Complexity Of Constraint Satisfaction Problems Research Articles

Fine-Grained Time Complexity of Constraint Satisfaction Problems

We study the constraint satisfaction problem (CSP) parameterized by a constraint language Γ (CSPΓ) and how the choice of Γ affects its worst-case time complexity. Under the exponential-time hypothesis (ETH), we rule out the existence of subexponential algorithms for finite-domain NP-complete CSPΓ problems. This extends to certain infinite-domain CSPs and structurally restricted problems. For CSPs with finite domain D and where all unary relations are available, we identify a relation S D such that the time complexity of the NP-complete problem CSP({ S D }) is a lower bound for all NP-complete CSPs of this kind. We also prove that the time complexity of CSP({ S D }) strictly decreases when |D| increases (unless the ETH is false) and provide stronger complexity results in the special case when |D|=3.

Two-element structures modulo primitive positive constructability

Primitive positive constructions have been introduced in recent work of Barto, Opršal, and Pinsker to study the computational complexity of constraint satisfaction problems. Let {mathfrak {P}}_{mathrm {fin}} be the poset which arises from ordering all finite relational structures by pp-constructability. This poset is infinite, but we do not know whether it is uncountable. In this article, we give a complete description of the restriction {mathfrak {P}}_{mathrm {Boole}} of {mathfrak {P}}_{mathrm {fin}} to relational structures on a two-element set. We use {mathfrak {P}}_{mathrm {Boole}} to present the various complexity regimes of Boolean constraint satisfaction problems that were described by Allender, Bauland, Immerman, Schnoor and Vollmer.

A Characterization of Hard-to-Cover CSPs

We continue the study of covering complexity of constraint satisfaction problems (CSPs) initiated by Guruswami, Hastad and Sudan [SIAM J. Computing, 31(6):1663--1686, 2002] and Dinur and Kol [In Proc. $28$th IEEE Conference on Computational Complexity, 2013]. The covering number of a CSP instance $\Phi$, denoted by $\nu(\Phi)$ is the smallest number of assignments to the variables of $\Phi$, such that each constraint of $\Phi$ is satisfied by at least one of the assignments. We show the following results regarding how well efficient algorithms can approximate the covering number of a given CSP instance. - Assuming a covering unique games conjecture, introduced by Dinur and Kol, we show that for every non-odd predicate $P$ over any constant sized alphabet and every integer $K$, it is NP-hard to distinguish between $P$-CSP instances (i.e., CSP instances where all the constraints are of type $P$) which are coverable by a constant number of assignments and those whose covering number is at least $K$. Previously, Dinur and Kol, using the same covering unique games conjecture, had shown a similar hardness result for every non-odd predicate over the Boolean alphabet that supports a pairwise independent distribution. Our generalization yields a complete characterization of CSPs over constant sized alphabet $\Sigma$ that are hard to cover since CSP's over odd predicates are trivially coverable with $|\Sigma|$ assignments. - For a large class of predicates that are contained in the $2k$-LIN predicate, we show that it is quasi-NP-hard to distinguish between instances which have covering number at most two and covering number at least $\Omega(\log\log n)$. This generalizes the $4$-LIN result of Dinur and Kol that states it is quasi-NP-hard to distinguish between $4$-LIN-CSP instances which have covering number at most two and covering number at least $\Omega(\log \log\log n)$.

Biased landscapes for random constraint satisfaction problems

The typical complexity of constraint satisfaction problems (CSPs) can be investigated by means of random ensembles of instances. The latter exhibit many threshold phenomena besides their satisfiability phase transition, in particular a clustering or dynamic phase transition (related to the tree reconstruction problem) at which their typical solutions shatter into disconnected components. In this paper we study the evolution of this phenomenon under a bias that breaks the uniformity among solutions of one CSP instance, concentrating on the bicoloring of k-uniform random hypergraphs. We show that for small k the clustering transition can be delayed in this way to higher density of constraints, and that this strategy has a positive impact on the performances of simulated annealing algorithms. We characterize the modest gain that can be expected in the large k limit from the simple implementation of the biasing idea studied here. This paper contains also a contribution of a more methodological nature, made of a review and extension of the methods to determine numerically the discontinuous dynamic transition threshold.

Gap theorems for robust satisfiability: Boolean CSPs and beyond

A computational problem exhibits a “gap property” when there is no tractable boundary between two disjoint sets of instances. We establish a Gap Trichotomy Theorem for a family of constraint problem variants, completely classifying the complexity of possible NP-hard gaps in the case of Boolean domains. As a consequence, we obtain a number of dichotomies for the complexity of specific variants of the constraint satisfaction problem: all are either polynomial-time tractable or NP-complete. Schaefer's original dichotomy for SAT variants is a notable particular case.Universal algebraic methods have been central to recent efforts in classifying the complexity of constraint satisfaction problems. A second contribution of the article is to develop aspects of the algebraic approach in the context of a number of variants of the constraint satisfaction problem. In particular, this allows us to lift our results on Boolean domains to many templates on non-Boolean domains.

Complexity and polymorphisms for digraph constraint problems under some basic constructions

The role of polymorphisms in determining the complexity of constraint satisfaction problems is well established. In this context, we study the stability of CSP complexity and polymorphism properties under some basic graph theoretic constructions. As applications we observe a collapse in the applicability of algorithms for CSPs over directed graphs with both a total source and a total sink: the corresponding CSP is solvable by the “few subpowers algorithm” if and only if it is solvable by a local consistency check algorithm. Moreover, we find that the property of “strict width” and solvability by few subpowers are unstable under first-order reductions. The analysis also yields a complete characterization of the main polymorphism properties for digraphs whose symmetric closure is a complete graph.

Distance constraint satisfaction problems

We study the complexity of constraint satisfaction problems for templates Γ over the integers where the relations are first-order definable from the successor function. In the case that Γ is locally finite (i.e., the Gaifman graph of Γ has finite degree), we show that Γ is homomorphically equivalent to a structure with one of two classes of polymorphisms (which we call modular max and modular min) and the CSP for Γ can be solved in polynomial time, or Γ is homomorphically equivalent to a finite transitive structure, or the CSP for Γ is NP-complete. Assuming a widely believed conjecture from finite domain constraint satisfaction (we require the tractability conjecture by Bulatov, Jeavons and Krokhin in the special case of transitive finite templates), this proves that those CSPs have a complexity dichotomy, that is, are either in P or NP-complete.

Necessary Conditions for Tractability of Valued CSPs

The connection between constraint languages and clone theory has been a fruitful line of research on the complexity of constraint satisfaction problems. In a recent result, Cohen et al. [SIAM J. Comput., 42 (2013), pp. 915--1939] have characterized a Galois connection between valued constraint languages and so-called weighted clones. In this paper, we study the structure of weighted clones. We extend the results of Creed and Živný from [Proceedings of the 17th International Conference on Principles and Practice of Constraint Programming, 2011, pp. 210--224] on types of weightings necessarily contained in every nontrivial weighted clone. This result has immediate computational complexity consequences as it provides necessary conditions for tractability of weighted clones and thus valued constraint languages. We demonstrate that some of the necessary conditions are also sufficient for tractability, while others are provably not.

Characterising the complexity of constraint satisfaction problems defined by 2-constraint forbidden patterns

Although the CSP (constraint satisfaction problem) is NP-complete, even in the case when all constraints are binary, certain classes of instances are tractable. We study classes of binary CSP instances defined by excluding subproblems. This approach has recently led to the discovery of novel tractable classes. The complete characterisation of all tractable classes defined by forbidding patterns (where a pattern is simply a compact representation of a set of subproblems) is a challenging problem. We demonstrate a dichotomy in the case of forbidden patterns consisting of either one or two constraints. This has allowed us to discover several new tractable classes including, for example, a novel generalisation of 2SAT. We then extend this dichotomy to existential patterns which are only forbidden on specific domain values.

The expressibility of functions on the boolean domain, with applications to counting CSPs

An important tool in the study of the complexity of Constraint Satisfaction Problems (CSPs) is the notion of a relational clone, which is the set of all relations expressible using primitive positive formulas over a particular set of base relations. Post's lattice gives a complete classification of all Boolean relational clones, and this has been used to classify the computational difficulty of CSPs. Motivated by a desire to understand the computational complexity of (weighted) counting CSPs, we develop an analogous notion of functional clones and study the landscape of these clones. One of these clones is the collection of log-supermodular (lsm) functions, which turns out to play a significant role in classifying counting CSPs. In the conservative case (where all nonnegative unary functions are available), we show that there are no functional clones lying strictly between the clone of lsm functions and the total clone (containing all functions). Thus, any counting CSP that contains a single nontrivial non-lsm function is computationally as hard to approximate as any problem in #P. Furthermore, we show that any nontrivial functional clone (in a sense that will be made precise) contains the binary function “implies”. As a consequence, in the conservative case, all nontrivial counting CSPs are as hard to approximate as #BIS, the problem of counting independent sets in a bipartite graph. Given the complexity-theoretic results, it is natural to ask whether the “implies” clone is equivalent to the clone of lsm functions. We use the Möbius transform and the Fourier transform to show that these clones coincide precisely up to arity 3. It is an intriguing open question whether the lsm clone is finitely generated. Finally, we investigate functional clones in which only restricted classes of unary functions are available.

Constraint satisfaction problems: Convexity makes AllDifferent constraints tractable

We examine the complexity of constraint satisfaction problems that consist of a set of AllDiff constraints. Such CSPs naturally model a wide range of real-world and combinatorial problems, like scheduling, frequency allocations, and graph coloring problems. As this problem is known to be NP-complete, we investigate under which further assumptions it becomes tractable. We observe that a crucial property seems to be the convexity of the variable domains and constraints. Our main contribution is an extensive study of the complexity of Multiple AllDiff CSPs for a set of natural parameters, like maximum domain size and maximum size of the constraint scopes. We show that, depending on the parameter, convexity can make the problem tractable even though it is provably intractable in general. Interestingly, the convexity of constraints is the key property in achieving fixed parameter tractability, while the convexity of domains does not usually make the problem easier.

Computing hypergraph width measures exactly

Hypergraph width measures are important in studying the complexity of constraint satisfaction problems (CSPs). We present a general exact exponential algorithm for a large variety of these measures. As a consequence, we obtain algorithms which, for a hypergraph H on n vertices and m hyperedges, compute its generalized hypertree-width in time O ⁎ ( 2 n ) and its fractional hypertree-width in time O ( 1.734601 n ⋅ m ) . 3 3 We omit factors polynomial in n whenever the base of the exponent is rounded. This is justified as c n ⋅ n O ( 1 ) = O ( ( c + ϵ ) n ) for every ϵ > 0 . We also use the notation O ⁎ to suppress polynomial factors.

Retractions to Pseudoforests

For a fixed graph H, let $\textsc{Ret}(H)$ denote the problem of deciding whether a given input graph is retractable to H. We classify the complexity of $\textsc{Ret}(H)$ when H is a graph (with loops allowed) where each connected component has at most one cycle, i.e., a pseudoforest. In particular, this result extends the known complexity classifications of $\textsc{Ret}(H)$ for reflexive and irreflexive cycles to general cycles. Our approach is based mainly on algebraic techniques from universal algebra that previously have been used for analyzing the complexity of constraint satisfaction problems.

Tractable Structures for Constraint Satisfaction with Truth Tables

The way the graph structure of the constraints influences the complexity of constraint satisfaction problems (CSP) is well understood for bounded-arity constraints. The situation is less clear if there is no bound on the arities. In this case the answer depends also on how the constraints are represented in the input. We study this question for the truth table representation of constraints. We introduce a new hypergraph measure adaptive width and show that CSP with truth tables is polynomial-time solvable if restricted to a class of hypergraphs with bounded adaptive width. Conversely, assuming a conjecture on the complexity of binary CSP, there is no other polynomial-time solvable case. Finally, we present a class of hypergraphs with bounded adaptive width and unbounded fractional hypertree width.

A rendezvous of logic, complexity, and algebra

An emerging area of research studies the complexity of constraint satisfaction problems under restricted constraint languages. This article gives a self-contained, contemporary presentation of Schaefer's theorem on Boolean constraint satisfaction, the inaugural result of this area, as well as analogs of this theorem for quantified formulas. Our exposition makes use of and may serve as an introduction to logical and algebraic tools that have recently come into focus.

Maximal infinite-valued constraint languages

We systematically investigate the computational complexity of constraint satisfaction problems for constraint languages over an infinite domain. In particular, we study a generalization of the well-established notion of maximal constraint languages from finite to infinite domains. If the constraint language can be defined with an ω -categorical structure, then maximal constraint languages are in one-to-one correspondence to minimal oligomorphic clones. Based on this correspondence, we derive general tractability and hardness criteria for the corresponding constraint satisfaction problems.

Computational complexity of auditing finite attributes in statistical databases

We study the computational complexity of auditing finite attributes in databases allowing statistical queries. Given a database that supports statistical queries, the auditing problem is to check whether an attribute can be completely determined or not from a given set of statistical information. Some restricted cases of this problem have been investigated earlier, e.g. the complexity of statistical sum queries is known by the work of Kleinberg et al. (J. Comput. System Sci. 66 (2003) 244–253). We characterize all classes of statistical queries such that the auditing problem is polynomial-time solvable. We also prove that the problem is coNP-complete in all other cases under a plausible conjecture on the complexity of constraint satisfaction problems (CSP). The characterization is based on the complexity of certain CSP problems; the exact complexity for such problems is known in many cases. This result is obtained by exploiting connections between auditing and constraint satisfaction, and using certain algebraic techniques. We also study a generalization of the auditing problem where one asks if a set of statistical information imply that an attribute is restricted to K or less different values. We characterize all classes of polynomial-time solvable problems in this case, too.

The complexity of soft constraint satisfaction

Over the past few years there has been considerable progress in methods to systematically analyse the complexity of constraint satisfaction problems with specified constraint types. One very powerful theoretical development in this area links the complexity of a set of constraints to a corresponding set of algebraic operations, known as polymorphisms. In this paper we extend the analysis of complexity to the more general framework of combinatorial optimisation problems expressed using various forms of soft constraints. We launch a systematic investigation of the complexity of these problems by extending the notion of a polymorphism to a more general algebraic operation, which we call a multimorphism. We show that many tractable sets of soft constraints, both established and novel, can be characterised by the presence of particular multimorphisms. We also show that a simple set of NP-hard constraints has very restricted multimorphisms. Finally, we use the notion of multimorphism to give a complete classification of complexity for the Boolean case which extends several earlier classification results for particular special cases.

Parameterized complexity of constraint satisfaction problems

We prove a parameterized analog of Schaefer's Dichotomy Theorem: we show that for every finite boolean constraint family ?, deciding whether a formula containing constraints from ? has a satisfying assignment of weight exactly k is either fixed-parameter tractable (FPT) or W[1]-complete. We give a simple characterization of those constraints that make the problem fixed-parameter tractable. The special cases when the formula is restricted to be bounded occurrence, bounded treewidth, or planar are also considered: it turns out that in these cases the problem is in FPT for every constraint family ?.

A new tractable class of constraint satisfaction problems

In this paper we consider constraint satisfaction problems where the set of constraint relations is fixed. Feder and Vardi (1998) identified three families of constraint satisfaction problems containing all known polynomially solvable problems. We introduce a new class of problems called para-primal problems, incomparable with the families identified by Feder and Vardi (1998) and we prove that any constraint problem in this class is decidable in polynomial time. As an application of this result we prove a complete classification for the complexity of constraint satisfaction problems under the assumption that the basis contains all the permutation relations. In the proofs, we make an intensive use of algebraic results from clone theory about the structure of para-primal and homogeneous algebras.