Coloring Circular Arc Graphs · Coloring Circular Arc Graphs Revisiting Tucker’s algorithm Alex...

Coloring Circular Arc GraphsRevisiting Tucker’s algorithm

Alex Angelopoulos

µ∏λ ∀

July 22, 2014

Outline

Introduction

Analyzing Tucker’s algorithm

Alex Angelopoulos (MPLA) Coloring Circular Arc Graphs-• Introduction 2/15

The problem

Input: a family F of circular arcs

χ(F ) = 3

Output: is there a proper coloring with ≤ k colors?what is the minimum k s.t. F has a proper coloring?

The problem

χ(F ) = 3

The problem

χ(F ) = 3

Some quantities

L(F ) =3

l(F ) = 4

= ω(F )

u Load of F : L

u circular-cover: l

u max. clique of F : ω(as usual)

u We also discretize anduse the -at most- 2|F |points defining thearcs.

Some quantities

L(F ) =3

l(F ) = 4

= ω(F )

u Load of F : L

u circular-cover: l

Some quantities

L(F ) =3

l(F ) = 4

= ω(F )

u Load of F : L

u circular-cover: l

Some quantities

L(F ) =3

l(F ) = 4

= ω(F )

u Load of F : L

u circular-cover: l

Some quantities

L(F ) =3

l(F ) = 4

= ω(F )

u Load of F : L

u circular-cover: l

Results

Theorem 1 (Tucker (1975)).

Let F be a family of circular arcs with load L = L(F ) andcircular-cover l = l(F ). If l(F ) ≥ 4, then

⌊32L⌋

colors suffice toproperly color F .

u This is actually a 2-approximation algorithm.

u Tucker, [4] conjecture that χ(F ) ≤ 32ω(F ).

u Karapetian (1980) proves the above.

u Garey et al. (1980) show NP-completeness for Circular Arc

u Many exact algos for subfamilies of graphs (≥ O(|F |1.5)).

Results

⌊32L⌋

Results

⌊32L⌋

Results

⌊32L⌋

Results

More recently:

Theorem 2 (Valencia-Pabon (2003)).

Consider F with load L(F ) and circular-cover l(F ) ≥ 5. Then⌈l−1l−2

colors suffice to color F , bound being tight.

That’s this presentation about!

It is based exactly on the algorithm proposed by Tucker, [4].

Results

More recently:

Results

More recently:

Outline

Introduction

Analyzing Tucker’s algorithm

Alex Angelopoulos (MPLA) Coloring Circular Arc Graphs-• Analyzing Tucker’s algorithm 7/15

Tucker’s algorithm

u Select p so that L(F ) arcscontain it

u Assign color #1 to the arcwhich extends at least onthe counterclockwise sideof p

u Move clockwise, assigncurrent color to the firstarc to begin after theprevious ends

u Unless it is not possible,so use next color

Notations needed

u starting point: t

u first arc colored with i:Ai

u arcs colored until kth

round: Fk

Notations needed

u starting point: t

round: Fk

Notations needed

u starting point: t

round: Fk

Notations needed

u starting point: t

round: Fk

Notations needed

u starting point: t

round: Fk

Properties

Base of induction

1. Ai intersects Ai−1or else no new color is needed

2. L(F \ Fi) ≤ L(F )− iOK, verify again later*

3. For 1 ≤ i ≤ l− 2, Fi

is colored with at mosti+ 1 colors by theAlgorithm...Proof?

Properties

Base of induction

3. For 1 ≤ i ≤ l− 2, Fi

Properties

Base of induction

3. For 1 ≤ i ≤ l− 2, Fi

is colored with at mosti+ 1 colors by theAlgorithm...

Proof?

Properties

Base of induction

3. For 1 ≤ i ≤ l− 2, Fi

Proving Property 3

1 ≤ i ≤ l− 2

A 3=Af3

|{A2, A3, ..., Ai, Afi }| = i+ 1

There is a circular cover of size ≤ l− 1

Contradiction!

Suppose Fi−1 is properlycolored with i colors. Let:

u Ai the first to getcolor i in round i− 1

u Afi the last of round

i− 1 to traverse tthese may be the same arc

u Now think aboutAi+1 and Af

i+1Same color except...

In this case:A2, A3, ..., Ai, A

cover the circle!

Proving Property 3

1 ≤ i ≤ l− 2

A 3=Af3

|{A2, A3, ..., Ai, Afi }| = i+ 1

Contradiction!

i− 1 to traverse t

these may be the same arc

cover the circle!

Proving Property 3

1 ≤ i ≤ l− 2

A 3=Af3

|{A2, A3, ..., Ai, Afi }| = i+ 1

Contradiction!

cover the circle!

Proving Property 3

1 ≤ i ≤ l− 2

A 3=Af3

|{A2, A3, ..., Ai, Afi }| = i+ 1

Contradiction!

Same color except...

cover the circle!

Proving Property 3

1 ≤ i ≤ l− 2

A 3=Af3

|{A2, A3, ..., Ai, Afi }| = i+ 1

Contradiction!

cover the circle!

Proving Property 3

1 ≤ i ≤ l− 2

A 3=Af3

|{A2, A3, ..., Ai, Afi }| = i+ 1

Contradiction!

cover the circle!

Proving Property 3

1 ≤ i ≤ l− 2

A 3=Af3

|{A2, A3, ..., Ai, Afi }| = i+ 1

Contradiction!

cover the circle!

Proving Property 3

1 ≤ i ≤ l− 2

A 3=Af3

|{A2, A3, ..., Ai, Afi }| = i+ 1

Contradiction!

cover the circle!

Proving Property 3

1 ≤ i ≤ l− 2

A 3=Af3

|{A2, A3, ..., Ai, Afi }| = i+ 1

Contradiction!

cover the circle!

Property 2. L(F \ Fl−2) ≤ L− (l − 2)

Property 3. Fl−2 are properly colored with l − 1 colors.

But how to use some induction here?

Maintain little arcs (p, p+ 1) to create a constant load of L(F )around the circle. Neither χ(F ) nor the Algorithm’s output ischanged! *Now check again Property 2!

Now: F ′ = F \ Fl−2 has lF ′ ≥ lF≥ 5, so use induction!...

u L rounds, every l− 2 rounds need at most l− 1 colors

u ⇒⌈l−1l−2

is the output of the algorithm.

Property 2. L(F \ Fl−2) ≤ L− (l − 2)

u ⇒⌈l−1l−2

Property 2. L(F \ Fl−2) ≤ L− (l − 2)

u ⇒⌈l−1l−2

Property 2. L(F \ Fl−2) ≤ L− (l − 2)

u ⇒⌈l−1l−2

Property 2. L(F \ Fl−2) ≤ L− (l − 2)

u ⇒⌈l−1l−2

Property 2. L(F \ Fl−2) ≤ L− (l − 2)

u ⇒⌈l−1l−2

Property 2. L(F \ Fl−2) ≤ L− (l − 2)

u ⇒⌈l−1l−2

Some final interaction

u What if every arc in F spans at most n/k “points” of the circle?

l ≥ k+ 1⇒ SOL =⌈

kk−1

u What if every arc spans more than a semi-circle?

The circular arc graph is complete!

u To show tightness, Valencia-Pabon uses a result of Stahl, [3] regardingr-tuple colorings...

Major open problem: better than 32 -approximation?

l ≥ k+ 1⇒ SOL =⌈

kk−1

l ≥ k+ 1⇒ SOL =⌈

kk−1

l ≥ k+ 1⇒ SOL =⌈

kk−1

l ≥ k+ 1⇒ SOL =⌈

kk−1

l ≥ k+ 1⇒ SOL =⌈

kk−1

That’s all folks!

Thank you!

Alex Angelopoulos (MPLA) Coloring Circular Arc Graphs-• The end 14/15

Bibliography I

[1] M. R. Garey, D. S. Johnson, G. L. Miller, and C. H.Papadimitriou. The complexity of coloring circular arcs andchords. SIAM J. Alg. Disc. Meth., 1(2):216–227, June 1980.

[2] I. Karapetian. Coloring of arc graphs. Akad. Nauk Armyan. SSRDokl., 70:306–311, 1980.

[3] S. Stahl. n-tuple colorings and associated graphs. Journal ofComb. Theory, Series B, 20(2):185 – 203, 1976.

[4] A. Tucker. Coloring a family of circular arcs. SIAM J. Appl.Math., 29:493–502, 1975.

[5] M. Valencia-Pabon. Revisiting tucker’s algorithm to color circulararc graphs. SIAM J. Comput., 32(4):1067–1072, 2003.

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