Linear Time Approximation Schemes for the Gale-Berlekamp Game and Related Minimization Problems Marek Karpinski (Bonn) Warren Schudy (Brown) STOC 2009 Please see for the most current version of the paper.

Minimize number of lit light bulbs NP hard [Roth & Viswanathan ’08] PTAS runtime n O(1/ε²) [Bazgan, Fernandez de la Vega, & Karpinski ’03] We give PTAS linear runtime O(n 2 )+2 O(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: n O(1/ε²) [Arora, Karger, & Karpinski ’95] –We give PTAS with linear runtime O(n 2 )+2 O(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 complete bipartite 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(n k-1 ) constraints Dense MIN UNCUT GB Game We give first PTAS for all fragile dense MIN-k-CSPs, which has linear runtime O(n k )+2 O(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: n O(1/ε²) [Arora, Karger, & Karpinski ’95] –We give PTAS with runtime O(n 2 )+2 O(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] PreviousThis work Fragile MIN-k-CSP-O(n k )+2 O(1/ε²) MIN-UNCUT, Multiway cutn O(1/ε²) [AKK 95]O(n 2 )+2 O(1/ε²) Gale-Berlekamp Gamen O(1/ε²) [BFK 03]O(n 2 )+2 O(1/ε²) MIN-k-SAT, Nearest codewordn O(1/ε²) [BFK 03]O(n k )+2 O(1/ε²) Rigid MIN-2-CSP-n 2 ·2 O(1/ε²) Correlation clustering w/constant number of clusters n O(1/ε²) [GG 06]n 2 ·2 O(1/ε²) Hierarchical Clust. w/const…-n 2 ·2 O(1/ε²) Runtimes for 1+ε approximation on [everywhere-] dense instances: Essentially optimal

Additive error algorithms Whenever OPT ≥ f(ε)·n k we have f(ε)·ε·n k = O(ε·OPT), so existing algorithms achieving additive error f(ε)·ε·n k 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(n k )+2 O(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 x 0 of S –partial coloring x 2 ← if margin of v w.r.t. x 0 is large then color v greedily w.r.t. x 0, else label v “ambiguous” –Extend x 2 to a complete coloring x 3 greedily Return the best coloring x 3 found Let x 0 = x* restricted to S – analysis version Assumes OPT ≤ ε κ 0 n 2 where κ 0 is a constant Animating… Runtime: 2 |S| = 2 O(1/ε²)·log n = n O(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 x 2 agrees with the optimal coloring x* 2.There are few ambiguous vertices partial coloring x 2 ← if margin of v w.r.t. x 0 is large then color v greedily w.r.t. x 0 else label v “ambiguous” Sample x 0 of OPT C AB DE F OPT Animating… C AB DE F C AB DE 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 2 to 0

x2x2 Previous algorithm (3/3) x 3 extends x 2 greedily C AB DE F C AB DE F

Previous algorithm Let S be random sample of V of size O(1/ε²)·log n For each coloring x 0 of S –partial coloring x 2 ← if margin of v w.r.t. x 1 is large then color v greedily w.r.t. x 1 else label v “ambiguous” –Extend x 2 to a complete coloring x 3 greedily Return the best coloring x 3 found Our κ2κ2 –x 1 ← greedy w.r.t. x 0 using an algorithm with additive error at most Err=κ 3 ε n · (# ambiguous) Runtime: n O(1/ε²) O(n 2 )+2 O(1/ε 4 ) O(n 2 )+2 O(1/ε²) Intermediate Assume OPT ≤ ε κ 0 n 2 Third conceptual contribution: use additive error algorithm to color ambiguous vertices. κ 1 n 2 Second conceptual contribution: two greedy phases before assigning ambiguity allows constant sample size Animating…

Sample x 0 of OPT C A BD E F OPT More Algorithm (1/2) C A B D E F x 1 is greedy w.r.t. (with respect to) x 0 C A BD 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 BD E F C A BD E F Blue 2 to 1 – margin is too small Blue 3 to 0 Red 2 to 1 – margin is too small x1x1 x 2 is greedy w.r.t. x 1 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 x 2 agrees with the optimal coloring x* 2.The additive error Err=κ 3 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 x 1 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. x 1 Proof (2/3): relating x 1 to OPT coloring Case 1: balanced vertices By Lemma 1 #(balanced) ≤ 6 OPT / (δ n) ≤ 6 (k 1 n 2 ) / (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 x 2 agrees with the optimal coloring x* –Assume v is colored by x 2 –Then v has a big margin w.r.to x 1 –Then by Corollary v is colored by x* in the same way as by x 2 Proof that the additive error Err=κ 3 ε n · (# ambiguous) is at most ε OPT –Assume v is not colored by x 2 (ambiguous) –Then v has a small margin w.r.to x 1 –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 n O(1/ε²) [Giotis & Guruswami ’06] We give PTAS with runtime n 2 ·2 O(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(n 2 )·2 O(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-UNCUT Uncut (monochromatic) edge 10n 2 vert. General MIN-2-UNCUT instance Dense MIN-3-UNCUT instance Reduction 10n 2 vert. n vertices Complete tripartite graph n vertices