Venkatesan Guruswami (CMU) Yury Makarychev (TTI-C) Prasad Raghavendra (Georgia Tech)

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

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

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

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

5. 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

6. 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 .

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

8. Approach

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

10. Our approach

11. 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.

12. 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

13. 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.

14. 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)

15. 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.

16. 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

17. 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.

18. 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

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

20. The End.Any questions?