Approximate Counting CSP Seen from the Other Side

In this paper we study the complexity of counting Constraint Satisfaction Problems (CSPs) of the form #CSP(C,-), in which the goal is, given a relational structure A from a class C of structures and an arbitrary structure B, to find the number of homomorphisms from A to B. Flum and Grohe showed that #CSP(C,-) is solvable in polynomial time if C has bounded treewidth [FOCS\u2702]. Building on the work of Grohe [JACM\u2707] on decision CSPs, Dalmau and Jonsson then showed that, if C is a recursively enumerable class of relational structures of bounded arity, then assuming FPT != #W[1], there are no other cases of #CSP(C,-) solvable exactly in polynomial time (or even fixed-parameter time) [TCS\u2704]. We show that, assuming FPT != W[1] (under randomised parametrised reductions) and for C satisfying certain general conditions, #CSP(C,-) is not solvable even approximately for C of unbounded treewidth; that is, there is no fixed parameter tractable (and thus also not fully polynomial) randomised approximation scheme for #CSP(C,-). In particular, our condition generalises the case when C is closed under taking minors

Hybrid VCSPs with crisp and conservative valued templates

A constraint satisfaction problem (CSP) is a problem of computing a homomorphism ${\bf R} \rightarrow {\bf \Gamma}$ between two relational structures. Analyzing its complexity has been a very fruitful research direction, especially for fixed template CSPs, denoted $CSP({\bf \Gamma})$, in which the right side structure ${\bf \Gamma}$ is fixed and the left side structure ${\bf R}$ is unconstrained. Recently, the hybrid setting, written $CSP_{\mathcal{H}}({\bf \Gamma})$, where both sides are restricted simultaneously, attracted some attention. It assumes that ${\bf R}$ is taken from a class of relational structures $\mathcal{H}$ that additionally is closed under inverse homomorphisms. The last property allows to exploit algebraic tools that have been developed for fixed template CSPs. The key concept that connects hybrid CSPs with fixed-template CSPs is the so called "lifted language". Namely, this is a constraint language ${\bf \Gamma}_{{\bf R}}$ that can be constructed from an input ${\bf R}$. The tractability of that language for any input ${\bf R}\in\mathcal{H}$ is a necessary condition for the tractability of the hybrid problem. In the first part we investigate templates ${\bf \Gamma}$ for which the latter condition is not only necessary, but also is sufficient. We call such templates ${\bf \Gamma}$ widely tractable. For this purpose, we construct from ${\bf \Gamma}$ a new finite relational structure ${\bf \Gamma}'$ and define $\mathcal{H}_0$ as a class of structures homomorphic to ${\bf \Gamma}'$. We prove that wide tractability is equivalent to the tractability of $CSP_{\mathcal{H}_0}({\bf \Gamma})$. Our proof is based on the key observation that ${\bf R}$ is homomorphic to ${\bf \Gamma}'$ if and only if the core of ${\bf \Gamma}_{{\bf R}}$ is preserved by a Siggers polymorphism. Analogous result is shown for valued conservative CSPs.Comment: 21 pages. arXiv admin note: text overlap with arXiv:1504.0706

Connectedness in Cayley Graphs and P/NP Dichotomy for Quay Algebras

This senior thesis attempts to determine the extent to which the P/NP dichotomy of finite algebras (as proven by Bulatov, et.al in 2017) can be cast in terms of connectedness in Cayley graphs. This research is motivated by Prof. Robert McGrail\u27s work ``CSPs and Connectedness: P/NP-Complete Dichotomy for Idempotent, Right Quasigroups published in 2014 in which he demonstrates the strong correspondence between tractability and total path-connectivity in Cayley graphs for right, idempotent quasigroups. In particular, we will introduce the notion of total V-connectedness and show how it could be potentially used to phrase the dichotomy in terms of connectivity for another class of algebras, namely for Quay algebras

The Complexity of Combinations of Qualitative Constraint Satisfaction Problems

The CSP of a first-order theory $T$ is the problem of deciding for a given finite set $S$ of atomic formulas whether $T \cup S$ is satisfiable. Let $T_1$ and $T_2$ be two theories with countably infinite models and disjoint signatures. Nelson and Oppen presented conditions that imply decidability (or polynomial-time decidability) of $\mathrm{CSP}(T_1 \cup T_2)$ under the assumption that $\mathrm{CSP}(T_1)$ and $\mathrm{CSP}(T_2)$ are decidable (or polynomial-time decidable). We show that for a large class of $\omega$-categorical theories $T_1, T_2$ the Nelson-Oppen conditions are not only sufficient, but also necessary for polynomial-time tractability of $\mathrm{CSP}(T_1 \cup T_2)$ (unless P=NP)

Global Cardinality Constraints Make Approximating Some Max-2-CSPs Harder

Assuming the Unique Games Conjecture, we show that existing approximation algorithms for some Boolean Max-2-CSPs with cardinality constraints are optimal. In particular, we prove that Max-Cut with cardinality constraints is UG-hard to approximate within ~~0.858, and that Max-2-Sat with cardinality constraints is UG-hard to approximate within ~~0.929. In both cases, the previous best hardness results were the same as the hardness of the corresponding unconstrained Max-2-CSP (~~0.878 for Max-Cut, and ~~0.940 for Max-2-Sat). The hardness for Max-2-Sat applies to monotone Max-2-Sat instances, meaning that we also obtain tight inapproximability for the Max-k-Vertex-Cover problem

Relational Width of First-Order Expansions of Homogeneous Graphs with Bounded Strict Width

Solving the algebraic dichotomy conjecture for constraint satisfaction problems over structures first-order definable in countably infinite finitely bounded homogeneous structures requires understanding the applicability of local-consistency methods in this setting. We study the amount of consistency (measured by relational width) needed to solve CSP(?) for first-order expansions ? of countably infinite homogeneous graphs ? := (A; E), which happen all to be finitely bounded. We study our problem for structures ? that additionally have bounded strict width, i.e., for which establishing local consistency of an instance of CSP(?) not only decides if there is a solution but also ensures that every solution may be obtained from a locally consistent instance by greedily assigning values to variables, without backtracking. Our main result is that the structures ? under consideration have relational width exactly (2, ?_?) where ?_? is the maximal size of a forbidden subgraph of ?, but not smaller than 3. It beats the upper bound: (2 m, 3 m) where m = max(arity(?)+1, ?, 3) and arity(?) is the largest arity of a relation in ?, which follows from a sufficient condition implying bounded relational width given in [Manuel Bodirsky and Antoine Mottet, 2018]. Since ?_? may be arbitrarily large, our result contrasts the collapse of the relational bounded width hierarchy for finite structures ?, whose relational width, if finite, is always at most (2,3)