Towards a Characterization of Constant-Factor Approximable Finite-Valued CSPs

  title={Towards a Characterization of Constant-Factor Approximable Finite-Valued CSPs},
  author={V{\'i}ctor Dalmau and Andrei A. Krokhin and Rajsekar Manokaran},

Optimal polynomial-time compression for Boolean Max CSP

It is shown that obtaining a running time of the form $O(2^{(1-\epsilon)n})$ for particular classes of Max CSPs is as hard as breaching this barrier for Max $d$-SAT for some $d$.

Algebraic approach to promise constraint satisfaction

A new class of problems that can be viewed as algebraic versions of the (Gap) Label Cover problem are introduced, and it is shown that every PCSP with a fixed constraint language is equivalent to a problem of this form.

Fractional homomorphism, Weisfeiler-Leman invariance, and the Sherali-Adams hierarchy for the Constraint Satisfaction Problem

A combinatorial characterization of the Sherali-Adams hierarchy applied to the homomorphism problem in terms of fractional isomorphism is given and a description of the families of CSPs that are closed under Weisfeiler-Leman invariance in Terms of their polymorphisms as well as decidability by the first level of theSherali- Adams hierarchy is obtained.

Toward a Dichotomy for Approximation of $H$-coloring

Given two (di)graphs G, H and a cost function $c:V(G)\times V(H) \to \mathbb{Q}_{\geq 0}\cup\{+\infty\}$, in the minimum cost homomorphism problem, MinHOM(H), goal is finding a homomorphism

91 : 2 Toward a Dichotomy for Approximation of H-Coloring 1

Given two (di)graphs G, H and a cost function c : V (G)×V (H)→ Q≥0∪{+∞}, in the minimum cost homomorphism problem, MinHOM(H), we are interested in finding a homomorphism f : V (G)→ V (H) (a.k.a

The Sherali-Adams Hierarchy for Promise CSPs through Tensors

We study the Sherali-Adams linear programming hierarchy in the context of promise constraint satisfaction problems (PCSPs). We characterise when a level of the hierarchy accepts an instance in terms

Examples, counterexamples, and structure in bounded width algebras

It is shown that minimal boundedwidth algebras can be arranged into a pseudovariety with one basic ternary operation, and a structure theorem is proved for minimal bounded width algeBRas which have no majority subalgebra, which form a Pseudovarieties with a commutative binary operation.

Hierarchies of Minion Tests for PCSPs through Tensors

It is shown that the hierarchies of minion tests obtained in this way are general enough to capture the (combinatorial) bounded width and also the Sherali-Adams LP, Sum-of-Squares SDP, and affine IP hierarchies.

Promise Constraint Satisfaction and Width

The main technical finding is that the template of every PCSP that is solvable in bounded width satisfies a certain structural condition implying that its algebraic closure-properties include weak near unanimity polymorphisms of all large arities.

Notes on CSPs and Polymorphisms

The theory of algebraic structures with few subpowers, the theory of absorbing subalgebras and its applications to studying CSP templates which can be solved by local consistency methods, and the dichotomy theorem for conservative C SP templates are covered.



The Complexity of General-Valued CSPs

It is proved that if a constraint language satisfies this algebraic necessary condition for tractability of a general-valued CSP with a fixed constraint language, then the VCSP is tractable.

The Power of Linear Programming for Valued CSPs

This work obtains tractability of several novel and previously widely-open classes of VCSPs, including problems over valued constraint languages that are: sub modular on arbitrary lattices, bisubmodular on arbitrary finite domains, and weakly (and hence strongly) tree-sub modular on arbitrarily trees.

Robust algorithms with polynomial loss for near-unanimity CSPs

Two randomized robust algorithms with polynomial loss for CSPs with a constraint language having a near-unanimity polymorphism are given: one works for any near- Unanimous polymorphism and the parameter k in the loss depends on the size of the domain and the arity of the relations in Γ, while the other works for a special ternary near- unanimity operation.

On LP-based approximability for strict CSPs

A generic linear program (LP) is presented for a large class of strict-CSPs and a systematic way to convert integrality gaps for this LP into UGC-based inapproximability results is given.

The approximability of MAX CSP with fixed-value constraints

Any MAX CSP problem with a finite set of allowed constraint types, which includes all fixed-value constraints (i.e., constraints of the form x = a), is either solvable exactly in polynomial time or else is APX-complete, even if the number of occurrences of variables in instances is bounded.

Combinatorial Optimization Algorithms via Polymorphisms

This work designs a randomized algorithm to minimize a function f that admits a fractional polymorphism which is measure preserving and has a transitive symmetry in the value-oracle model.

Asking the Metaquestions in Constraint Tractability

This article systematically studies—for various classes of polymorphisms—the computational complexity of deciding whether or not a given structure ℍ admits a polymorphism from the class, and proves the NP-completeness of deciding a condition conjectured to characterize the tractable problems CSP(ℍ).

The Power of Linear Programming for General-Valued CSPs

The main result is a precise algebraic characterization of valued constraint languages whose instances can be solved exactly by the basic linear programming relaxation (BLP).

Tight bounds on the approximability of almost-satisfiable Horn SAT and exact hitting set

The hardness results are proved by constructing integrality gap instances for a semidefinite programming relaxation for the problems, and using Raghavendra's result [Rag08] to conclude that no algorithm can do better than the SDP assuming the UGC.

A characterization of strong approximation resistance

This work presents a characterization of strongly approximation resistant predicates under the Unique Games Conjecture, and presents characterizations in the mixed linear and semi-definite programming hierarchy and the Sherali-Adams linear programming hierarchy.