# Linear Time Approximation Schemes for the Gale-Berlekamp Game and Related Minimization Problems

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Linear Time Approximation Schemes for the

Gale-Berlekamp Game and Related Minimization Problems

Marek Karpinski (Bonn)Warren Schudy (Brown)

STOC 2009

Please see http://www.cs.brown.edu/~ws/papers/gb.pdf for the most current version of the paper.

• Minimize number of lit light bulbs• NP hard [Roth & Viswanathan ’08]• PTAS runtime nO(1/ε²) [Bazgan, Fernandez de la

Vega, & Karpinski ’03]• We give PTAS linear runtime O(n2)+2O(1/ε²)

Gale-Berlekamp Game (1960s)

n/2

Animating…

• “Approximate” 2-coloring• General case:

– O(√ log n) approx is best known– no PTAS unless P=NP

• [Everywhere-] dense case, i.e. every vertex has degree Ω(n)– Previous best PTAS: nO(1/ε²) [Arora, Karger, & Karpinski ’95]– We give PTAS with linear runtime O(n2)+2O(1/ε²)

– If three colors no PTAS unless P=NP

• Average degree Ω(n) is insufficient for PTAS unless P=NP

Dense MIN-UNCUT

Uncut (monochromatic) edge

Added completebipartite graph

Animating…

Generalization: Fragile dense MIN-k-CSP•n variables taking values from constant-sized domain

•GB-Game: switches•MIN UNCUT: vertices

•Soft constraints, which each depend on k variables

•GB Game: lightbulbs•MIN UNCUT: edges

•These constraints are fragile, i.e. changing value of a variable makes all satisfied constraints it participates in unsatisfied. (For all assignments.)•Dense, i.e. each variable appears in Ω(nk-1) constraints

Den

se M

IN U

NC

UT

GB

Gam

e

We give first PTAS for all fragile dense MIN-k-CSPs, which has linear runtime O(nk)+2O(1/ε²)

First conceptual contribution: unifying these PTASs (and others) using new “fragile” framework

Another fragile problem: Multiway cut

• General case has O(1) approx. but no PTAS• Dense case:

– Previous best PTAS: nO(1/ε²) [Arora, Karger, & Karpinski ’95]

– We give PTAS with runtime O(n2)+2O(1/ε²) (linear-time)

Vertices are variables

Edges are soft constraints

These constraints are fragile, i.e. changing value of a variable makes all satisfied constraints it participates in unsatisfied

Animating…

Summary of results

Reference key:– [AKK 95]=[Arora, Karger, & Karpinski ’95]– [BFK 03]=[Bazgan, Fernandez de la Vega, & Karpinski ’03]– [GG 06]=[Giotis & Guruswami ’06]

Previous This work

Fragile MIN-k-CSP - O(nk)+2O(1/ε²)

MIN-UNCUT, Multiway cut nO(1/ε²) [AKK 95] O(n2)+2O(1/ε²)

Gale-Berlekamp Game nO(1/ε²) [BFK 03] O(n2)+2O(1/ε²)

MIN-k-SAT, Nearest codeword nO(1/ε²) [BFK 03] O(nk)+2O(1/ε²)

Rigid MIN-2-CSP - n2·2O(1/ε²)

Correlation clustering w/constant number of clusters

nO(1/ε²) [GG 06] n2·2O(1/ε²)

Hierarchical Clust. w/const… - n2·2O(1/ε²)

Runtimes for 1+ε approximation on [everywhere-] dense instances:

Ess

enti

ally

op

tim

al

Additive error algorithms• Whenever OPT ≥ f(ε)·nk we have f(ε)·ε·nk =

O(ε·OPT), so existing algorithms achieving additive error f(ε)·ε·nk suffice for a PTAS. [Arora, Karger, & Karpinski ‘95, Fernandez de la Vega ‘96, Goldreich, Goldwasser & Ron ’98, Frieze & Kannan ’99, Alon, Fernandez de la Vega, Kannan, & Karpinski ’02, Mathieu & Schudy ’08]

• Typical runtime: O(nk)+2O(1/ε²)

• Rest of talk focuses on:– OPT small and– MIN-UNCUT

Previous algorithm (1/3)

• Let S be random sample of V of size O(1/ε²)·log n • For each coloring x0 of S

– partial coloring x2 ←if margin of v w.r.t. x0 is largethen color v greedily w.r.t. x0,else label v “ambiguous”

– Extend x2 to a complete coloring x3 greedily• Return the best coloring x3 found

Let x0 = x* restricted to S

– analysis version

Assumes OPT ≤ ε κ0 n2 where κ0 is a constant

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Runtime: 2|S|= 2O(1/ε²)·log n = nO(1/ε²)

Previous Algorithm (2/3)

• Define the margin of vertex v w.r.t. coloring x to be|(number of green neighbors of v in x) - (number of red neighbors of v in x)|.

• Key facts: (recall dense assumption)

1. Partial coloring x2 agrees with the optimal coloring x*

2. There are few ambiguous vertices

partial coloring x2 ←if margin of v w.r.t. x0 is largethen color v greedily w.r.t. x0 else label v “ambiguous”

Sample x0 of OPT

C

A B

D E

FOPT

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C

A B

D E

F

C

A B

D E

F

Blue 1 to 0 – margin is too small

Blue 2 to 1 – margin is too small

Blue 2 to 0

Blue 1 to 0 –

margin is too small

Blue 1 to 0 – margin is too small

Blue 2 to 0

x2

Previous algorithm (3/3)

x3 extends x2 greedily

C

A B

D E

F

C

A B

D E

F

Previous algorithm

• Let S be random sample of V of size O(1/ε²)·log n• For each coloring x0 of S

– partial coloring x2 ←if margin of v w.r.t. x1 is largethen color v greedily w.r.t. x1

else label v “ambiguous”– Extend x2 to a complete coloring x3 greedily

• Return the best coloring x3 found

Our

κ2

– x1 ← greedy w.r.t. x0

using an algorithm with additive error at most Err=κ3 ε n · (# ambiguous)

Runtime: nO(1/ε²) O(n2)+2O(1/ε4) O(n2)+2O(1/ε²)

IntermediateAssume OPT ≤ ε κ0 n2

Third conceptual contribution: use additive error algorithm to color ambiguous vertices.

κ1 n2

Second conceptual contribution: two greedy phases before assigning ambiguity allows constant sample size

Animating…

Sample x0 of OPT

C

A

B D

E

FOPT

More Algorithm (1/2)

C

A

B D

E

F

x1 is greedy w.r.t. (with respect to) x0

C

A

B D

E

F

Me too

C is Blue so I like being red

C is blue so I like being red

E is red so I’ll go blue

E is red so I like being blue

My reasoning exactly

More Algorithm (2/2)

C

A

B D

E

F

C

A

B D

E

F

Blue 2 to 1 – margin is too small

Blue 3 to 0

Red 2 to 1 – margin is too small

x1 x2 is greedy w.r.t. x1

Ambiguous – run additive

error algorithm to color

Blue 4 to 0

Red 2 to 1 – margin is too small

Red 2 to 0

Plan of analysis

• Main Lemma: (≈ Lemma 16)1. Coloring x2 agrees with the optimal coloring x*

2. The additive error Err=κ3 ε n · (# ambiguous) is at most ε OPT

Proof (1/3): Bounding OPT• Assume all degrees are at least δ n

• Vertex v is balanced if its margin w.r.t. x* is at most δ n / 3.

• Lemma 12: #(balanced vert.) ≤ 6 OPT / (δ n)

• Proof:– If v is balanced then v is incident in x* to at

least δ n / 3 uncut edges

– OPT = ½∑v #(uncut edges incident to v) ≥ ½∑v balanced #(uncut edges incident to v) ≥ ½ #(balanced vert.) (δn / 3)

C

A

B

D

E

F

G

Optimum assignment x*

Balanced: 1≈3

• Lemma 14: with probability at least 90% at most δ n / 24 vertices are colored different colors in x1 and x*

• Proof:

• Corollary: with probability at least 90% all vertices have margin w.r.t. x* within δ n / 12 of margin w.r.t. x1

Proof (2/3): relating x1 to OPT coloring

Case 1: balanced vertices

By Lemma 1 #(balanced) ≤ 6 OPT / (δ n) ≤ 6 (k1 n2) / (δ n) = δ n / 48.

Case 2: unbalanced vertices

Chernoff and Markov bounds imply that the number unbalanced vertices is at most δ n / 48.

Proof (3/3): Proof of main lemma• Proof that x2 agrees with the optimal coloring x*

– Assume v is colored by x2 – Then v has a big margin w.r.to x1

– Then by Corollary v is colored by x* in the same way as by x2

• Proof that the additive errorErr=κ3 ε n · (# ambiguous) is at most ε OPT

– Assume v is not colored by x2 (ambiguous)– Then v has a small margin w.r.to x1

– Then by Corollary v has small margin w.r.to x* (balanced)

– So (# ambiguous) ≤ (# balanced)– Bound (# ambiguous) by (# balanced) in Err, and use

Lemma 12 to get Err ≤ ε OPT.

• Previous best PTAS runtime nO(1/ε²) [Giotis & Guruswami ’06]• We give PTAS with runtime n2·2O(1/ε²) (linear time)• Cor. Clust. constraints not fragile for d>2, but it satisfies a

generalization we call rigidity

Correlation Clustering with ≤ d clusters

• Definition of rigid CSP: in any assignment, a vertex in a large cluster is either incident to many incorrect edges or would be incident to many if moved to any other cluster.

• Fragility implies rigidity• Key additional algorithmic technique (also used

in [GG 06]): after identifying some clear-cut variables fix them and recurse on the remaining variables

=

= ==

Correlation Clustering and Rigidity

= =

v

Directions• More applications of the fragility and

rigidity methods for other minimization problems. Might require generalizing the notion of rigidity to k-CSP problems.

• Improving runtimes for Correlation Clustering, replacing "·" with "+" in O(n2)·2O(1/ε²)

• Designing linear time (1 + ε)-approximation algorithms for the k-Clustering (MIN-SUM) problem.

Bonus slides

• MIN-3-UNCUT constraints are not fragile• Dense MIN-3-UNCUT is at least as hard as general MIN-

2-UNCUT so no PTAS unless P=NP

MIN-3-UNCUTUncut (monochromatic)

edge

10n2 vert.

GeneralMIN-2-UNCUT instance

Dense MIN-3-UNCUT instance

Reduction

10n2 vert.

10n2 vert.n vertices

Complete tripartite graph

n vertices