It has been shown that for a general-valued constraint language Γ the following statements are equivalent: (1) any instance of VCSP(Γ) can be solved to optimality using a constant level of the Sherali-Adams LP hierarchy; (2) any instance of VCSP(Γ) can be solved to optimality using the third level of the Sherali-Adams LP hierarchy; (3) the support of Γ satisfies the "bounded width condition", i.e., it contains weak near-unanimity operations of all arities. We show that if the support of Γ violates the bounded with condition then not only is VCSP(Γ) not solved by a constant level of the Sherali-Adams LP hierarchy but it is also not solved by Ω(n) levels of the Lasserre SDP hierarchy (also known as the sum-of-squares SDP hierarchy). For Γ corresponding to linear equations in an Abelian group, this result follows from existing work on inapproximability of Max-CSPs. By a breakthrough result of Lee, Raghavendra, and Steurer [STOC'15], our result implies that for any Γ whose support violates the bounded width condition no SDP relaxation of polynomial-size solves VCSP(Γ). We establish our result by proving that various reductions preserve exact solvability by the Lasserre SDP hierarchy (up to a constant factor in the level of the hierarchy). Our results hold for general-valued constraint languages, i.e., sets of functions on a fixed finite domain that take on rational or infinite values, and thus also hold in notable special cases of 0, ∞ -valued languages (CSPs), 0, 1 -valued languages (Min-CSPs Max-CSPs), and Q-valued languages (finite-valued CSPs).
In an important series of papers @cite_18 @cite_6 @cite_28 @cite_47 it was established that the BWC captures precisely the decision CSPs that are solved by a certain natural local propagation algorithm @cite_43 .
@cite_18 @cite_28 @cite_6 @cite_43 @cite_47
1988961015 2531021157 2034079307 2150339067
A k-ary weak near-unanimity operation (or k-WNU) on A is an operation that satisfies the equations w(x, . . . x) ≈ x and w(y, x, . . . , x) ≈ w(x, y, . . . , x) ≈ · · · ≈ w(x, x, . . . , x, y) . If an algebra A has a k-NU (or a k-WNU) term operation, we say that A satisfies NU(k) (or WNU(k), respectively). Likewise, a variety is said to satisfy NU(k) (or WNU(k), respectively), it it has a k-variable term satisfying these equations. It has been conjectured that a finite idempotent algebra A has finite relational width if and only if V(A) (the variety generated by A) has meet semi-distributive congruence lattices. The concept of “finite relational width” arises in the theory of complexity of algorithms, in the algebraic study of constraint-satisfaction problems. Actually, there are several different definitions of this concept and it is not known if they are equivalent. One version of the concept and the conjecture mentioned above are due to B. Larose and L. Zadori [10]. The important family of varieties with meet semi-distributive congruence lattices has various known characterizations. There is a characterization by a certain Maltsev condition; also, it is known that a locally finite variety has this property iff it omits congruence covers of types 1 and 2 (defined in the tame congruence theory of D. Hobby, R. McKenzie [6]). E. Kiss showed that a finite idempotent algebra of relational width k must have an m-WNU term operation for every m ≥ k. E. Kiss and M. Valeriote then observed that a finite algebra with a k-WNU term operation, k > 1, must omit congruence covers of type 1. These observations led M. Valeriote to make two conjectures: any locally finite variety omits congruence covers of type 1 iff it satisfies WNU(k) for some k > 1; any locally finite variety has meet semi-distributive congruence lattices if and only if for some k, it satisfies WNU(m) for all m ≥ k. In this paper, we prove both of these conjectures of M. Valeriote. The family of locally finite varieties omitting type 1 is the largest family of locally finite varieties defined by a nontrivial idempotent Maltsev condition. For this In our LICS 2004 paper we introduced an approach to the study of the local structure of finite algebras and relational structures that aims at applications in the Constraint Satisfaction Problem (CSP). This approach involves a graph associated with an algebra @math or a relational structure A, whose vertices are the elements of @math (or A), the edges represent subsets of @math such that the restriction of some term operation of @math is ‘good’ on the subset, that is, act as an operation of one of the 3 types: semilattice, majority, or affine. In this paper we significantly refine and advance this approach. In particular, we prove certain connectivity and rectangularity properties of relations over algebras related to components of the graph connected by semilattice and affine edges. We also prove a result similar to 2-decomposition of relations invariant under a majority operation, only here we do not impose any restrictions on the relation. These results allow us to give a new, somewhat more intuitive proof of the bounded width theorem: the CSP over algebra @math has bounded width if and only if @math does not contain affine edges. Actually, this result shows that bounded width implies width (2,3). We also consider algebras with edges from a restricted set of types. In particular, it can be proved that type restrictions are preserved under the standard algebraic constructions. Finally, we prove that algebras without semilattice edges have few subalgebras of powers, that is, the CSP over such algebras is also polynomial time. Let ( A ) be finite relational structure of finite type, and let CSP ( ( A) ) denote the following decision problem: if ( I ) is a given structure of the same type as ( A ) , is there a homomorphism from ( I ) to ( A )? To each relational structure ( A ) is associated naturally an algebra ( A ) whose structure determines the complexity of the associated decision problem. We investigate those finite algebras arising from CSP’s of so-called bounded width, i.e., for which local consistency algorithms effectively decide the problem. We show that if a CSP has bounded width then the variety generated by the associated algebra omits the Hobby-McKenzie types 1 and 2. This provides a method to prove that certain CSP’s do not have bounded width. We give several applications, answering a question of Nesetřil and Zhu [26], by showing that various graph homomorphism problems do not have bounded width. Feder and Vardi [17] have shown that every CSP is polynomial-time equivalent to the retraction problem for a poset we call the Feder − Vardi poset of the structure. We show that, in the case where the structure has a single relation, if the retraction problem for the Feder-Vardi poset has bounded width then the CSP for the structure also has bounded width. This is used to exhibit a finite order-primal algebra whose variety admits type 2 but omits type 1 (provided P ≠ NP). This paper starts with the project of finding a large subclass of NP which exhibits a dichotomy. The approach is to find this subclass via syntactic prescriptions. While the paper does not achieve this goal, it does isolate a class (of problems specified by) "monotone monadic SNP without inequality" which may exhibit this dichotomy. We justify the placing of all these restrictions by showing, essentially using Ladner's theorem, that classes obtained by using only two of the above three restrictions do not show this dichotomy. We then explore the structure of this class. We show that all problems in this class reduce to the seemingly simpler class CSP. We divide CSP into subclasses and try to unify the collection of all known polytime algorithms for CSP problems and extract properties that make CSP problems NP-hard. This is where the second part of the title, "a study through Datalog and group theory," comes in. We present conjectures about this class which would end in showing the dichotomy.