We give a precise algebraic characterization of the power of Sherali--Adams relaxations for solvability of valued constraint satisfaction problems (CSPs) to optimality. The condition is that of bounded width, which has already been shown to capture the power of local consistency methods for decision CSPs and the power of semidefinite programming for robust approximation of CSPs. Our characterization has several algorithmic and complexity consequences. On the algorithmic side, we show that several novel and well-known valued constraint languages are tractable via the third level of the Sherali--Adams relaxation. For the known languages, this is a significantly simpler algorithm than those previously obtained. On the complexity side, we obtain a dichotomy theorem for valued constraint languages that can express an injective unary function. This implies a simple proof of the dichotomy theorem for conservative valued constraint languages established by Kolmogorov and Živný [J. ACM, 60 (2013), 10], and also a ...
It could be that either SA @math and SA @math , or SA @math and SA @math have the same power. The former happens in case of relational width. Dalmau proved that if a crisp language has relational width @math then it has relational width @math @cite_52 . Together with Theorem and the analogue of Proposition for relational width established in @cite_55 , this gives a trichotomy for relational width.
@cite_55 @cite_52
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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. In 2013 Belmonte and Vatshelle used mim-width, a graph parameter bounded on interval graphs and permutation graphs that strictly generalizes clique-width, to explain existing algorithms for many domination-type problems, known as LC-VSVP problems. We focus on chordal graphs and co-comparability graphs, that strictly contain interval graphs and permutation graphs respectively. First, we show that mim-width is unbounded on these classes, thereby settling an open problem from 2012. Then, we introduce two graphs (K_t K_t ) and (K_t S_t ) to restrict these graph classes, obtained from the disjoint union of two cliques of size t, and one clique of size t and one independent set of size t respectively, by adding a perfect matching. We prove that ((K_t S_t) )-free chordal graphs have mim-width at most (t-1 ), and ((K_t K_t) )-free co-comparability graphs have mim-width at most (t-1 ). From this, we obtain several algorithmic consequences, for instance, while Dominating Set is NP-complete on chordal graphs, it can be solved in time ( O (n^ t ) ) on chordal graphs where t is the maximum among induced subgraphs (K_t S_t ) in the given graph. We also show that classes restricted in this way have unbounded rank-width which validates our approach. Relational phrases (e.g., “got married to”) and their hypernyms (e.g., “is a relative of”) are central for many tasks including question answering, open information extraction, paraphrasing, and entailment detection. This has motivated the development of several linguistic resources (e.g. DIRT, PATTY, and WiseNet) which systematically collect and organize relational phrases. These resources have demonstrable practical benefits, but are each limited due to noise, sparsity, or size. We present a new general-purpose method, RELLY, for constructing a large hypernymy graph of relational phrases with high-quality subsumptions using collective probabilistic programming techniques. Our graph induction approach integrates small highprecision knowledge bases together with large automatically curated resources, and reasons collectively to combine these resources into a consistent graph. Using RELLY, we construct a high-coverage, high-precision hypernymy graph consisting of 20K relational phrases and 35K hypernymy links. Our evaluation indicates a hypernymy link precision of 78 , and demonstrates the value of this resource for a document-relevance ranking task. 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).