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

Bipartite graph recognition Depth-first search/breadth-first search With some noise? Given a bipartite graph with 1% noisy edges, can we remove a small fraction of edges (10% say) to get a bipartite graph, i.e. can we divide the vertices into two parts, so that 90% of the edges go accross the two parts?

MaxCut G=(V,E) cut(A, B) = edges(A, B) / |E| B = V - A where B = V - A exact one of i, j in A : edge (i, j) "on the cut" MaxCut: find A, B such that cut(A, B) is maximized Bipartite graph recognition: MaxCut = 1 ? Robust bipartite graph recognition: given MaxCut ≥ 0.99, to find cut(A, B) ≥ 0.9 B = V - A A cut(A, B) = 4/5 subject to

c vs. s approximation for MaxCut Given a graph with MaxCut value at least c, can we find a cut of value at least s ? Robust bipartite graph recognition: given MaxCut ≥ 0.99, to find cut(A, B) ≥ 0.9 0.99 vs 0.9 approximation "approximating almost perfect MaxCut"

Robust bipartite graph recognition Task: given MaxCut ≥ 0.99, find cut(A, B) ≥ 0.9 We can always find cut(A, B) ≥ 1/2. Assign each vertex -1, 1 randomly For any edge (i, j), E[(1 - xixj)/2] = 1/2 vi vj

Robust bipartite graph recognition (cont'd) Task: given MaxCut ≥ 0.99, find cut(A, B) ≥ 0.9 We can always find cut(A, B) ≥ 1/2. Better than 1/2? DFS/BFS/greedy? Linear Programming? No combinatorial algorithm known until very recent [KS11] Natural LPs have big Integrality Gaps [VK07, STT07, CMM09]

Robust bipartite graph recognition (cont'd) Task: given MaxCut ≥ 0.99, find cut(A, B) ≥ 0.9 We can always find cut(A, B) ≥ 1/2. Better than 1/2? The GW Semidefinite Programming relaxation [GW95] 0.878-approximation Given MaxCut , can find a cut vs approximation, tight under Unique Games Conjecture [Kho02, KKMO07, MOO10] subject to

Robust satisfiability algorithms Given an instance which can be satisfied by removing ε fraction of constraints, to make the instance satisfiable by removing g(ε) fraction of constraints g(ε) -> 0 as ε -> 0 Examples vs. algorithm for MaxCut [GW95] vs. algorithm for Max2SAT [Zwick98] vs. algorithm for MaxHorn3SAT [Zwick98]

MaxBisection G = (V, E) Objective: A B subject to

MaxBisection (cont'd) 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: 0.6514 [FJ97], 0.699 [Ye01], 0.7016 [HZ02], 0.7027 [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, where ? 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 .

Extension to MinBisection minimize edges(A, B)/|V|, s.t. B = V - A, |B| = |A| Our result Theorem. There is a vs approximation algorithm for MaxBisection. Theorem. Given a MinBisection instance of value , it is easy to find a (.49, .51)-balanced cut of value .

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

Previous algorithms for MaxBisection

The GW algorithm for (almost perfect) MaxCut [GW95] MaxCut objective SDP relaxation subject to MaxCut = 2/3 -1 0 1 subject to SDP ≥ MaxCut In this example: SDP = 3/4 > MaxCut

The "rounding" algorithm Lemma. We can (in poly time) get a cut of value when Algorithm. Choose a random hyperplane, the hyperplane divides the vertices into two parts. Analysis. subject to

The "rounding" algorithm (cont'd) Lemma. We can (in poly time) get a cut of value when Algorithm. Choose a random hyperplane, the hyperplane divides the vertices into two parts. Analysis. implies for most edges (i, j), their SDP contribution is large Claim. If , then Therefore, the random hyperplane cuts many edges (in expectation) subject to

The "rounding" algorithm (cont'd) Claim. If , then Proof. vi, vj seperated by the hyperplane vi vj vi, vj not seperated by the hyperplane

Known algorithms for MaxBisection The standard SDP (used by all the previous algorithms) Gives non-trivial approximation gaurantee But does not help find almost perfect MaxBisection , subject to Bisection condition

Known algorithms for MaxBisection (cont'd) The standard SDP (used by all the previous algorithms) The "integrality gap" , subject to OPT < 0.9 SDP = 1

Known algorithms for MaxBisection (cont'd) The standard SDP (used by all the previous algorithms) The "integrality gap" : instances that OPT < 0.9, SDP = 1 Why is this a bad news for SDP? Instances that OPT > 1 - ε, SDP > 1 - ε Instances that OPT < 0.9, SDP > 1 - ε SDP cannot tell whether an instance is almost satisfiable (OPT > 1 - ε) or not. , subject to

Our approach

Theorem. There is a vs approximation algorithm for MaxBisection.

A simple fact Fact. -balanced cut of value bisection of value . Proof. Get the bisection by moving fraction of random vertices from the large side to the small side. fraction of cut edges affected : at most in expectation Only need to find almost bisections.

Almost perfect MaxCuts on expanders λ-expander: for each , such that , we have , where G=(V,E) S

Almost perfect MaxCuts on expanders (cont'd) 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 (cont'd) 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) Step 2: find MaxCut Step 3: combine pieces Step 1: decompose into expanders G=(V,E)

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 small parts by sparsest cut (cheeger's inequality). Process the small parts recursively. G=(V,E) λ-expander

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

Theorem. There is a vs approximation algorithm for MaxBisection. Proved: Theorem. There is a vs approximation algorithm for MaxBisection. Will prove: short story

Eliminating the factor Recall. Only need to find almost bisections ( -close to a bisection) Observation. Subset sum is "flexible with small items" Making small items more biased does not change the solution too much. (101, 304) (397, 201) (8, 0) (3, 5) (8, 0) (6, 2) (6, 0) (5, 1) (5, 0) (3, 2) sum (515, 515)

Eliminating the factor Recall. Only need to find almost bisections ( -close to a bisection) Observation. Subset sum is "flexible with small items" Making small items more biased does not change the solution too much. (101, 304) (397, 201) (8, 0) (8, 0) (6, 0) (5, 0) sum (498, 505)

Eliminating the factor Recall. Only need to find almost bisections ( -close to a bisection) Observation. Subset sum is "flexible with small items" Making small items more biased does not change the solution too much. (101, 304) (397, 201) (8, 0) (8, 0) (6, 0) (5, 0) sum (506, 505)

Eliminating the factor Recall. Only need to find almost bisections ( -close to a bisection) Observation. Subset sum is "flexible with small items" Making small items more biased does not change the solution too much. (101, 304) (397, 201) (8, 0) (0, 8) (6, 0) (5, 0) sum (506, 513)

Eliminating the factor Recall. Only need to find almost bisections ( -close to a bisection) Observation. Subset sum is "flexible with small items" Making small items more biased does not change the solution too much. (101, 304) (397, 201) (8, 0) (0, 8) (6, 0) (5, 0) sum (512, 513)

Eliminating the factor Recall. Only need to find almost bisections ( -close to a bisection) Observation. Subset sum is "flexible with small items" Making small items more biased does not change the solution too much. (101, 304) (397, 201) (8, 0) (0, 8) (6, 0) (5, 0) sum (517, 513)

Eliminating the factor Recall. Only need to find almost bisections ( -close to a bisection) Observation. Subset sum is "flexible with small items" Making small items more biased does not change the solution too much. However, making small items more balanced might be a bad idea. (200, 0) (0, 2) (0, 2) 100 copies (0, 2) sum (200, 200)

Eliminating the factor Recall. Only need to find almost bisections ( -close to a bisection) Observation. Subset sum is "flexible with small items" Making small items more biased does not change the solution too much. However, making small items more balanced might be a bad idea. (200, 0) (1, 1) (1, 1) 100 copies (1, 1) sum (300, 100)

Eliminating the factor (cont'd) Idea. Terminate early in the decomposition process. Decompose the graph into -expanders (large items), or subgraphs of vertices (small items). Corollary. Only need to discard edges. Lemma. We can find an almost bisection if the MaxCuts we get for small sets are more biased than those in OPT.

Finding a biased MaxCut To find a cut that is as biased as OPT and as good as OPT (in terms of cut value). 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 . MaxBisection Biased MaxCut

The algorithm Decompose the graph into -expanders or small parts. Lose edges. Apply GW algorithm on each expander to approximate OPT. Lose edges, different from OPT Find biased MaxCuts in small parts. Lose edges, at most less biased than OPT Combine the cuts on the expanders and small parts (subset sum). -balanced cut of value a bisection of value

Finding a biased MaxCut -- A simpler task 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. Claim. SDP ≥ |X|/|V| --- Bias maximize subject to --- Cut value

Rounding algorithm (sketch) Goal: given SDP solution, to find a cut (A, B) such that For most ( fraction) edges (i, j), we have vi, vj are almost opposite to each other: vi ≈ - vj, Indeed,

Rounding algorithm (sketch) (cont'd) for most edges (i, j): Project all vectors to v0 Divide v0 axis into intervals length = Most ( fraction ) edges' incident vertices fall into opposite intervals (good edges) Discard all bad edges v0 I(-4) I(-3) I(-2) I(-1) I(1) I(2) I(3) I(4)

Rounding algorithm (sketch) (cont'd) Let the cut (A, B) be for each pair of intervals I(k) and I(-k), let A include the one with more vertices, B include the other (A, B) cuts all good edges v0 -4 -3 -2 -1 1 2 3 4

Rounding algorithm (sketch) (cont'd) Let the cut (A, B) be for each pair of intervals I(k) and I(-k), let A include the one with more vertices, B include the other For each i in I(k) For each i in I(-k)

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. --- Bias maximize subject to --- Cut value -triangle inequality

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

The End. Any questions?

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. MaxBisection Biased MaxCut

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?