Yuichi Yoshida (NII & PFI) Yuan Zhou (CMU)

Approximation Schemes via Sherali-Adams Hierarchy for Dense Constraint Satisfaction Problems and Assignment Problems Yuichi Yoshida (NII & PFI) Yuan Zhou (CMU)

Constraint satisfaction problems (CSPs) • In Max-kCSP, given: • a set of variables: V = {v1, v2, v3, …, vn} • the domain of variables: D • a set of arity-k “local” constraints: C • Goal: find an assignment α : V D to maximize #satisfied constraints in C

Constraint satisfaction problems (CSPs) • In Max-kCSP, given: • a set of variables: V = {v1, v2, v3, …, vn} • the domain of variables: D • a set of arity-k “local” constraints: C • Goal: find an assignment α : V D to maximize #satisfied constraints in C • Example: MaxCut • D = {0, 1} • p(i,j) = 1[vi ≠ vj] , Max-3SAT, UniqueGames, …

Assignment problems (APs) • In Max-kAP, given • a set of variables V = {v1, v2, v3, …, vn} • a set of arity-k “local” constraints C • Goal: find a bijectionπ : V  {1, 2, …, n} (i.e. permutaion) to maximize #satisfied constraints in C

Assignment problems (APs) • Examples • MaxAcyclicSubgraph(MAS) • π(u) < π(v) • Betweenness • π(u) < π(v) < π(w) or π(w) < π(v) < π(u) • MaxGraphIsomorphism(Max-GI) • (π(u), π(v)) ∈ E(H), where H is a fixed graph • DensekSubgraph(DkS) • (π(u), π(v)) ∈ E(Kk), where Kk is a k-clique

Approximate schemes • Max-kCSPand Max-kAPare NP-Hard in general • Polynomial-time approximation scheme (PTAS): for any constant ε > 0, the algorithm runs in nO(1) time and gives (1-ε)-approximation • Quasi-PTAS: the algorithm runs in nO(log n)time • Max-kCSP/Max-kAPadmits PTAS or quasi-PTAS when the instance is “dense” or “metric”

PTAS for dense/metric Max-kCSP • Max-kCSPis dense: has Ω(nk) constraints. • PTAS for dense MaxCut[dlV96] • PTAS for dense Max-kCSP[AKK99, FK96, AdlVKK03] • Max-2CSP is metric: edge weight ω satisfies ω(u, v) ≤ ω(u, w)+ω(w, v) • PTAS for metric MaxCut[dlVK01] • PTAS for metric MaxBisection[FdlVKK04] • PTAS for locally dense Max-kCSP(a generalized definition of “metric”) [dlVKKV05]

Quasi-PTAS for dense Max-kAP • Max-kAPis dense: • roughly speaking, the instance has Ω(nk) constraints • In [AFK02] • (1-ε)-approximate dense MAS, Betweennessin nO(1/ε^2) time • (1-ε)-approximate dense DkS, Max-GI, Max-kAPin nO(log n/ε^2) time

Previous techniques • Exhaustive search on a small set of variables [AKK99] • Weak Szemerédi’s regularity lemma [FK96] • Copying important variables [dlVK01] • A variant of SVD [dlVKKV05] • Linear programming relaxation for “assignment problems with extra constraints” [AFK02] • In this paper, we show: The standard Sherali-Adams LP relaxation hierarchy is a unified approach to all these results!

Sherali-Adams LP relaxation hierarchy • A systematic way to write tighter and tighter LP relaxations: [SA90] • In an r-round SA LP relaxation, • For each set S = {v1, …, vr} of r variables, we have a distribution of assignments μS= μ{v1, …, vr} • For any two sets S and T, marginal distributions are consistent: μS(S∩T) = μT(S∩T) • Solving an r-round LP relaxation takes nO(r) time.

Our results • Sherali-Adams LP-based proof for known results • O(1/ε2)-round SA LP relaxation gives (1-ε)-approximation to dense or locally dense Max-kCSP, and Max-kCSPwith global cardinality constraints such as MaxBisection • O(log n/ε2)-round SA LP relaxation gives (1-ε)-approximation to dense or locally dense Max-kAP • New algorithms • Quasi-PTAS for Maxk-HypergraphIsomorphismwhen one graph is dense and the other one is locally dense

Our techniques • Solve the Sherali-Adams LP relaxation for sufficiently many rounds (Ω(1/ε2) or Ω((log n)/ε2)) • Randomized conditioning operation to bring down the pair-wise correlations • Independent rounding for Max-kCSP • Special rounding for Max-kAP

Conditioning operation • Randomly choose v from V, sample a ~μv • For each local distribution μ{v1, …, vr}, generate the new local distribution μ{v1, …, vr}|v=a • r-round SA solution  (r-1)-round SA solution • Essentially from [RT12]: • After t steps of conditioning, • on average, μ{v1, …, vk} is only -far from μ{v1} x … x μ{vk}

Independent rounding for Max-kCSP After Ω(1/ε2)steps of conditioning, on average, μ{v1, …, vk} is only ε-far from μ{v1} x … x μ{vk} Sample each v from μ{v}, and we have Therefore, This is a (1-O(ε))-(multiplicative) approximation because of the density

Rounding for Max-kAP • Independent sampling does not work: • objective value is good, but resulting assignment might not be permutation because of collisions • Our special rounding: • View {μ{v}(w)}v,was a doubly stochastic matrix, therefore a distribution of permutations • Distribution supported on one permutation  ✔ • Two permutations?  Merge them • Even more permutations?  Pick arbitrary two, merge them, and iterate Similar operation in [AFK02]

Merging two permutations • View the two permutations as disjoint cycles • Break long cycles (length > n1/2) into short ones (length ≤ n1/2) • In each cycle, choose Permutation 1/Permutation 2 independently Analysis • Step 2: modified O(n1/2) entries of Permutation 2, affecting O(n-1/2)- fraction of the constraints n1/2

Merging two permutations • View the two permutations as disjoint cycles • Break long cycles (length > n1/2) into short ones (length ≤ n1/2) • In each cycle, choose Permutation 1/Permutation 2 independently Analysis • Step 3: value of the constraints where each variable from a distinct cycle is preserved because of independence – all but n-1/2-fraction of them n1/2

Merging two permutations • View the two permutations as disjoint cycles • Break long cycles (length > n1/2) into short ones (length ≤ n1/2) • In each cycle, choose Permutation 1/Permutation 2 independently Analysis • Conclusion: In this way, we get a permutation whose objective value is at least (1 – O(n-1/2)) * [Indep. Sampling] ≥ (1 – O(n-1/2)) (1 – O(ε)) [Val of LP] n1/2

Future directions • Can we solve the Sherali-Adams LP faster (as in [GS12]) to get a PTAS for dense assignment problems?

Thanks

Yuichi Yoshida (NII & PFI) Yuan Zhou (CMU)