Finding Almost-Perfect Graph Bisections Venkatesan Guruswami (CMU) Yury Makarychev (TTI-C) Prasad Raghavendra (Georgia Tech) David Steurer (MSR) Yuan Zhou (CMU)

MaxCut and Goemans-Williamson alg. The GW SDP relaxation [GW95] 0.878-approximation vs approximation G = (V, E) Objective: A B subject to

Finding almost-perfect MaxCut vs approximation Bipartite graph recognition algorithm (robust version against noise) Optimal under Unique Games Conjecture [KKMO07, MOO10]

MaxBisection Approximating MaxBisection? No easier than MaxCut Reduction: take two copies of the MaxCut instance G = (V, E) Objective: A B

MaxBisection (cont'd) Approximating MaxBisection? No easier than MaxCut Strictly harder than MaxCut? Approximation ratio: 0.7028 [FJ97, Ye01, HZ02, FL06] Approximating almost perfect solutions? Not known G = (V, E) Objective: A B

Finding almost-perfect MaxBisection Question Is there a vs approximation algorithm for MaxBisection? Answer. Yes. Our result. Theorem. There is a vs approximation algorithm for MaxBisection. Theorem. Given a satisfiable MaxBisection instance, it is easy to find a (.49, .51)-balanced cut of value .

The rest of this talk... Theorem. There is a vs approximation algorithm for MaxBisection.

Approach

? Approach -- SDP The standard SDP (used by all the previous algorithms) Integrality gap , subject to OPT < 0.9 SDP = 1

Our approach

A simple fact Fact. -balanced cut of value bisection of value . Proof. Get the bisection by moving fraction of random vertices from the right side to the left side. Only need to find almost bisections.

Almost perfect MaxCuts on expanders λ-expander: for each , such that , we have , where Key Observation. The (volume of) difference between two cuts on a λ-expander is at most . Proof. C X A B Y D

Almost perfect MaxCuts on expanders λ-expander: for each , such that , we have , where Key Observation. The (volume of) difference between two cuts on a λ-expander is at most . Approximating almost perfect MaxBisection on expanders is easy. Just run the GW alg. to find the MaxCut.

The algorithm (sketch) Decompose the graph into expanders Discard all the inter-expander edges Approximate OPT's behavior on each expander by finding MaxCut (GW) Discard all the uncut edges Combine the cuts on the expanders Take one side from each cut to get an almost bisection. (subset sum)

Expander decomposition Cheeger's inequality. Can (efficiently) find a cut of sparsity if the graph is not a -expander. Corollary. A graph can be (efficiently) decomposed into -expanders by removing edges (in fraction). Proof. If the graph is not an expander, divide it into two parts by sparsest cut (cheeger's inequality). Process the two parts recursively.

The algorithm Decompose the graph into -expanders. Lose edges. Apply GW algorithm on each expander to approximate OPT. OPT(MaxBisection) = GW finds cuts on these expanders different from behavior of OPT Lose edges. Combine the cuts on the expanders (subset sum). -balanced cut of value a bisection of value

Eliminating the factor Another key step. Idea. Terminate early in the decomposition process. Decompose the graph into -expanders or subgraphs of vertices. Corollary. Only need to discard edges. Lemma. We can find an almost bisection if the MaxCuts for small sets are more biased than those in OPT.

Finding a biased MaxCut Lemma. Given G=(V,E), if there exists a cut (X, Y) of value , then one can find a cut (A, B) of value , such that . SDP. Rounding. A hybrid of hyperplane and threshold rounding. maximize subject to -triangle inequality

Future directions vs approximation? "Global conditions" for other CSPs. Balanced Unique Games?

The End. Any questions?