The complexity of the list homomorphism problem for graphs

We completely classify the computational complexity of the list H-colouring problem for graphs (with possible loops) in combinatorial and algebraic terms: for every graph H the problem is either NP-complete, NL-complete, L-complete or is first-order definable; descriptive complexity equivalents are given as well via Datalog and its fragments. Our algebraic characterisations match important conjectures in the study of constraint satisfaction problems.Comment: 12 pages, STACS 201

Functors on relational structures which admit both left and right adjoints

This paper describes several cases of adjunction in the homomorphism preorder of relational structures. We say that two functors $\Lambda$ and $\Gamma$ between thin categories of relational structures are adjoint if for all structures $\mathbf A$ and $\mathbf B$, we have that $\Lambda(\mathbf A)$ maps homomorphically to $\mathbf B$ if and only if $\mathbf A$ maps homomorphically to $\Gamma(\mathbf B)$. If this is the case $\Lambda$ is called the left adjoint to $\Gamma$ and $\Gamma$ the right adjoint to $\Lambda$. In 2015, Foniok and Tardif described some functors on the category of digraphs that allow both left and right adjoints. The main contribution of Foniok and Tardif is a construction of right adjoints to some of the functors identified as right adjoints by Pultr in 1970. We generalise results of Foniok and Tardif to arbitrary relational structures, and coincidently, we also provide more right adjoints on digraphs, and since these constructions are connected to finite duality, we also provide a new construction of duals to trees. Our results are inspired by an application in promise constraint satisfaction -- it has been shown that such functors can be used as efficient reductions between these problems

09441 Abstracts Collection -- The Constraint Satisfaction Problem: Complexity and Approximability

From 25th to 30th October 2009, the Dagstuhl Seminar 09441 ``The Constraint Satisfaction Problem: Complexity and Approximability\u27\u27 was held in Schloss Dagstuhl~--~Leibniz Center for Informatics. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

Maximum constraint satisfaction on diamonds

In this paper we study the complexity of the (weighted) maximum constr aint satisfaction problem (Max CSP) over an arbitrary finite domain. In this pro blem, one is given a collection of weighted constraints on overlapping sets of v ariables, and the goal is to find an assignment of values to the variables so as to maximize the total weight of satisfied constraints. Max Cut is a typical exa mple of a Max CSP problem. Max CSP is NP-hard in general; however, some restrict ions on the form of constraints may ensure tractability. Recent results indicate that there is a connection between tractability of such restricted problems and supermodularity of the allowed constraint types with respect to some lattice or dering of the domain. We prove several results confirming this. Diamonds are the smallest lattices in terms of the number of comparabilities, and so are as unor dered as a lattice can possibly be. In the present paper, we study Max CSP on di amond-ordered domains. We show that if all allowed constraints are supermodular with respect to such an ordering then the problem can be solved in polynomial (i n fact, in cubic) time. We also prove a partial converse: if the set of allowed constraints includes a certain small family of binary supermodular constraints on such a lattice, then the problem is tractable if and only if all of the allowed constraints are supermodular; otherwise, it is NP-hard

Submodularity and the Complexity of Constraint Satisfaction

IMA, Google Corp., Microsoft Corp., Yandex Corp