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 bijection π : 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(K∈ k), where Kk is a k-clique

Approximate schemes• Max-kCSP and Max-kAP are 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-kAP admits PTAS or quasi-PTAS when the instance is “dense” or “metric”

PTAS for dense/metric Max-kCSP

• Max-kCSP is 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-kAP is dense:– roughly speaking, the instance has Ω(nk)

constraints

• In [AFK02]

– (1-ε)-approximate dense MAS, Betweenness in nO(1/ε^2) time

– (1-ε)-approximate dense DkS, Max-GI, Max-kAP in 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-kCSP with 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-HypergraphIsomorphism when 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,w as 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 permutations1. View the two permutations as disjoint cycles2. Break long cycles (length > n1/2) into short ones (length ≤ n1/2)

3. 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 permutations1. View the two permutations as disjoint cycles2. Break long cycles (length > n1/2) into short ones (length ≤ n1/2)

3. 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 permutations1. View the two permutations as disjoint cycles2. Break long cycles (length > n1/2) into short ones (length ≤ n1/2)

3. In each cycle, choose Permutation 1/Permutation 2 independently

Analysis• Conclusion: In this way, we get a permutationwhose 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