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

1. Linear Time Approximation Schemesfor theGale-Berlekamp Gameand RelatedMinimization 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.

2. Gale-Berlekamp Game (1960s) • 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) n/2 Animating…

3. Dense MIN-UNCUT • “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 Uncut (monochromatic) edge Added complete bipartite graph Animating…

4. 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 GB Game Dense MIN UNCUT First conceptual contribution: unifying these PTASs (and others) using new “fragile” framework • We give first PTAS for all fragile dense MIN-k-CSPs, which has linear runtime O(nk)+2O(1/ε²)

5. Another fragile problem: Multiway cut 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 • 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) Animating…

6. Summary of results Runtimes for 1+ε approximation on [everywhere-] dense instances: Reference key: • [AKK 95]=[Arora, Karger, & Karpinski ’95] • [BFK 03]=[Bazgan, Fernandez de la Vega, & Karpinski ’03] • [GG 06]=[Giotis & Guruswami ’06] Essentially optimal

7. 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

8. Previous algorithm (1/3) – analysis version Assumes OPT ≤ εκ0 n2 where κ0 is a constant • 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 • Runtime: 2|S|= 2O(1/ε²)·log n = nO(1/ε²) Animating…

9. Previous Algorithm (2/3) Blue 1 to 0 – margin is too small Blue 2 to 0 A • 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) • Partial coloring x2 agrees with the optimal coloring x* • There are few ambiguous vertices B A B A B Blue 1 to 0 – margin is too small Blue 1 to 0 – margin is too small D E D E D E C C C OPT F F F Blue 2 to 1 – margin is too small Sample x0 of OPT • partial coloring x2←if margin of v w.r.t. x0 is largethen color v greedily w.r.t. x0else label v “ambiguous” Blue 2 to 0 Animating…

10. Previous algorithm (3/3) A B A B D E D E C C F F x2 x3 extends x2 greedily

11. Previous algorithm Our Intermediate Assume OPT ≤ εκ0 n2 κ1 n2 κ2 • 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. x1else label v “ambiguous” • Extend x2 to a complete coloring x3 greedily • Return the best coloring x3 found Second conceptual contribution: two greedy phases before assigning ambiguity allows constant sample size • x1← greedy w.r.t. x0 Third conceptual contribution: use additive error algorithm to color ambiguous vertices. • using an algorithm with additive error at most Err=κ3 ε n · (# ambiguous) O(n2)+2O(1/ε4) O(n2)+2O(1/ε²) • Runtime: nO(1/ε²) Animating…

12. More Algorithm (1/2) C is blue so I like being red E is red so I like being blue My reasoning exactly Me too A A A C C B D B D C D E B E E OPT F F C is Blue so I like being red F Sample x0 of OPT x1 is greedy w.r.t. (with respect to) x0 E is red so I’ll go blue

13. More Algorithm (2/2) Blue 2 to 1 – margin is too small Ambiguous – run additive error algorithm to color Red 2 to 1 – margin is too small Blue 4 to 0 A A Red 2 to 1 – margin is too small C C B D B D E Blue 3 to 0 E Red 2 to 0 F F x1 x2 is greedy w.r.t. x1

14. Plan of analysis • Main Lemma: (≈ Lemma 16) • Coloring x2 agrees with the optimal coloring x* • The additive error Err=κ3 ε n · (# ambiguous) is at most ε OPT

15. Proof (1/3): Bounding OPT Optimum assignment x* • 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 leastδ 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) F D C B A E G Balanced: 1≈3

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

17. 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.

18. Correlation Clustering with ≤ d clusters • 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

19. Correlation Clustering and Rigidity • 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 = = = = = = v

20. 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.

21. Bonus slides

22. MIN-3-UNCUT Uncut (monochromatic) edge • 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 General MIN-2-UNCUT instance Dense MIN-3-UNCUT instance 10n2vert. Reduction 10n2 vert. n vertices n vertices 10n2vert. Complete tripartite graph