ÕªAáÔ_ #Q" _¹èכ ÂÒìrs zH

sH rÐ ÷&#Q 9, sìrÕªAáÔsl M:ëH\ \Oõ áÔÐ[j"f °ú H Ãº rÐ )a. "f, \OõáÔÐ[j"f\¦ z\Os tÖ¦ Ãº e.



N±Ó§hæ

Äº T_ #30A\¦ ½KÐ. y \O¦ þjè Ãº' rçß¦ °úH áÔÐ[j"f\@/6£xrvH K\¦ ÒqtyK Ð. #Q*ôÇ +þA¢-ao K_ 8úx \Orçß_ ½+Ë sK_ âÄºÐ ¦ Ãº \O. ôÇ¼# s K_ makespan¦ α¦ , s KHpruning¦ #î'ôÇ +þA¢-aoëH]j LP(α)_ 0pxKs. "f, T ∗

H ½çß

[α/m, α]_ #3_ îß\ e.

·ú¦o1pu 15.7

1. ½çß [α/m, α]_ sìrÃÐÒo¦ :x # LP(T ) 0pxK\¦ °úH þjè T\¦½ôÇ. s\¦ T ∗¦ ;2. LP(T ∗)_ =Gt&hK x\¦ ½ôÇ;3. &ñÃº°úכ¦ °úH Ãº[þtÉr @/6£x H \O[þtõ áÔÐ[j"f[þt¦ tîr;4. ·ú¡\"f [O"îôÇ õכ °ú s ÕªAáÔ H\¦ ½$í ¦ H \O¦ &!Q Hf±l\¦ ¹1ÔH:5. F_ \O[þt¦ s f±l\ C&ñôÇ.



&ño 15.8

·ú¦o1pu 15.7Ér 2-H\¦ Ð©ôÇ.

¤ ÃZ: T ∗ ≤ OPTe¦ ~1> ·ú Ãº e. =Gt&hK x makespanT ∗\¦ °ú¦ el M:ëH\, éß> 3t H áÔÐ[j"fH T ∗îß\

\O¦ =åQ?/> )a. ¢ôÇ H_ ñ[þtÉr ¿º \Orçßs T ∗\¦

Åt ·ú§H. ÕªX< éß> 5\"f y áÔÐ[j"f þj@/ ôÇ>h_\O¦ ÆÒ C&ñ ~ÃÎÜ¼Ù¼Ð, þj7áx K_ makespanÉr 2T ∗

\¦ Åt

·ú§H.



Tight example

m>h_ áÔÐ[j"fü< 1 + m(m − 1)>h_ \O¦ Òqty . %6£§ ôÇ>h_ \OÉr H áÔÐ[j"f\"f m_ rçßs ¹כ9. Qt\OÉr, ¿º, #Q" áÔÐ[j"f\"fHt éß0Arçßs ¹כ9¦ . s âÄº, þj&hKH OPT = m.

·ú¦o1pu 15.7\¦ &h6 x # Ð. T < ms LP(T )H 0pxK>rF t ·ú§H. "f T ∗ = m. LP(T )H 6£§õ °ú Ér=Gt&hK\¦ °úH. %6£§ ôÇ >h_ \OÉr H áÔÐ[j"f\1mëßpu ½+É©÷&¦, Qt \OÉr ¿º m − 1>hm m>h_áÔÐ[j"f\ 1Ð ½+É©. (=Gt&he¦ SX ½+É .כ _þvëH]j 17.2)s M:, ·ú¦o1pu 15.7s Òqt$íôÇ K_ makespanÉr 2m− 1s )a.


Multicut and Integer Multicommodity Flow in Trees

Part X

Chapter 18


Multicut and Integer Multicommodity Flow in Trees Multicutõ &ñÃº¾¡§3lqâì2£§ëH]j: Áº_ âÄº

Multicutõ &ñÃº¾¡§3lqâì2£§ëH]j: Áº_ âÄº

&ñ_ 16.1

multicut]: Áº¾ÓÕªAáÔ G = (V ,E), ñ 6 x|¾Ó ce ∈ Q+, e ∈ E , k>h_ n © |9½+Ë,(s1, t1), . . . , (sk , tk).i¦\B: ]j÷& H © (si , ti ) "fÐ ìro÷&H ñ |9½+Ë ×æ þjè 6 x|¾Ó¦ .כ

þjèq6 x 'pV, ]Xéß ëH]jH multicutëH]jÐ ~1> 8)a: s1, s2, s3\¦

¿º "fÐ ìro÷&2¤ H ,Érכ 6£§õ °ú Ér n©[þts ¿º ìro H õכ°ú : (s1, s2), (s2, s3), (s3, s1). "f, NP-hardëH]js.

multicutÉr k ≥ 2s NP-hards. k ≥ 2s max-SNP-hards 9 "f #Q

" c-HKZOÉr Ô¦0pxôÇ ©Ãº c >rFôÇ. 9ìøÍ&h âÄº, &³F

O(log k)_ H>Ãº\¦ °úH ·ú¦o1pus >rFôÇ. G Áº âÄº\

max-SNP-hard. ÕªQ s âÄº 2-HKZOs 0px .



$í|9 16.2

G Zs 1s¦ éß0A 6 x|¾Ó_ ñ\¦ Áº âÄº\ multicutëH]jHNP-hards.

7£x"î: n&!QëH]j\¦ 0A_ $í|9¦ multicutëH]jÐ 8½+É Ãº e.

ÈÁ multicut %KV· Dø5 ÇaÊÁNÌ(¿ÌfC

min∑

e∈E cede

sub. to∑

e∈Pide ≥ 1, i = 1, . . . , k

de ∈ 0, 1.

min∑

e∈E cede

sub. to∑

e∈Pide ≥ 1, i = 1, . . . , k

de ≥ 0.



¿º ëH]j s\H 6£§_ \V\"f ¦ Ãº eH 3!%כ &ñÃºçßs >rFôÇ.

+þA¢-aoëH]j_ ©@/ëH]jH 6£§õ °ú . 5Åq&h °úכ¦ °úH ¾¡§3lqâì2£§ëH]jÐ K$3½+É Ãº e. s M:, y ñ\"f, ~½Ó¾Ó\ 'a> \Os, H ¾¡§3lqâì2£§_âì2£§_ ½+ËÉr 6 x|¾Ó |¦ ëß7á¤ # ôÇ.

¥o>¢4 âéÛ%KV

max∑k

i=1 fi (16.18)

sub. to∑

i :e∈Pifi ≤ ce , e ∈ E

fi ≥ 0 i ∈ 1, . . . , k.



ëH]j 16.3

ÇaÊÁ âéÛ%KV

]: multicutëH]jü< 1lx9. y (si , ti )\¦ ¾¡§3lq i_ ‘source’ü<‘sink’Ð ÒqtyôÇ. éß, 6 x|¾ÓÉr ¿º &ñÃº.·]: H ¾¡§3lq âì2£§ ß¼l_ ½+Ës þj@/ ÷&H &ñÃº âì2£§.

&ñÃº¾¡§3lqâì2£§ëH]jH e__ 6 x|¾Ó¦ t, Zs 3 Áº_ âÄº NP-hard )a. 5Åq Õªo¦ &ñÃº¾¡§7áxâì2£§ëH]js\H &ñÃº çß (integrality gap)s >rF H ¦כ 0A_ Áº_ âÄº\"f ·ú Ãº e.



"é¶-©@/ HKZO

6£§õ °ú Ér H©Ð#Ä»$í¦ ÆÒ½ôÇ.

"é¶©Ð$í: de > 0 ⇒∑

i :e∈Pifi = ce . 7£¤, multicut\ í<Ê÷&H

ñ_ âì2£§Ér Tq ôÇ.

H©@/©Ð$í: fi > 0 ⇒ (1 ≤)∑

e∈Pide ≤ 2.

7£¤, âì2£§¦ °úH (si , ti )-âÐ ×æ\"f multicut\ í<Ê÷&H ñH¿º >h\¦ Åt ·ú§H.

G_ e__ n\¦ ÍÒoÐ t&ñôÇ. y nÐÂÒ' ÍÒotâÐ_ Us\¦ n_ U·sÐ &ñ_ôÇ. ¿º n u, v ∈ V\¦9H âÐ ×æ\"f © ·û Ér U·s_ n\¦ lca(u, v)Ð³ðl .



·ú¦o1pu 16.4

1. f ← 0, D ← ∅;2. U·s ±úÉr íHÜ¼Ð y n v\ @/ # 6£§¦ ìøÍ4¤ôÇ: lca(si , ti ) = vH (si , ti )\ @/ # âì2£§¦ þj@/ 0pxôÇ ëßpu C&ñôÇ. s M:, ío)a ñ[þt¦ D\ íH"f@/Ð 'ôÇ. °ú Ér ìøÍ4¤éß>\ ')a ñ[þtÉr íH"f\¦e__ &ñôÇ.3. D\ ')a ñ[þt¦ íH"f@/Ð e1, e2, . . ., els¦ .4. ')a %iíHÜ¼Ð y ñ e\ @/ # 6£§¦ Ãº'ôÇ: ëß9 D \ emulticutÜ¼Ð Ä»t÷& D ← D \ eÐ Ãº&ñôÇ.5. fü< D\¦ Ø¦§4ôÇ.

l:r&ño 16.5

(si , ti )\¦ 0Ð H âì2£§¦ °úH ©, Õªo¦ lca(si , ti ) = v¦ . ÕªQsi\"f v, Õªo¦ v\"f tit_ âÐ yy\"f þj@/ ôÇ >h ñ D\í<Ê)a.



¤ ÃZ: si\"f vt âÐ_ ¿º >h_ ñ eü< e ′s D\ í<Ê÷&%3¦ . Õªo¦ e e ′Ð U·s e¦ . 0A_ ·ú¦o1pu éß> 4\"f e\¦ ¦9 H íHçß¦ ÒqtyKÐ. e ]j÷&t ·ú§¤Ü¼Ù¼Ð, D\"f e\¦ Ä»9 > Õª s âÐ\ °úH n©(sj , tj) >rFôÇ. s M:, e ′Ér lca(sj , tj)Ð 0A\ e#Q lM:ëH\, u := lca(sj , tj)H v = lca(si , ti ) Ð U·s >rFôÇ.



u éß> 2\"f ¦9)a Êê\ DH (sj , tj)\¦ H âÐ 0A_ #Q" ñ e′′¦D\ í<Êr~ .sכ &ñ\"f v\¦ ¦9½+É M:, (si , ti )\¦ H âÐ 0A\0Ð H âì2£§¦ Ðèq Ãº e%3H ,Érכ eH Õª tH ío©I\ et·ú§¤H ¦כ _p Ù¼Ð, v\¦ éß> 2\"f ¦9ôÇ Êê\ D\ í<Ê÷&%3H כ¦ _p ôÇ. uH vÐ 8 U·Ér 0Au\ eÜ¼Ù¼Ð $ ¦9÷& 9, "fe′′Ér eÐ $ D\ í<Ê ÷&%3l M:ëH\, éß> 4\"f e ¦9|c M:, e′′ÉrD\ >rFôÇ. sH e (sj , tj) âÐ 0A_ Ä»9ôÇ D_ ñH \כ íH.

&ño 16.6

·ú¦o1pu 16.4H multicutëH]j_ 2-H\¦, &ñÃº¾¡§3lqëH]j\H 12-H\¦

Ð©ôÇ.

¤ ÃZ: ·ú¦o1pu_ K yy multicutõ &ñÃº¾¡§3lqâì2£§ëH]j_ 0pxK )a

H Érכ ~1> ·ú Ãº e. ¢ôÇ y âÐ\"f âì2£§s ío÷&H ñ[þt¦ D\ V,

%3l M:ëH\ "é¶©Ð$ís $íwnôÇ. ¢ôÇ l:r&ño 16.5\ _ # H©@/©

Ð$í $íwn H ¦כ ·ú Ãº e. "f l:r&ño 13.1\"f α = 1, β = 2

ëß7á¤ Ù¼Ð c(D) ≤ 2|f |s $íwn. |f |H þj&h multicut_ ôÇs¦, c(D)H þj&h

&ñÃº¾¡§3lqâì2£§ëH]j_ ©ôÇs÷&Ù¼Ð &ño $íwnôÇ.



sÐÂÒ' 6£§õ °ú Ér H&h min-max 'a>\¦ %3H.

2£§&ño 16.7

ñ_ 6 x|¾Ós &ñÃº Áº\"fH 6£§õ °ú Ér H&h þj@/

&ñÃº¾¡§3lqâì2£§ þjè multicut 6 x|¾Ó 'a> $íwnôÇ:

max&ñÃºâì2£§ f

|f | ≤ minmulticut C

c(C ) ≤ 2 max&ñÃºâì2£§ f

|f |.

9ìøÍ&h ÕªAáÔ\"fH O(log k)-HKZOs 0pxôÇX< s âÄº, ôÇ¦ ìrÃº multicut¦ 6 x > )a. ÕªQ, &ñÃº¾¡§7áxâì2£§ëH]j_ âÄº, ÁºÐ 9ìøÍ&h ÕªAáÔ\ f #Q*ôÇ HKZO ·ú94R et ·ú§. z]jÐ 6£§_ \VH î(planar) ÕªAáÔ\"f ¾¡§3lqâì2£§ëH]j_ +þA¢-aoëH]j_ &ñÃºçßs þj

è n2H ¦כ Ð#ïr.


Multicut in General Graph

Part XI

Chapter 20


Multicut in General Graph 9ìøÍ ÕªAáÔ_ multicut ëH]j

9ìøÍ ÕªAáÔ_ multicut ëH]j

þj@/âì2£§-þjè]Xéß &ño(max-flow min-cut theorem)_ ¾¡§3lq 9ìøÍo.

1. þj@/ ¾¡§3lq âì2£§½+Ë ëH]j: 1lxr\ âìØÔH ¾¡§3lq[þt_ âì2£§_ ½+Ë¦ þj@/o -multicutëH]jü< 'a.2. þj@/ ¾¡§3lq âì2£§Ö¦ ëH]j: y ¾¡§3lq i_ Ãº¹כ di\¦ po &ñK Z~¦ H ¾¡§3lq

s f · diëßpu 1lxr âì\¦ Ãº eH âì2£§Ö¦(throughput) f\¦ þj@/o- sparsestcutëH]jü< 'a

þj@/ ¾¡§3lq âì2£§½+Ë ëH]j

ëH]j 17.1

]: Áº¾Ó ÕªAáÔ G = (V ,E), ñ 6 x|¾Ó ce ∈ Q+, "fÐ Ér n©(s1, t1), . . . , (sk , tk) (y ©Ér "fÐ Ér ¾¡§3lq_ sourceü< sink).i¦\B: G\"f H ¾¡§3lq[þt_ âì2£§ß¼l ½+Ë¦ þj@/o. s M:, y ¾¡§3lq_âì2£§Ð>rõ y ñ_ 6 x|¾Ó ]jôÇ¦ ¿º ëß7á¤KôÇ. s M:, °ú Ér ~½Ó¾Ó÷rm ìøÍ@/ ~½Ó¾ÓÜ¼Ð âìØÔH ¾¡§3lq[þt_ âì2£§ ß¼l_ ½+Ës ñ 6 x|¾Ó¦

Å#Q"fH îß)a.



þjy ¾¡§3lq i_ H (si , ti )-âÐ |9½+Ë¦ Pi , Õªo¦ P =⋃k

i=1 Pi¦ .P_ y âÐ P\ âìØÔH âì2£§_ ß¼l\¦ fP¦ .

max∑

p∈P fP (17.19)

s.t.∑

P:e∈P fP ≤ ce , e ∈ E (17.20)

fP ≥ 0, P ∈ P (17.21)

©@/ëH]j Ãº\¦ de , e ∈ E¦ . sM: de\¦ ñ e_ oÐ K$3 .

min∑

e∈E cede (17.22)

s.t.∑

e∈P de ≥ 1, P ∈ Pde ≥ 0, e ∈ E

©@/K d 0pxK |c ¹כ9 |Ér, s o °úכ¦ &ñ_)a (si , ti ) þjéßâÐþjè 1,s ÷&H .sכ ("f ìroëH]j\¦ þjéßâÐëH]jÐ ½Órçß\ Û¦Ãº eÜ¼Ù¼Ð ©@/ëH]jH ½Órçß\ Û¦ Ãº e.) s ©@/ëH]j_ &ñÃº þj&hKH Ð þjèq6 x multicuts )a.

þj@/ ¾¡§3lq âì2£§½+Ë ëH]j_ þj&hK_ O(log k) C ÷&H &ñÃºK\¦ ½ H ·ú

¦o1pu¦ /BNÂÒ > ÷&HX<, sH nulticutõ þj@/ ¾¡§3lq âì2£§½+Ë ëH]j_ ©@/

çßs O(log k)\¦ Åt ·ú§H ¦כ _pôÇ.



LP-(Pþb N±Ó§hæ

þjèq6 x multicut_ LP-¢-aoëH]j (17.22)_ þj&hK de_ 3lq&h<ÊÃº °úכ

F =∑

e∈E cede\¦ Òqty . Û¼XO> F\¦ ôÇÜ¼Ð, s\ qK ß¼t ·ú§Érq6 x¦ °úH multicut D\¦ ¹1ÔH HKZO¦ Òqty½+É Ãº e.

(\V\¦ [þt#Q o °úכs ªÃº ñ\¦ ¿º í<Ê multicutÉr ÷&tëß F\qK BÄº H q6 x¦ |9 Ãº e. (_þvëH]j 20.3, 20.4))

s\¦ 0AK multicut ÕªAáÔ G = (V ,E)\"f y ñ e_ o\¦ deÐ, ×æu\¦cedeÐ &ñ_ . ¢ôÇ &³F o °úכ de (e ∈ E)\ @/ # n uü< vçß_þjéß âÐ_ Us\¦ d(u, v)Ð ³ðl .

Äºo_ HKZOÉr 6£§_ ¿º |¦ ëß7á¤ H, "fÐ è n |9½+Ë[þt, S1,S2, . . ., Sl l ≤ k\¦ ¹1ÔH. y |9½+Ë¦ %ò%is¦ ÂÒØÔ.

#Q" i siü< ti ¿º °ú Ér %ò%i\ í<Ê÷&t ·ú§H. H iH, siü< ti×æ þjè #Q" %ò%i\ í<Ê)a.

y %ò%i Si\ _ # &ñ_)a ]Xéß δ(Si )_ ]Xéß_ 6 x|¾Ó c(Si )H,#Q" ε > 0\ @/K c(Si ) ≤ εw(Si )¦ ëß7á¤ôÇ.



#l"f, w(S)H %ò%i S_ ×æuÐ, Äº @/|ÄÌ&hÜ¼Ð ú S\ ª =åQ ×¼\¦ °úH ñ_ ×æu_ ½+ËÜ¼Ð &ñ_)a.'Í P: |Ér M = δ(S1) ∪ δ(S2) ∪ · · · ∪ δ(Sl)s multicute¦ Ð7£xôÇ. ¿ºP: |Ér, @/|ÄÌ&hÜ¼Ð úK, multicut M_ 6 x|¾Ós O(ε)F (&ñ) Hd¦ úôÇ.

ÆZ] È*× øÇa

%ò%i S1, S2, . . ., Sl l ≤ kH 6£§õ °ú Ér SX© õ&ñ¦ 5g ½$í)a. y

%ò%iÉr #Q" i\ @/K si ti ×æ _ "é¶èÐ rôÇ. s\¦ %ò%i_

ÍÒo¦ ôÇ. ÍÒo, \V\¦ [þt#Q, s1s¦ . s ÍÒo\¦ ×ædÜ¼Ð

%ò%i_ ìøÍt2£§¦ 7£xr. 7£¤, y r ≥ 0\ @/K, S(r)¦ ÍÒoÐÂÒ'_

o r¦ Åt ·ú§H n[þt_ |9½+Ës¦ : S(r) = v : d(s1, v) ≤ r.S(0) = s1Ð r # r¦ 5Åq&hÜ¼Ð 7£xrv, S(r)Ér, s1Ü¼ÐÂÒ'

o Ér íHÜ¼Ð n '÷&#Q (Ô¦5Åq&hÜ¼Ð) SX© > )a.



l:r&ño 17.2

rs 12s ÷&l \ %ò%i SX© õ&ñs 7áx«Ñ÷& S(r)Ér #Q*ôÇ i\ @/K"f

siü< ti¿º í<Ê½+É Ãº \O.

¤ ÃZ: S(r)_ H n çß_ oH 2r¦ Å¦ Ãº \O. ]j|\"f Hi\ @/K d(si , ti ) ≥ 1s.

ä»n w(S)

s]j n |9½+Ë S_ ×æu w(S)\¦ &ñSXy &ñ_ . Äº w(s1) = F/kÐZ~H. S\ ôÇ =åQ n í<Ê÷&H ñ e[þtÉr Õª í<Ê÷&H qÖ¦ qe ëßpu

×æu\¦ S\ í<Êr. 7£¤, ëß e_ ª=åQ n ¿º S\ í<Ê÷&qe = 1Ð &ñ_ôÇ. ëß e = (u, v)s¦ u ∈ S(r), v /∈ S(r)s,

qe =r − d(s1, u)

d(s1, v)− d(s1, u).

Õªo¦ s\ , S_ ×æuH 6£§õ °ú s &ñ_ôÇ:

w(S(r)) = w(s1) +∑

cedeqe .



]jr H ·ú¦o1puÉr r < 12#30A\"f c(S(r)) ≤ εw(S(r))¦ ëß7á¤ H εõ

r¦ ÆÒ½ôÇ.

l:r&ño 17.3

ε = 2 ln(k + 1)Ü¼Ð Z~Ü¼ #Q" r < 12\ @/ # c(S(r)) ≤ εw(S(r))s

$íwn > )a.

¤ ÃZ: H r ∈ [0, 12]\"f c(S(r)) > εw(S(r))s $íwnôÇ¦ . H &h\

"f dw(S(r))/dr =∑

e cede (dqe/dr)s $íwnôÇ. (#l"f Äº\"fH, Óüt:r,S(r)\ ôÇ Aá¤ =åQëß¦ ñ[þtëß¦ ½+ËôÇ.) e = (u, v)\¦ ÕªQôÇ ñ ×æ\ ¦ : u ∈ S(r), v /∈ S(r). ÕªQ,

cededqe =cede

d(s1, v)− d(s1, u)dr .

ÕªX<, de ≥ d(s1, v)− d(s1, u)sÙ¼Ð cededqe ≥ cedrs $íwn 9 "f,

dw(S(r)) ≥ c(S(r))dr > εw(S(r))dr . (17.23)

w(S(0)) = Fk, w(S( 1

2)) ≤ F + F

k. Õªo¦,∫ f (b)

f (a)

g(f )df =

∫ b

a

g(f (x))f ′(x)dx .



s\¦ (17.23)ü< ½+Ë ∫ F+ Fk

Fk

1

wdw >

∫ 12

0

1

w(S(r))εw(S(r))dr =

∫ 12

0

εdr .

ÕªQÙ¼Ð, ln(k + 1) > 12ε. sÉrכ &ñ\ íH.

N±Ó§hæ

·ú¦o1pu_ òÖ¦$í¦ 0A #, %ò%i¦ SX© H õ&ñÉr síß&h õ&ñÜ¼Ð Ãº&ñ)a. S = s1Ü¼Ð r #, s1Ð ÂÒ' o Ér nÐÂÒ' YVÐ

S\ 'ôÇ. S_ ×æu w(S)H 6£§õ °ú s Ãº&ñ)a:

w(S(r)) = w(s1) +∑

cede .

éß, Äº_ ½+Ë\"fH S\ þjèôÇ ôÇ =åQ¦ °úH ñ\¦ ¿º í<Ê 9,

w(s1) = FkÐ &ñ_ôÇ. s õ&ñÉr c(S) ≤ εw(S)s %6£§ ëß7á¤÷&H íHçß 7áx

«Ñ)a. &ñ_\ °ú Ér S\ @/K 5Åq&h õ&ñ_ ×æuÐ síß&h õ&ñ

_ ×æu °ú ß¼. "f síß&h õ&ñ\"f Òqt$í÷&H SH 5Åq&h õ&ñ

Ð 9þt Ãº \OÜ¼ 9, "f #Q" i\ @/K"f siü< ti\¦ ¿º í<Ê½+É Ãº \O.



sQôÇ síß&h SX© õ&ñ¦ 6 x H ·ú¦o1puÉr 6£§õ °ú .éß, A ' HH ÂÒìrÕªAáÔ\"f &ñ_)a y °úכ¦ _pôÇ.

·ú¦o1pu 17.4

þjè multicut HKZO1. +þA¢-aoëH]j (17.22)¦ Û¦#Q G_ y ñ_ o °úכ¦ ½ôÇ;2. ε← 2 ln(k + 1), H ← G , M ← ∅;3. H\ siü< ti ¿º í<Ê÷&H #Q" i >rF½+É M:t 6£§¦ìøÍ4¤ôÇ:

e__ sj\¦ ×þôÇ;cH(S) ≤ εwH(S) ëß7á¤|c M:t %ò%i S\¦SX©ôÇ;M ← M ∪ δH(S), H ← H \ S ;

4. M¦ Ø¦§4;



l:r&ño 17.5

·ú¦o1pu 17.4s Ø¦§4 H MÉr multicut s.

¤ ÃZ: #Q" %ò%i source-sink ©¦ í<Ê t ·ú§HH ¦כ Ðs )a. s\¦ 0AK, l:r&ño 17.2õ 17.3 H ìøÍ4¤éß>\"f $íwn<Ê¦ Ðs )a. e__ ìøÍ4¤éß>_ ÂÒìrÕªAáÔ H\¦Òqty . Äº H\"f_ oH G\"fH oÐ °ú U#Qtl M:ëH\ l:r&ño 17.2 $íwn H Érכ ~1> ·ú Ãº e.¢ôÇ, H\"f SX©÷&H %ò%i SH ôÇ F

kü< ©ôÇFk +F\¦ °úH.

"f l:r&ño 17.3_ 7£x"îÉr H\"f Õª@/Ð $íwnôÇ.



l:r&ño 17.6

c(M) ≤ 2εF = 4 ln(k + 1)F .

¤ ÃZ: iP: ìøÍ4¤éß>_ Hü< S\¦ yy Giü< SiÐ ³ðl .(G = G1.) ÕªQ ·ú¡_ 7£x"î\"f /åLôÇ õכ °ú s,cGi

(Si ) ≤ εwGi(Si ) H ìøÍ4¤éß>\"f $íwnôÇ. ¢ôÇ

wGi(Si )_ ñ[þtÉr K© iP: ìøÍ4¤éß> 7áx«Ñ ¿º ]j)a

. y ìøÍ4¤éß> þjè ôÇ©_ source-sink ©s ìro÷&l M:ëH\, Ãº'÷&H ìøÍ4¤éß>_ SÃºH k\¦ Åt ·ú§H. "f,

c(M) =∑

i cGi(Si ) ≤ ε (

∑i wGi

(Si ))

≤ ε(k F

k +∑

e cede

)= 2εF .



&ño 17.7

·ú¦o1pu 17.4H O(log k)-HK\¦ Ð©ôÇ.

2£§&ño 17.8

k->h_ source-sink ©¦ þj@/ ¾¡§3lq âì2£§½+Ë ëH]jH 6£§õ °ú ÉrH&h þj@/¾¡§3lqâì2£§-þjèmulticut6 x|¾Ó 'a>\¦ °úH:

maxâì2£§ f

|f | ≤ minmulticut C

c(C) ≤ O(log k) maxâì2£§ f

|f |.

Tight example

\V 17.9

+þA¢-aoëH]j (17.22) Ω(log k)_ &ñÃºçß¦ °úH \V\¦ ëß[þt Ãº e. sH·ú¦o1pu_ ìr$3s 0A\ lÕüt)a þj@/¾¡§3lqâì2£§-þjèmulticut6 x|¾Ó 'a>z]jÐ ©ÃºCsÐ tight H ¦כ _pôÇ. ”Expander” ÕªAáÔ.



multicut+ £ £

\V 17.10

þjèq6 x 2CNF≡]X ]jëH]j.2CNF≡ formula FH uü< v yy o'XO9 M:, (u ≡ v)_ +þAI_ ]X[þt_conjunction¦ úôÇ. y ]X\ ×æu w ÅÒ#Q&¦ M:, ]j÷& Fëß7á¤÷&H ]X[þt_ ÂÒìr|9½+Ë ×æ\"f ×æu_ ½+Ës þjè ÷&H ¦כ ¹1ÔH

ëH]j\¦ Òqty . F n>h_ ÃºÐ sÀÒ#Q&¦ .

6£§õ °ú s, y Ãº_ ¿º >h_ o'XO\ yy ôÇ >h_ n\¦ @/6£xr&,

¿º 2n>h_ n\¦ °úH ÕªAáÔ G(F )\¦ Òqty . y ]X (p ≡ q)\ ¿º >h_

ñ (p, q), (p, q)\¦ @/6£xr. ¿º >h_ ñ ¿º, ]X (p ≡ q)_ ×æuü< °ú Ér

6 x|¾Ó¦ ½+É©ôÇ. 1lx1pxôÇ ]X (p ≡ q)ü< (p ≡ q) Ñüt >rF ½+Ë5g"f

Ð @/½+É Ãº e. "f, F_ y ]X\ G(F )_ ¿º >h_ ñ

@/6£x)a¦ &ñ½+É Ãº e.



(x1 ≡ x2) ∧ (x1 ≡ x3) ∧ (x2 ≡ x3) ∧ (x2 ≡ x4) ∧ (x3 ≡ x5).

l:r&ño 17.11

F ëß7á¤|c ¹Øæìכ9r|Ér G(F )_ #Q" ¹èכ ôÇ Ãºü< Õª_ÂÒ&ñ¦ í<Ê t ·ú§H .sכ

¤ ÃZ: Øæìr |¦ ×æ"î # Ð. o'XO pü< q ôÇ >h_ \¹èכ ìøÍ×¼r Õª ÂÒ&ñ+þA[þt ôÇ >h_ \¹èכ èß. "f, #Q*ôÇ ¹èכ ôÇ Ãºü< Õª ÂÒ&ñ+þA¦ í<Ê t ·ú§H, H ,¦¹è[þtכ #Q" o'XO |9½+Ëõ ÕªÂÒ&ñ+þAÜ¼Ð sÀÒ#Q ¿º >h_ ¹èzoכ tÖ¦ Ãº e. s ôÇ ©\ 5Åq H ¿º ]X_ ÂÒ&ñ+þAÜ¼Ð @/6£x H o'XO[þtÉr o°úכ¦ ìøÍ@/Ð °ú>

H ]X¦ ëß7á¤ H o°úכ¦ ½ > )a.

(0A\"f ôÇ>h_ ]X\ ¿º>h_ ñ\¦ ¿º @/6£xrvH sכ = x½+É Ãº \OH כ

t\¦ Û¼Û¼Ð SX½+É (.כ



þjèq6 x 2CNF≡]X ]jëH]j\ 6£§õ °ú Ér G (F )_ þjèq6 xmulticutëH]j\¦ @/6£xr&Ð: y Ãº\ @/6£x÷&H ¿º >h_ n[þt¦ ¿º source-sink ©Ü¼Ð t&ñ . M¦ s multicutëH]j_þj&hK¦ ¦, C\¦ 2CNF≡]X ]jëH]j_ þj&hK¦ .(MÉr 9ìøÍ&hÜ¼Ð y ]X\ @/6£x÷&H ¿º >h_ ñ×æ\"f ôÇ >hëß¦ í<Ê½+É Ãº eH \כ Ä»_.) ÕªQ

l:r&ño 17.12

w(C ) ≤ c(M) ≤ 2w(C ).

¤ ÃZ: M¦ ]j ìr"î H Ãºü< Õª ÂÒ&ñ+þAÉr °ú Ér \¹èכet ·ú§> )a. l:r&ño 17.11\ , FH ëß7á¤÷& 9 'ÍP:ÂÒ1pxds 7£x"î)a. ¿º P: ÂÒ1pxdÉr C_ ]X\ ¿º >hm @/6£x H G (F )_ ñ|9½+Ës multicuts 9 Õª ×æu ½+Ës 2w(C )Hf"\כ ·ú Ãº e.


Semidefinite Programming

Part XII

Chapter 26



SDP¢-ao\¦ s6 xôÇ HKZOSDP¢-ao\¦ s6 xôÇ HKZOSDP¢-ao\¦ s6 xôÇ HKZO0-1 2<ÊÃº>S\ZO¦ 0AôÇ SDP ¢-ao

þj@/]XéßëH]j\¦ 0AôÇ 0.878-HKZO

Goemansü< Williamson_ þj@/]XéßëH]j HKZO½H, Ér]XHZOÜ¼Ð ²ú$í½+É Ãº eH þj@/]XéßëH]j_ þj@/ 0px H

°úכ(approximation ratio)¦, SDP\¦ 6 x S\l&hÜ¼Ð >h½+ÉÃº eH ¦כ Ð þjí_ ½s. ]jîß)a KZOÉr SXÒ¦·ú¦o1pus 9, ·ú¡_ ìrÀÓ\"f Áº0Aíî lZO\ K©ôÇ.s ½H, p²DG Ed¦êøÍ °¹t /BN@/\"f \P2;, 2000 Ãºo>S\<Ær (mathematical programming society)\"f Û¦&H©¦Ãº© %i.




þj@/]XéßëH]j

ÅÒ#Q Áº~½Ó¾Ó_ ÕªAáÔ\¦ G = (V ,E )¦ ¦, ñ_ ×æu\¦wij , (i , j) ∈ EÐ ³ðr lÐ . G_ ]Xéß(cut)sêøÍ, ×¼|9½+Ë(node set)¦ ªìrÙþ¡¦ M:, ¿º ×¼|9½+Ë\ uH ñ(edge)[þt_ |9½+Ë¦ úôÇ.

Figure: þj@/]Xéß(cut)_ \V




þj@/]XéßëH]jH ñ_ ×æu_ ½+Ës þj@/ ÷&H ]Xéß¦ ½ H ëH]js

([ÕªaË>.1]ÃÐ). 7£¤,

w(S) ≡∑

i∈S, j∈S

wij (18.24)

\¦ þj@/o H ×¼_ ÂÒìr|9½+Ë S ⊂ V\¦ ½ H ëH]js.

þj@/]XéßëH]jH ½Órçß îß\ &ñSXôÇ K\¦ ½ H ·ú¦o7£§Ér >rF t

·ú§H Ü¼Ðכ Òqty÷&H NP-hard ëH]js. :£¤y ½ÓdHKZOçHs Ô¦0pxôÇ max-NP-hard ëH]js.

&ñ_ 18.1

#Q" þjèo (þj@/o) ëH]j P_ H \V(instance)\ @/K, þj&h3lq&h<ÊÃº°úכ_ δ C\¦ Åt ·ú§H (ÅH) K\¦ Ð© H ½Órçß KZO A\¦δ-HKZO (δ-approximation algorithm) s¦ ÂÒÉr.




_þvëH]j 18.2

þj@/]XéßëH]j_ ÅÒ çßéßôÇ 12 -HKZO: y ×¼ Z>Ð

1lx¦ ~. ·ú¡s V1\ +' V2\ 5Åq 2¤

×¼|9½+Ë¦ Áº0AÐ ªìrôÇ:V = V1·∪ V2.

Ztîr zÉr 6£§õ °ú Ér HKZOs >hµ1Ï÷&l 20 1lxîß, 0A_ çßéßôÇ HKZO Ð, %3x9ôÇ _p\"f 8 Ér KZOs >rF t ·ú§¤H .sכ

Óüt:r H NP-hard ½+Ëþj&hoëH]j_ H°úכ¦ &ñSXôÇ þj&hK\ K© H 1\ ÁºôÇy ¾ú> >h½+É Ãº eH Érכ m. sQôÇ ½+Ëþj&hoëH]jH èÃº\ K© 9, ú§Ér ëH]j[þts ·ú¦o1pu_ Ãº'rçß¦ tÃº&hÜ¼Ð(exponentially) Zþtst ·ú§HôÇ,#Q" °úכÐ a%~Ér K\¦ Ð© H כ NP-hard âÄºú§. sQôÇ °úכ¦ Ô¦0px H°úכ(impossible approximation ratio)s¦ ÂÒÉr.




&³Ft ·ú9 þj@/]XéßëH]j_ Ô¦0px H°úכÉr 0.94s.7£¤ þj&hK_ 94%_ 3lq&h<ÊÃº °úכ¦ °úH K\¦ Ð© H Érכ Ô¦0px H .sכ Goemansü< Williamson_ Áº0A íîKZOÉr 88%_ l@/°úכ¦ Ð© 9, Õª_ 50%\ qK Ô¦0pxH°úכ\ S\l&hÜ¼Ð 0> ¦כ ·ú Ãº e.




SDP¢-ao\¦ s6 xôÇ HKZOþj@/]XéßëH]j_ 2<ÊÃº+þA

þj@/]XéßëH]j 0− 1 2<ÊÃº>S\ZOõ °ú Ér ëH]jeÉr ú ·ú9 zs. Goemansü< WilliamsonÉr 6£§õ °ú Ér −1,+1 2<ÊÃº>S\ZO +þA\"f Ø¦

µ1Ï %i. V = V1

·∪ V2Ð ªìr½+É M:, i ∈ V1s xi = −1, i ∈ V2s

xi = 1Ð x ∈ R|V |\¦ &ñ_ . ÕªQ þj@/]XéßëH]jH 6£§_ −1,+1 2

<ÊÃº>S\ZOëH]jü< °ú .

ëH]j 18.3

q@/gA+þA

max 12

∑i<j wij(1− xixj)

s.t. xi ∈ −1, 1 ∀i ∈ V




SDP¢-ao\¦ s6 xôÇ HKZO 7'¢-ao

·ú¡]X_ −1,+1 2<ÊÃº>S\ZO +þAÉr 0− 1 2<ÊÃº>S\ZO_ çß_ +þA9÷r, Õª Ð Ð KZO\ e#Q"f Áº :£¤Z>ôÇ ©&hs \O. Goemansü<Williamson_ HKZO_ 'ÍP: Ér s −1,+1 2<ÊÃº>S\ZO +þA¦ 6£§õ °ú s ‘ 7'>S\ZO’ëH]jÐ ¢-aoÙþ¡HX< e.

7£¤, y 1 Ãº xj\¦ m ≥ 2 m-"é¶ éß0A 7', vjÐ @/ # 6£§õ °ú Ér

ëH]j\¦ &ñ_ %i.

Figure: y xjH \V\¦ [þt#Q 3"é¶ /BNçß_ ½ S3\ 5Åq H éß0A 7'

vjÐ @/ Ãº e.




ëH]j 18.5

max 12

∑i<j wij(1− vT

i vj)

s.t. vi ∈ Sm ∀i ∈ V

ëH]j (18.3)_ e__ 0pxK xj j ∈ V ÅÒ#Q&¦ M:,vj = (xj , 0, · · · , 0) j ∈ VÉr (18.5)_ 0pxK ÷& 9, ¿º K_ 3lq&h<ÊÃº °úכÉr °ú 6£§¦ ·ú Ãº e. "f ëH]j 18.5_ þj&h3lq&h<ÊÃº°úכÉr þj@/]XéßëH]j_ ©ôÇs Hd¦ ·ú Ãº e. s _p\"fëH]j (18.5)H ëH]j (18.3)_ ¢-aoëH]j(relaxation) )a.

éßíHy ¢-aoëH]j\¦ ëß×¼H sכ 3lq&hs 0A\"f :r ,3!%כm≥ 2 #Q" m 0px . ÕªQ Goemansü< Williamson_ KZO¦ 0AK"fH m = nÜ¼Ð Z~ôÇ.




ëH]j 18.6

7'>S\ZOëH]j

max 12

∑i<j wij(1− vT

i vj) (18.25)

s.t. vi ∈ Sn ∀i ∈ V

ÄºoH #l"f Û¼XO> 6£§õ °ú Ér ¿ºt |9ëH¦ > )a

.

1 7'>S\ZOëH]j 18.6H òÖ¦&hÜ¼Ð Û¦ Ãº eH?

2 ÕªXO, ëH]j 18.6_ þj&hKH þj@/]XéßëH]j_ ‘a%~Ér’K\¦ ½ HX< 6 x|c Ãº eH?




'ÍP: |9ëH_ ²úÉr sp A\ ·ú94R e%3. 6£§õ °ú Ér '§>=[þt_ YL¦ Òqty .

−− vT

1 −−−− vT

2 −−...

−− vTn −−

| | |v1 v2 · · · vn

| | |

(18.26)

=

vT

1 v1 vT1 v2 vT

1 v3 · · · vT1 vn

vT2 v1 vT

2 v2 vT2 v3 · · · vT

2 vn

......

vTn v1 vT

n v2 vTn v3 · · · vT

n vn

. (18.27)

s M:, yij ≡ vTi vj(= vi · vj)Ð &ñ_

%3#Q (Gram) '§>=,

Y ≡

y11 y12 · · · y1n

y21 y22 · · · y2n

.... . .

...yn1 yn2 · · · ynn

(18.28)

Ér (@/gA) PSD(positive semidefinite)'§>=s )a.<ª $í 9 þj&hoHKZO



"f ëH]j 18.6Ér 6£§õ °ú Ér SDP )a.

ëH]j 18.7

max 12

∑i ,j wij(1− yij)

s.t. yii = 1 ∀i = 1, · · · , nY = (yij)H PSD.

PSD '§>=Ér &ñ~½Ó'§>=_ YLÜ¼Ð ³ðr|c Ãº eÜ¼Ù¼Ð, Y =W TWÐ jþt Ãº e¦, W_ y 'Ü¼Ð &ñ_)a 7'[þtÉr ëH]j18.6_ K )a. "f ëH]j 18.6H SDP ëH]j 18.7õ 1lx1pxôÇëH]j ÷& 9, sÉrכ ëH]j 18.6 ½Órçß\ Û¦ Ãº eH _pÐ"f 'ÍP: |9ëH_ ²ús )a.

s]j ¿ºP: |9ëH\ ²úKÐ.




SDP¢-ao\¦ s6 xôÇ HKZOSXÒ¦·ú¦o1pu

6£§õ °ú Ér SXÒ¦·ú¦o1pu(randomized algorithm)¦ ÒqtyKÐ.

1 ëH]j 18.6_ þj&hK, v1, v2, · · · , vn\¦ ½ôÇ.

2 éß0A ½ SnÜ¼Ð ÂÒ' çH9ôÇ SXÒ¦Ð 7', r¦ Áº0AÐ×þôÇ.

3 rõ Ãºfs¦ "é¶&h¦ tH íîÜ¼Ð v1, v2, · · · , vn\¦

¿º ÕªÒÜ¼Ð èHÊê, s\ ×¼[þt¦ ªìr # G_]Xéß¦ Òqt$íôÇ. 7£¤, ëß rTvi ≥ 0s, xi ← 1Ð Z~¦,ÕªXOt ·ú§Ü¼, xi ← −1Ð Z~H.

&ño 18.8

0A_ KZOÉr þj@/]XéßëH]j_ 0.878-HKZOs )a.




¤ ÃZ: HKZOÜ¼Ð %3#Q K_ l@/°úכÉr 6£§õ °ú s ÅÒ#Q.

E [w(S)] (18.29)

=∑

i<j wijPr [viü< vj rõ Ãºf íîÜ¼Ð ¾º#Q.]

=∑

i<j wijPr[sgn(rTvi ) 6= sgn(rTvj)

]ÕªQ, sgn(rTvi ) 6= sgn(rTvj)9 ¹כ9 Øæìr|Ér, r¦ viü< vjÐ Òqt$í÷&H î

\ ÈÒ%òÙþ¡¦ M:, ôÇ 7'ü<H \Vy¦ Ér 7'ü<H éHy¦ sÀÒH .sכ(ÕªaË>3 ÃÐ¦.)

Figure: viü< vj ìro÷&H âÄº.




sH "é¶ÅÒÖ¦ π ×æ\, viü< vj_ e±y_ ¿º C\ K© H ÂÒìr

\ rs ÈÒ%ò|c SXÒ¦õ °ú . "f,

Pr [viü< vj rõ Ãºf íîÜ¼Ð ¾º#Q.] (18.30)

= 1π arccos (vT

i vj).

ÕªQÙ¼Ð,

E [w(S)] =∑

i<j wij1π arccos (vT

i vj) (18.31)

≥ 0.87812

∑i<j wij(1− vT

i vj). (18.32)

(18.32)H 6£§õ °ú Ér zM:ëH\ $íwn2³. (ÕªaË> 4Harccos x

π /(1/2)(1− x)_ °úכ¦ −1 ≤ x ≤ 1_ #30A\"f rôÇ כs.)




1

πarccos x ≥ 0.878

1

2(1− x) ∀ − 1 < x < 1 (18.33)

Figure: arccos xπ /(1/2)(1− x).

ÕªQ, 7'¢-aoZOëH]j 18.6_ þj&h3lq&h<ÊÃº°úכÉr, þj@/]XéßëH]j_ 3lq&h<ÊÃº°úכÐ ß¼Ù¼Ð, 7£x"îs =åQèß.




0-1 2<ÊÃº>S\ZO¦ 0AôÇ SDP ¢-ao

0− 1 2 <ÊÃº>S\ZOëH]jÉr © ~1> SDPÐ ¢-ao|c Ãº e6£§¦ ·ú Ãº eH ëH]js. 1997 Nesterov ]jîßôÇ, H°úכs ÕªÊê\ >h)a KZO[þt ÐH åÔtëß ·ú¡]X_ ~½ÓZO¦ çßéß >

SX©ôÇ ~½ÓZO¦ è>hôÇ.

0− 1 2 <ÊÃº>S\ZOëH]jH çßéßôÇ Ãºu8¦ 5g, 6£§õ°ú Ér 1lx1pxôÇ −1,+1 2 &ñÃº >S\ëH]jÐ ~1> 8½+É Ãº e.

ëH]j 18.9

max∑

i ,j aijxixj

s.t. xi ∈ −1, 1 ∀ i




#l"f A = (aij) ∈ SnH PSD¦ &ñ½+É Ãº e. Õª sÄ»H,x2i = 1sÙ¼Ð A_ @/y\ K© H ÂÒìrÉr ©Ãº ½Ó\ K© l M:ëH\ ¹כ9 \OHt H ©Ãº\¦ @/y\ 8½+É Ãº e

l M:ëHs. "f þj&h3lq&h<ÊÃº °úכs q6£§s¦ &ñ½+É Ãºe.

·ú¡_ ~½ÓZOõ Ä» > xi\¦ éß0A 7' vi ∈ RnÐ u8 #, 6£§õ °ú Ér 7'>S\ZOëH]j\¦ %3H.

ëH]j 18.10

max∑

i ,j aijvTi vj

s.t. vTi vi = 1 ∀ i

vi ∈ Rn




6£§õ °ú Ér SXÒ¦·ú¦o1pu(randomized algorithm)¦ ÒqtyKÐ.

1 0A_ 7'>S\ZO ëH]j_ þj&hK, v1, v2, · · · , vn\¦ ½ôÇ.

2 éß0A ½ SnÜ¼Ð ÂÒ' çH9ôÇ SXÒ¦Ð 7', r¦ Áº0AÐ×þôÇ.

3 þj&hK v∗Ð ÂÒ' 6£§õ °ú s x\¦ ½ôÇ.

xi =

1 if rTv∗i ≥ 0−1 if rTv∗i < 0

ÕªQ 6£§õ °ú Ér H°úכ\¦ %3H.

&ño 18.11

0A_ SXÒ¦·ú¦o1puÉr 2π -HKZOs )a.




i 6= j âÄº, 6£§s $íwnôÇ:

Pr[sgn(rTv∗i 6= rTv∗i )] = Pr[xi xj = −1] =1

πarccos v∗i

Tv∗j .

(18.34)"f, 6£§õ °ú Ér l@/°úכ¦ %3H:

E [xi xj ]

= 1(1− 1π arccos v∗i

Tv∗j ) + (−1) 1π arccos v∗i

Tv∗j

= 1− 2π arccos v∗i

Tv∗j

= 1− 2π (π

2 − arcsin v∗iTv∗j )

= 2π arcsin v∗i

Tv∗j .




l:r&ño 18.12

ëß zij = arcsin xij − xijs¦ |xij | ≤ 1 for all i , js X 09 M:,Z 0s )a.

l:r&ño 18.12\¦ 6 x ,

E [∑

i ,j aij xi xj ]− 2π

∑i ,j aijv

∗iTv∗j

= 2π

∑i ,j aij(arcsin v∗i

Tv∗j − v∗iTv∗j )

= 2π

∑i ,j aijzij

= 2πX · Z

≥ 0 (Lemma 18.12).

"f &ño 7£x"î)a.


Counting Problems

Part XIII

Chapter 28


Counting Problems H&h Counting

H&h Counting

&ñ_ 19.1

#P: L¦ NP_ ôÇ #Q¦ . M¦ s_ 7£x(verifier)q&ñÈÓaAl>, Õªo¦ p\¦ \V-H_ Us_ ½Ó<ÊÃº ©ôÇs¦ .ëHH x ∈ Σ∗\ @/K, f (x)\¦ |y | ≤ p(|x |)s 9 M(x , y) ~ÃÎ[þtsH y_>hÃº¦ . sQôÇ <ÊÃº f : Σ∗ −→ Z+_ |9½+Ë¦ #P¦ ôÇ.

f'a&hÜ¼Ð ú #PêøÍ NP ëH]j_ ëH]j\V e¦ M:,‘\V’ ÷&H “K”_ >hÃº\¦ [jH ëH]j¦ ½+É Ãº e. \V\¦ [þt#Q, Áº¾ÓÕªAáÔ_ Kx9:r âÐ_ >hÃº, SAT_ Øæ7á¤ o°úכ ½+É©_ >hÃº 1pxs Õª \Vs. "f #PH þjèôÇNP ëßpu #Q9îr ëH]j¦ ½+É Ãº e.

f'a&hÜ¼Ð úK"f, H #P ëH]j[þts #Q" #P ëH]jÐ ½Ó8 0px½+É M:,

Êê\¦ #P-completes¦ ôÇ. NP-complete ëH]j_ 0pxK\¦ [jH ëH]jH

@/ÂÒìr #P-completes 9, <ÉªpÐîr Érכ ú§Ér P ëH]j_ 0pxK\¦ [jH ëH]j

%ir #P-completesH .sכ



&ñ_ 19.2

FPRAS: counting ëH]j #P-complete P\ 5Åq H ëH]j e¦ .s M:, 6£§õ °ú Ér ·ú¦o1pu A\¦ fully polynomial time randomizedapproximation scheme (FPRAS)¦ ôÇ: H ëHH x ∈ Σ∗ü<

ε > 0\ @/K,

Pr[|A(x)− f (x)| ≤ εf (x)] ≥ 3

4.

ëH]j 19.3

DNF K_ >hÃº f = C1 ∨ C2 ∨ · · · ∨ Cm¦ n>h_ ÂÒÖ¦Ãº x1, . . ., xn_ DNF¦ . li\¦ o'XOs¦ ½+É M:, 6£§õ °ú s ³ð&³÷&H y ]XCi = l1 ∧ l2 ∧ · · · ∧ lriÉr 0pxK\¦ °úH¦ &ñ½+É Ãº e. (7£¤, #Q" Ãºü<s_ ÂÒ&ñs °ú s [þt#Q et ·ú§.) Äºo_ ëH]jH f_ 0pxK_ >hÃº #f\¦>íß H .sכ



l:r&h sn#QH #f\ @/ôÇ Ô¦¼#ÆÒ&ñ|¾Ó (unbiased estimator)s ÷&H SXÒ¦Ãº X\¦ &ñ_ H :sכ E [X ] = #f . ëß\ ÕªQôÇ X_ ³ðïr¼#σ[X ] E [X ]_ ½Ó<ÊÃº CÐ ]jôÇ÷& FPRAS\¦ ~1> ëß[þtÃº e: X_ °úכ¦ ëH]j 9§4 ß¼lü< 1

ε_ ½Ó<ÊÃº SÃº ëßpu ³ð:r¦ ÆÒØ¦ # îçH¦ Ø¦§4

)a.

~1> Òqty½+É Ãº eH ~½ÓZO: 2n>h_ o°úכ τ 0A\ çH9ìrí(uniformdistribution)\¦ °úH SXÒ¦Ãº Y\¦ 6£§õ °ú s &ñ_ôÇ: τ f\¦ ëß7á¤ Y (τ) = 2n, ÕªXOt ·ú§Ü¼ Y (τ) = 0. tëß s âÄºH ³ðïr¼# -Áº &"f FPRAS\¦ s=åJ#Q ?/t 3lwôÇ.

sQôÇ ëH]j&h¦ FG4¤ l 0AK, f\¦ ëß7á¤ H o°úכ[þtëß 0s SXÒ¦¦°ú2¤ SXÒ¦Ãº\¦ &ñ_ôÇ.

Si\¦ Ci\¦ ëß7á¤ H n>h Ãº_ o°úכ |9½+Ës¦ . ÕªQ |Si | = 2n−ri ,

#f = |⋃m

i Si |. c(τ)\¦ τ ëß7á¤ H ]X_ >hÃº¦ . M¦ Si[þts ‘×æ4¤

¦ ¿º í<Ê H’ ½+Ë|9½+Ës¦ : |M| =∑m

i |Si |. MÉr y τ\¦ c(τ) SÃº

í<Ê > )a.



f\¦ ëß7á¤ H o°úכ τ\¦ c(τ)/|M|_ SXÒ¦Ð ×þ÷&Ð2¤ ôÇ. Õªo¦X (τ) = |M|/c(τ)Ð &ñ_ .

l:r&ño 19.4

SXÒ¦Ãº X\¦ òÖ¦&hÜ¼Ð ³ð:rÃº|9s 0px 9 #f\ @/ôÇ Ô¦¼#ÆÒ&ñ|¾Ós)a.

¤ ÃZ: Äº y τ c(τ)/|M|_ SXÒ¦Ð ×þ÷&2¤ òÖ¦&h ³ð:rÃº|9s 0px H ¦כ Ðs. Äº y ]X, Ci |Si |/|M|_ SXÒ¦Ð ]X¦ ×þôÇ. Õªo¦ s ]X¦ ëß7á¤ H o°úכ |9½+Ë ×æ\ çH1pxôÇ SXÒ¦Ð \¦ ×þôÇ. ÕªQ, Äºo "é¶ H SXÒ¦_ ³ð:rÃº|9s )aH ¦כ ~1> Ö½+É Ãº e (K¦.(כ

¢ôÇ,

E [X ] =∑

τ Pr[r s ×þHd] · X (τ)

=∑

f \¦ ëß7á¤ H τ

c(τ)|M|

|M|c(τ)

= #f .



l:r&ño 19.5

m¦ ]X_ >hÃº¦ , σ(X )E [X ] ≤ m − 1.

¤ ÃZ: |M|m ¦ α¦ . ÕªQ E [X ] ≥ α $íwnôÇ. y o

°úכ\ @/K, 1 ≤ c(τ) ≤ m. "f X (τ) ∈ [α, mα], "f|X (τ)− E [X ]| ≤ (m − 1)α. "f, ³ðïr¼#, σ(X ) ≤ (m − 1)α.



l:r&ño 19.6

k>h_ ³ð:r_ îçH¦ Yk¦ e__ ε > 0\ @/ #,k = 4(m − 1)2/ε2s

Pr[|Yk −#f | ≤ ε#f ] ≥ 3

4.

¤ ÃZ: q[VáÔ_ ÂÒ1pxd, σ(Yk) = σ(X )/√

k\¦ 6 x ,

Pr[|Yk − E [Yk ]| ≥ εE [Yk ]]

≤(

σ(Yk )εE [Xk ]

)2=

(σ(X )

ε√

kE [X ]

)2≤ 1

4

sÐ"f FPRAS ¢-a$í)a.


NP-complete

Part XIV

Chapter 29


NP-complete

NP-completee¦ 7£x"î lL-8õ Max-SNP-hardnessMax-SNP-hardnessMST\¦ 6 xôÇ StT ·ú¦o1pu

NP-completee¦ 7£x"î l

From Computers and intractability by Garey and Johnson

&ñëH]j(decision problem) Π NP-completee¦ 7£x"î H éß>

(1) Π ∈ NPe¦ Ð: ²ús “Yes” 9 M:, s\¦ ½Órçß\ SX ½+É Ãº eH ~½ÓZO, 7£¤ yes-certificate (7£x"î, proof, ¢Hverifier)s >rF<Ê¦ Ð.

(2) Π\¦ sp NP-completee¦ ·ú¦ eH ëH]j Π′Ü¼Ð ½Órçß\ 8s 0pxôÇ ¦כ Ð.


NP-complete


Cook_ &ño\ _ # SAT\¦ Π′Ü¼Ð ×þ½+É Ãº e. s\¦ rÜ¼Ð 6£§_6>h ëH]j_ NP-completeness\¦ éß>&hÜ¼Ð 7£x"îôÇ.

ëH]j 20.1

SAT]: Ä»ôÇ>h_ ÂÒÖ¦Ãº |9½+Ë X = x1, x2, . . . , xn_ o'XO[þt_disjunctionÜ¼Ð ½$í)a ]X[þt_ |9½+Ë C = c1, c2, . . . , cm.%K: H ]X¦ ëß7á¤ H o°úכs >rF H?

ëH]j 20.2

3SAT]: Ä»ôÇ>h_ ÂÒÖ¦Ãº |9½+Ë X = x1, x2, . . . , xn_ [j >h_ o'XO[þt_disjunctionÜ¼Ð ½$í)a ]X[þt_ |9½+Ë C = c1, c2, . . . , cm.%K: H ]X¦ ëß7á¤ H o°úכs >rF H?


NP-complete


ëH]j 20.3

3DM]: y q>h_ "é¶è\¦ °úH "fÐ è |9½+Ë W , X , Õªo¦ Y .ÂÒìr|9½+Ë M ⊆W × X × Y .%K: MÉr ¢-af±l M ′

¦ °úH? 7£¤, °ú Ér ýa³ð\ °ú Ér"é¶è\¦ °út ·ú§Ér q>h_ íH"f© |9½+Ë M ′s M\ >rF H?

ëH]j 20.4

VC]: Áº¾ÓÕªAáÔ G = (V ,E )ü< ªÃº k(≤ |V |).%K: G\ ß¼l k s vertex cover C >rF H? 7£¤,E_ H ñ_ =åQ n ×æ &h#Q ôÇ >h\¦ í<Ê 9, ß¼l k\¦Åt ·ú§H V_ ÂÒìr|9½+Ë C >rF H?


NP-complete


ëH]j 20.5

CLIQUE]: Áº¾ÓÕªAáÔ G = (V ,E)ü< ªÃº k(≤ |V |).%K: G\ ß¼l k s 9þtaË: Q >rF H? 7£¤, QÐ Ä»)a G_ÂÒìrÕªAáÔ G [Q] ¢-a ñÕªAáÔ ÷& 9, ß¼l kü< °ú H V_ÂÒìr|9½+Ë Q >rF H?

ëH]j 20.6

HC (Hamiltonian circuit)]: Áº¾ÓÕªAáÔ G = (V ,E).%K: G Kx9 rÐ(Hamiltonian circuit)\¦ tH? 7£¤, H n\¦í<Ê H éßíHrÐ\¦ ÂÒìrÕªAáÔÐ tH ?

ëH]j 20.7

PARTITION]: Ä»ôÇ|9½+Ë Aü< Õª "é¶è a ∈ A_ ×æu w : A→ Z+.%K: ×æu_ ½+Ës °ú 2¤ A\¦ ¿º>h_ |9½+ËÜ¼Ð ÐütÃº eH ?


NP-complete


&ño 20.8

3SATÉr NP-completes.

¤ ÃZ: 3SAT ∈ NPeÉr ~1> ·ú Ãº e: H ]X¦ ëß7á¤ H o°úכ¦yes-certificateÜ¼Ð 6 x O(]X_ Ãº)_ rçß\ SXs 0px . (éß, ÕªQôÇ o°úכ¦ #QbG> ½ HtH hs&¹כ mH ¦כ "îd .)SAT ∝ 3SAT: ÅÒ#Q SAT_ ôÇ ]X c k>h_ o'XO z1, · · · , zkÐ sÀÒ#Q&

¦ : k ≥ 4 âÄº c := z1 ∨ z2 ∨ · · · ∨ zk . c\¦ o'XO¦ [j>hm ]XÐ 8 )a.c ′ = (z1 ∨ z2 ∨ y1) ∧ (y1 ∨ z3 ∨ y2) ∧ (y2 ∨ z4 ∨ y3) ∧ · · · (yk−4 ∨ zk−1 ∨ yk−3) ∧(yk−3 ∨ zk−1 ∨ ∨zk).cü< c ′_ satisfiability 1lxuH ¦כ A\¦ 7£x"î H Ü¼Ðכ r # ) ±ú&hÜ¼Ð SX ½+É .כ

z1 ∨ · · · ∨ zm ≡ (z1 ∨ · · · zl ∨ y) ∨ (zl+1 ∨ · · · ∨ zm ∨ y).


NP-complete


_þvëH]j 20.9

k ≤ 2 âÄº, c ′¦ #QbG> ½$í½+É Ãº eH?

_þvëH]j 20.10

2SATÉr ½Órçß\ Û¦ Ãº e6£§¦ Ð#.


NP-complete


&ño 20.11

3DMÉr NP-completes.

¤ ÃZ: 3SAT ∝ 3DM: ÂÒÖ¦Ãº |9½+Ë X = x1, x2, . . . , xnõ Õªo'XO[þtÐ ½$í)a ]X[þt_ |9½+Ë C = c1, c2, . . . , cm¦ °úH3SAT¦ Òqty . &ñSXy, 3SAT¦ ëß7á¤ H o°úכs >rF½+ÉM:, ¢-af±l\¦ °úH |9½+Ë W , X , Yü< ÂÒìr|9½+ËM ⊆W × X × Y\¦ ½$í )a.7á§8 ½&hÜ¼Ð úK, 3SAT (¢H SAT)_ H ]X¦ ëß7á¤ Ho°úכs >rFôÇH Érכ G±ÛÖR ÉÙ·£o>ÊÁ+ xjSÌfCø ÉÙÇaÌfCl

â kUc m Níâ¿ ÇÐ âUc" h'aÝ£· ø55ÛÏ ¤n>ÇaTÒ» ÊÁ

ÑÏÐM ¿ õ 1lxuH ¦כ "îd . s Érכ K© Ãº, 6 x)a, H ]X\"f True <ÊÉr False ôÇ t âÄºÐëß 6 x÷&#Q ôÇH õכ 1lxus.


NP-complete


sQôÇ |_ $íwnõ 9u H 3DM ëH]j íH"f©[þt_ |9½+Ë¦[j éß>Ð ½$í½+É .sכ

éß> I\"fH #Q" ÂÒÖ¦Ãº "é¶+þAõ ÂÒ&ñ+þAs 1lxr\ ¿º >hs©_ ]X\"f ×þs ÷&t ·ú§2¤ íH"f©¦ ½$íôÇ. éß> II\"fH éß> I_ ½©gË:\ úH y ]X ôÇ >h o'XO ×þs 0px H õכ 0pxôÇ íH"f©s >rFôÇH sכ 1lxu ÷&2¤ íH

"f© |9½+Ë¦ ½$í½+É .sכ tÜ¼Ð éß> III\"fH, éß> Iõ II_ íH"f©s >rF H âÄº, ìøÍ×¼r ¢-a f±lÐ SX©÷&2¤ íH"f©¦ ' ½+É .sכ


NP-complete


I. Äº y Ãº xi\ @/ # 6£§õ °ú ÉríH"f©[þt¦ H ]X

j = 1, 2, . . . ,m\ @/K ½$íôÇ:

T ti = (xij , aij , bij) : 1 ≤ j ≤ m

T fi = (xij , ai(j+1), bij) : 1 ≤ j < m ∪ (xim, ai1, bim)


NP-complete


aü< bH 0A\¦ ]jü@ôÇ íH"f©\H t ·ú§H ‘?/ÂÒ6 x’ Ãºs. "f, ¢-af±l\¦ 0AK"fH, y i\ @/ # 0A\"f &ñSXy m>h ×þ÷&#Q 9, sH ¿º T t

i _ "é¶è[þts

T fi _ "é¶è[þtsH ¦כ ·ú Ãº e.sH xi °úכ¦ H ]X\"f Trueü< FalseÐ Z~H \כ @/6£x > )a. sü< 1lxuÐ, H ]Xs ¦ ëß7á¤rvH o'XO¦ ôÇ >hm ×þ½+É M:, #Q" ÂÒÖ¦Ãº xi "é¶+þAõ ÂÒ&ñ+þAs 1lxr\ ¿º

>h s©_ ]X\"f ×þ÷&t ·ú§H ¦כ _pôÇ.


NP-complete


II. y ]X cj\ ¿º ?/ÂÒ6 x Ãº\¦ @/6£xr: s1j ∈ X , s2j ∈ Y . Õªo¦ 6£§_ íH"f©¦ ½$íôÇ: Cj = Cj1 ∪ Cj2,Cj1 = (xij , s1j , s2j) : xi ∈ cj, Cj1 = ∪(xij , s1j , s2j) : xi ∈ cj.

"f, ¢-af±l ÷&l 0AK"fH Cj\"fH f ôÇ >h_ íH

"f©¦ ×þ > )a. sH y ]Xs ¦ ëß7á¤rvH “@/³ð”o'XO¦ ×þ H \כ @/6£x)a. ëß ]X j xij\¦ ×þ

%i sH "é¶+þA o'XO xi\¦ ×þ %iH _ps 9, éß> I_ íH"f© ×æ\"f T t

i ×þ÷&H âÄº\ëß 0px . "f I_ ½©gË:\ , Ér ]X[þt, xi\¦ ×þôÇ %ir °ú Ér "é¶+þA

Ü¼Ðëß ×þ½+É Ãº e. ðøÍtÐ ëß Cj\"f xij ×þ)a

sH ÂÒ&ñ+þA xi\¦ ¦ ëß7á¤rvH “@/³ð” o'XOÐ ×þôÇH _ps 9, éß> I\"f T f

i ×þ)a âÄº\ëß 0px .Õªo¦ xiH Ér ]X\"f 6 x)a, f ÂÒ&ñ+þAÜ¼Ðëß 6 x)a.


NP-complete


III. ÅÒ#Q 3SAT¦ ëß7á¤ H o°úכs >rF H ,õכ I\"f ½$íôÇ |9½+Ë\"f&ñSXy mn>h Õªo¦ II\"f ½$íôÇ |9½+Ë\"fH m>h_ íH"f©¦, °ú Ér 0Au\°ú Ér "é¶è µ1ÏÒqt t ·ú§2¤, ×þ½+É Ãº eH õכ 1lxuH ¦כ ·ú Ãº e.s íH"f©_ 'ÍP: "é¶è_ |9½+Ë W\¦ xij ¢H xij_ |9½+ËÜ¼Ð Z~Ü¼ ¿º

2mn>hs. ¿º P: "é¶è_ |9½+Ë X aijü< s1j[þt¦ í<Ê > ¦, [j P:"é¶è_ |9½+Ë Y bijü< s2j[þt¦ í<Ê > , ·ú¡\"f /åLôÇ 3SAT¦ ëß7á¤ H o°úכ\ K© H íH"f©[þt_ |9½+ËÉr mn + m>h )a.

"f sQôÇ íH"f© |9½+Ë¦ ¢-a f±lÐ ½Ó© SX©½+É Ãº e> l 0AK

"f Qt 2mn −mn −m>h_ xij ¢H xij í<Ê÷&H íH"f©s ½Ó© >rF

2¤ 6£§õ °ú Ér íH"f©[þt¦ M\ í<Êrv )a. sH y xij ¢H xij\>

2mn−mn−m>h_ 2"é¶è íH"f© (g1k , g1k), 1 ≤ k ≤ m(n− 1) ¿º

0px 2¤ 6£§õ °ú Ér íH"f© |9½+Ë¦ M\ í<ÊrvH .sכ

G = (xij , g1k , g2k), (xij , g1k , g2k) : 1 ≤ k ≤ m(n − 1), 1 ≤ i ≤ m, 1 ≤ j ≤ n.sH X\H g1k , 1 ≤ k ≤ m(n − 1) Y\H g2k , 1 ≤ k ≤ m(n − 1) ÆÒ÷&

H ¦כ _pôÇ.


NP-complete


&ño 20.15

3SAT ∝ VC

¤ ÃZ: 3SAT ∝ 3DM: ÂÒÖ¦Ãº |9½+Ë X = x1, x2, . . . , xnõ Õªo'XO[þtÐ ½$í)a ]X[þt_ |9½+Ë C = c1, c2, . . . , cm¦ °úH3SAT¦ Òqty . y ]X_ [j >i_ o'XO[þt¦ e__ íH"fÐ¦&ñ ¦, j P: ]X_ l P: o'XO¦ zjlÐ ³ðl :c1 = z11 ∨ z12 ∨ z13, . . ., cm = zm1 ∨ zm2 ∨ zm3.k = n + 2m.(x1 ∨ x2 ∨ x4) ∧ (x1 ∨ x4 ∨ x5) ∧ (x2 ∨ x3 ∨ x5)


NP-complete


_þvëH]j 20.16

VC ∝ STABLE ∝ CLIQUE ∝ VC


NP-complete


&ño 20.17

VC ∝ HC.

¤ ÃZ: e__ VC ëH]j\V\¦ Òqty : Áº¾ÓÕªAáÔ G = (V ,E )ü<Ãº k ≤ |V |. s\ , &ñSXy G k\¦ Åt ·ú§H vertexcover\¦ |9 M:, Kx9 rÐ\¦ °úH ÕªAáÔ G ′ = (V ′,E ′)\¦ ½Órçß\ ½$íK ôÇ.


NP-complete



NP-complete


&ño 20.18

3DM ∝ PARTITION.

¤ ÃZ: e__ 3DM ëH]j \V, 7£¤, y q>h_ "é¶è\¦ °úH "fÐ è|9½+Ë W = w1,w2, . . . ,wq, X = x1, x2, . . . , xq, Õªo¦Y = y1, y2, . . . , yqü< ÂÒìr|9½+Ë T ⊆W × X × Y\¦ Òqty .|T | = k, T = t1, t2, . . . , tk¦ .&ñSXy Ms ¢-af±l\¦ tH âÄº ×æu_ ½+Ës °ú Ér ìr½+É¦ °úH PARTITION_ ëH]j \V\¦ ëß[þt#Q Ð:S = s(a) : a ∈ A.SH k + 2 >h_ "é¶è\¦ °úHX<, s ×æ k>h, a1, a2, . . . , akHT_ y "é¶è\ @/6£x÷& 9 6£§ ÕªaË>õ °ú s 3q × p>h_ aÅ@Ãº\¦ °úH 2ZO ×æu s(ai )\¦ °úH.


NP-complete


ëß ti = (wji , xki, yli )s s\ @/6£x H aiH ÕªaË>_ wji ñß,

xkiñß, Õªo¦ zli ñß[þt_ © Ér Aá¤, 7£¤ © Ér aÅ@Ãº[þt¦

¿º 1Ð ¦ QtH ¿º 0Ü¼Ð Z~Ér Ãº\¦ ×æu s(ai )Ð&ñ_ôÇ:

s(ai ) = 2p(3q−ji ) + 2p(2q−ki ) + 2p(q−li ).


NP-complete


"f T ′ = ti1 , ti2 , . . . , tiq ⊆ T ¢-af±l |c ¹Øæìכ9r|Ér T ′_

#Q" ¿º ×æu °ú Ér ñß © ÉrAá¤ aÅ@Ãº\ >á¤°ú s 1¦ °út ·ú§HH,Ü¼ 9כ ¢ôÇ H ñßs © ÉrAá¤ aÅ@Ãº\ 1¦ °úHH .sכy ñß_ aÅ@Ãº_ Ãº p = dlog2(k + 1)esl M:ëH\ k>h_ ×æu\¦ ¿º 8K #Q" ñß_ Õüw[þt_ ½+Ës ¢,aAá¤ ñßÜ¼Ð ¦H 9Ér µ1ÏÒqt t ·ú§H."f, ¢-af±l |c ¹Øæìכ9r|Ér Õª @/6£x H ×æu_ ½+Ës 6£§õ°ú H :sכ

q∑j=1

s(aij ) = B.

Qt ¿º >h_ "é¶è b1õ b2H 6£§õ °ú s &ñ_ :

s(b1) = 2k∑

i=1

s(ai )− B, s(b2) = 2k∑

i=1

s(ai ) + B.

ÕªQ, · · · .


NP-complete


Y>t _ß¼_

Restriction

Local Replacement

Component Design

Restriction

SET COVERHITTING SETSUBGRAPH ISOMORPHISMBOUNDED DEGREE SPANNING TREEMINIMUM EQUIVALENT DIGRAPHKNAPSACKMULTIPROCESSOR SCHEDULING


NP-complete


Local Replacement

ëH]j 20.19

ENSEMBLE COMPUTATION]: Ä»ôÇ|9½+Ë A, Õª ÂÒìr|9½+Ë[þt_ |9½+Ë C, Õªo¦ Ãº J.%K: 6£§_ |¦ ëß7á¤ H Aü< °ú Ér \P(sequence)s>rF H?

z1 = x1 ∪ y1, z2 = x2 ∪ y2, . . . , zj = xj ∪ yj .

1 j ≤ J,2 xiü< yiH yy A_ "é¶è aÐ ëß[þt#Q |9½+Ë as

k < i zks¦, xiü< yiH "fÐ ès 9,3 H c ∈ C zi ×æ\ èß.


NP-complete


&ño 20.20

VC ∝ ENSEMBLE COMPUTATION.

¤ ÃZ: ÕªAáÔ G = (V ,E )ü< Ãº KÐ &ñ_)a VC\¦ ECÐ 8 . By a “local replacement”:

A← V ∪ a0, C ← a0, u, v : uv ∈ E , J ← K + |E |.


NP-complete


ëH]j 20.21

PARTITION INTO TRIANGLES]: #Q" Ãº q\ @/K |V | = 3q ÕªAáÔ G = (V ,E ).%K: V\¦ "fÐ è 3"é¶è |9½+Ë(3-sets) V1,V2, . . . ,VqÐ

ìr½+É # ViÐ Ä»)a ÕªAáÔ ¿º y+þA(C3)s ÷&2¤ ½+ÉÃº eH?

&ño 20.22

X3C ∝ PIT.

¤ ÃZ: #Q" Ãº q\ @/K |X | = 3q |9½+Ë Xü< Õª ÂÒìr|9½+Ë[þt_ |9½+Ë CÐ &ñ_)a X3C\¦ Òqty .


NP-complete


C_ |9½+Ë ci = xi , yi , zi\¦ 6£§õ °ú Ér ÕªAáÔ EiÐ @/ #

ÕªAáÔ\¦ 6£§õ °ú s &ñ_,

V := X ∪|C|⋃i=1

aij : j = 1, . . . , 9, E =

|C|⋃i=1

Ei .


NP-complete


L-8õ Max-SNP-hardness

First-order logic+ Syntax

&ñ_ 20.23

l ñ|9½+Ë(vocabulary) Σ = (Φ,Π, r)Ér <ÊÃº l ñ (function symbols) |9½+Ë Φ,'a> l ñ(relation symbols) |9½+Ë Π, Õªo¦ y <ÊÃº l ñü< 'a> l ñ\>q6£§ &ñÃº\¦ @/6£xrvH ‘arity’ <ÊÃº rÐ sÀÒ#Q.‘arity’ <ÊÃº rÉr y <ÊÃº x9 'a> l ñ Y> >h_ Ãº\¦ 6 x H\¦·p. <ÊÃº f ∈ Φ r(f ) = ks k-ary <ÊÃº l ñ¦ ¦, 'a> l ñR ∈ Π r(R) = ks %ir k-ary¦ ôÇ. 0-ary <ÊÃºH©Ãº(constant)¦ ôÇ. 'a>l ñH 0-ary |c Ãº \O. Õªo¦ 'a>l ñ|9½+Ë Π ×æ\ ìøÍ×¼r 1px ñ = í<Ê÷&#Q e. Õªo¦ ¦&ñ)a íß |9½+ËÃº |9½+Ë x , y , z , . . .s ÅÒ#Qt 9, s[þtÉr âÄº\ &ñ_)a|9½+Ë(universe)\"f °úכ¦ 2[ôÇ.


NP-complete


&ñ_ 20.24

½Ó(terms) l ñ|9½+Ë(vocabulary) Σ\¦ 6 x # 6£§õ °ú s ½Ó(term)[þt¦&ñ_ôÇ. Äº y V_ y ÃºH ¿º ½Ós. ëß f ∈ Φ k-ary <ÊÃºl ñs¦, t1, t2, . . . , tk yy ½Ós, f (t1, . . . , tk)H ½Ós. (k = 0ÐúÜ¼, ©Ãº c ∈ Φ %ir ½Ós ÷&H ¦כ ·ú Ãº e.)½Ós &ñ_÷&, Σ ©_ 7Hod(expression)¦ &ñ_½+É Ãº e. ëß R ∈ Πk-ary 'a> l ñs¦ t1, t2, . . . , tk yy ½Ós, R(t1, . . . , tk)\¦ l:r7Hod(atomic expression) s¦ ôÇ. 9 7Hod(first-order expression)Ér6£§õ °ú s &ñ_ôÇ:

1 l:r7HodÉr ¿º 97Hods.

2 ëß φü< ψ 97Hods ¬φ, (φ ∨ ψ), Õªo¦ (φ ∧ ψ) %ir97Hods )a.

3 ëß φ 9 7Hods¦ x e__ Ãºs (∀xφ)H97Hods.


NP-complete


6£§õ °ú Ér ³ðlZO¦ &ñ_ôÇ.

∃xφ ≡ ¬(∀x¬φ)

\V 20.25

Ãº:r ΣN = (ΦN,ΠN, rN),ΦN = 0, σ, +,×, ↑.0Ér ©Ãº, σH unary <ÊÃºÐ 6£§ Ãº\¦ @/6£x r; ½+Ë +;, YL×; tÃº5px ↑ΠN = =, <.∀x < (+(x , σ(σ(0)))), σ(↑ (x , σ(σ(0))))¢H

∀x(x + 2) < σ((x ↑ 2))


NP-complete


Max-SNP-hardness

&ñ_ 20.26

6£§õ °ú s ³ð&³÷&H ½ G\ @/ôÇ H Õüt#Q7Hod(predicates)_ |9½+Ë¦NP¦ ôÇ:

∃Sφ(G ,S).

#l"f, SH _ ½(structure)s¦ φ 9(first order)s.

&ñ_ 20.27

6£§s $íwn þj@/oëH]j Π\¦ Max-SNP\ 5ÅqôÇ¦ ôÇ.

Π = maxS|x : Ψ(x ,G ,S) = True|

#l"f, ΨH Ãº|¾Ó ∃ ¢H ∀\¦ °út ·ú§H Õüt#Q7Hod (predicates)s.


NP-complete


\V 20.28

MAX 3SAT ∈ Max-SNP:

MAX 3SAT = maxS |(x1, x2, x3) :

(x1, x2, x3) ∈ C0 → (x1 ∈ S ∨ x2 ∈ S ∨ x3 ∈ S) ∧· · · ∧ (x1, x2, x3) ∈ C3 → (x1 6∈ S ∨ x2 6∈ S ∨ x3 6∈ S)|

(x1, x2, x3)H (Ãº m) y ]X_ o'XO¦ ©fç H 7',SH Ãº °úכs ÃÐ Ãº[þt_ |9½+Ë, CiH i>h_ ÂÒ&ñ+þA o'XO¦°úH H ]X_ |9½+Ës.


NP-complete


\V 20.29

MAX CUT ∈ Max-SNP:

MAX CUT = maxS |(u, v) :

(u < v) ∧ E (u, v) ∧ S(u) 6= S(v)|

SH ]Xéß\ _K *'H ¿º n |9½+Ë¦ ³ð&³ H 0-1 7'Ðu S\ 5Åq S(u) = 1, ÕªXOt ·ú§Ü¼ S(u) = 0. ñ uv>rF E (u, v) = 1, ÕªXOt ·ú§Ü¼ E (u, v) = 0.


NP-complete


&ño 20.30

Max-SNP_ þj@/oëH]jH #Q" ©Ãº ε\ @/K(1− ε)-HKZO¦ °úH.

&ñ_ 20.31

6£§õ °ú Ér ½Órçß ·ú¦o1pu fü< g , ©Ãº α, β > 0s >rF½+ÉM:, þj&ho(þj@/, ¢H þjè)ëH]j Π1s Π2Ð L-8)a¦ôÇ: Π1_ e__ \V I1\ @/ #,

1 fH, OPT(I2) ≤ αOPT(I1)\¦ ëß7á¤ H Π2_ \V I2\¦Òqt$íôÇ.

2 gH, I2_ e__ K x2\ @/ #,|c1(x1)−OPT(I1)| ≤ β|c2(x2)−OPT(I2)| I1_ K x1¦

Òqt$íôÇ.


NP-complete


$í|9 20.32

L-8_ ½+Ë$íÉr r L-8s )a.

$í|9 20.33

Π1s Π2Ð L-8÷&¦, Π2_ ε > 0 HKZOs>rF Π1Ér αβε HKZOs 0px . (#l"f,Π1õ Π2H yy þjè ¢H þj@/o ëH]j |c Ãº e.)


NP-complete


&ñ_ 20.34

Max-SNP_ H ëH]j þj&ho ëH]j ΠÐ L-8|c M:, Π\¦Max-SNP-hard¦ ôÇ. s M:, Π %ir Max-SNP\ 5Åq Max-SNP-completes¦ ôÇ.

&ño 20.35

MAX 3SATÉr Max-SNP-completes.


NP-complete


çßÐ>r8õ HÔ¦0px$í

SAT >\P_ ëH]jH HÔ¦0px$í_ 7£x"î\ ×æ¹כôÇ %i½+É¦ ôÇ. 6£§õ °ú Ér 3SAT_ þj&ho ëH]j\¦ Òqty .

&ñ_ 20.36

MAX3SAT%KVUd I : n>h_ ÂÒÖ¦Ãº x = (x1, x2, . . ., xn)_ [j>ho'XO[þt_ disjunctionÜ¼Ð sÀÒ#Q ]X(clause) c[þt_ |9½+ËC_ cnf,

f =∧c∈C

c .

i¦\B: f_ ëß7á¤÷&H ]X[þt_ qÖ¦ MAX3SAT(I ) þj@/÷&H x_ o°úכ.


NP-complete


&ño 20.37

6£§¦ ëß7á¤ H ©Ãº ε > 0õ SATÜ¼ÐÂÒ' MAX3SATÜ¼Ð_½Ó8 τ >rFôÇ: e__ SAT_ ëH]j \V I\ @/ #,

I ∈ SAT⇒ MAX3SAT(τ(I )) = 1,

I /∈ SAT⇒ MAX3SAT(τ(I )) < 11+ε .

¤ ÃZ: ×æ\.

2£§&ño 20.38

MAX3SAT_ (1 + ε)-H ·ú¦o1puÉr (P 6= NP s)Ô¦0px .

sQôÇ õH 6£§õ °ú Ér çßÐ>r8(gap-preservingreduction)¦ 6 x # ªôÇ þj&hoëH]j_ HÔ¦0px$í¦ 7£x"î½+É Ãº e.


NP-complete


&ñ_ 20.39

çßÐ>r8: Πü< Π′¦ þj@/oëH]j¦ . 6£§¦

ëß7á¤ H ΠÐ ÂÒ' Π′Ü¼Ð_ ½Ó8 f\¦ p' (c , ρ),(c ′, ρ′)¦ °úH çßÐ>r8s¦ ôÇ: e__ I ∈ Π\@/ #,

OPT(I ) ≥ c ⇒ OPT(f (I )) ≥ c ′,

OPT(I ) < cρ ⇒ OPT(f (I )) < c ′

ρ′ .

#l"f cü< ρH |I |_ c ′ü< ρ′Ér |f (I )|_ <ÊÃºs 9 ρ, ρ′ ≥ 1s.


NP-complete


ëß 0Aü< °ú Ér çßÐ>r8 f\¦ °úH Πü< Π′s e¦, 6£§õ°ú Ér SATÐ ÂÒ' ΠÐ ½Ó8 τ 7£x"î÷&%3¦ :

I ∈ SAT⇒ OPT(τ(I )) ≥ c ,

I /∈ SAT⇒ OPT(τ(I )) < cρ .

ÕªQ,

I ∈ SAT⇒ OPT(f (τ(I ))) ≥ c ′,

I /∈ SAT⇒ OPT(f (τ(I ))) < c ′

ρ′ ,

7£¤, Π′_ ρ′-H KZOÉr Ô¦0px H ¦כ _pôÇ.


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