Boolean Constraint Satisfaction Research Articles

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.

On memoryless provers and insincere verifiers

In this article, we introduce a Prover–Verifier model for analysing the computational complexity of a class of constraint satisfaction problems (CSPs) termed boolean binary constraint satisfaction problems (BBCSPs). BBCSPs represent an extremely general class of CSPs and find applications in a wide variety of domains including constraint programming, puzzle solving and program testing. The constraints in a BBCSP permit the combination of multiple theories as opposed to traditional constraint systems in which all constraints belong to the same theory. We establish that each instance of a BBCSP admits a coin-flipping Turing machine that halts in time polynomial in the size of the input. Furthermore, the algorithm is oblivious in that it never sees more than one constraint at a time. The prover, P, in the Prover–Verifier model is endowed with very limited powers. In particular, it has no memory and it can only pose restricted queries to the verifier. The verifier, on the other hand, is both omniscient in that it is cognisant of all the problem details and insincere in that it does not have to decide a priori on the intended proof. However, the verifier must stay consistent in its responses, i.e. it cannot rule out a certain possibility in one response to a query from the prover and then rule in the same possibility in response to a subsequent query. We note that the combination of the resources required by the prover and the type of certificate demanded of the verifier, determine the resources required by an algorithm. Inasmuch as our provers will be memoryless and our verifiers will be asked for extremely simple certificates, our work establishes the existence of a simple, randomised algorithm for BBCSPs. Our model itself serves as a basis for the design of zero-knowledge machine learning algorithms in that the prover ends up learning the proof desired by the verifier. Likewise, our work finds applications in the domain of certifying algorithm design, wherein the goal is to provide a proof of correctness of the algorithm on the input instance by providing an easily checkable certificate.

The complexity of weighted Boolean #CSP with mixed signs

We give a complexity dichotomy for the problem of computing the partition function of a weighted Boolean constraint satisfaction problem. Such a problem is parameterized by a set Γ of rational-valued functions, which generalize constraints. Each function assigns a weight to every assignment to a set of Boolean variables. Our dichotomy extends previous work in which the weight functions were restricted to being non-negative. We represent a weight function as a product of the form ( − 1 ) s g , where the polynomial s determines the sign of the weight and the non-negative function g determines its magnitude. We show that the problem of computing the partition function (the sum of the weights of all possible variable assignments) is in polynomial time if either every function in Γ can be defined by a “pure affine” magnitude with a quadratic sign polynomial or every function can be defined by a magnitude of “product type” with a linear sign polynomial. In all other cases, computing the partition function is FP #P -complete.

The Complexity of Weighted Boolean #CSP

This paper gives a dichotomy theorem for the complexity of computing the partition function of an instance of a weighted Boolean constraint satisfaction problem. The problem is parameterized by a finite set $\mathcal{F}$ of nonnegative functions that may be used to assign weights to the configurations (feasible solutions) of a problem instance. Classical constraint satisfaction problems correspond to the special case of 0,1-valued functions. We show that computing the partition function, i.e., the sum of the weights of all configurations, is $\text{{\sf FP}}^{\text{{\sf#P}}}$-complete unless either (1) every function in $\mathcal{F}$ is of “product type,” or (2) every function in $\mathcal{F}$ is “pure affine.” In the remaining cases, computing the partition function is in P.

The complexity of satisfiability problems: Refining Schaefer's theorem

Schaefer proved in 1978 that the Boolean constraint satisfaction problem for a given constraint language is either in P or is NP-complete, and identified all tractable cases. Schaefer's dichotomy theorem actually shows that there are at most two constraint satisfaction problems, up to polynomial-time isomorphism (and these isomorphism types are distinct if and only if P ≠ NP ). We show that if one considers AC 0 isomorphisms, then there are exactly six isomorphism types (assuming that the complexity classes NP, P, ⊕L, NL, and L are all distinct). A similar classification holds for quantified constraint satisfaction problems.

Satisfying Assignments of Random Boolean Constraint Satisfaction Problems: Clusters and Overlaps

Satisfying Assignments of Random Boolean Constraint Satisfaction Problems: Clusters and Overlaps

Solving topological constraints for declarative families of objects

Parametric and feature-based CAD models can be considered to represent families of similar objects. In current modelling systems, however, the semantics of such families are unclear and ambiguous. We present the Declarative Family of Objects Model (DFOM), which enables us to adequately specify and maintain family semantics. In this model, not only geometry, but also topology is specified declaratively, by means of constraints. A family of objects is modelled by a DFOM with multiple realizations. A member of the family is modelled by adding constraints, e.g. to set dimension variables, until a single realization remains. The declarative approach guarantees that the realization of a family member is also a realization of the family. The realization of a family member is found by solving first the geometric constraints, and then the topological constraints. From the geometric solution, a cellular model is constructed. Topological constraints indirectly specify which combinations of cellular model entities are allowed in the realization. The system of topological constraints is mapped to a Boolean constraint satisfaction problem. The realization is found by solving this problem using a SAT solver.

Expected number of locally maximal solutions for random Boolean CSPs

For a large number of random Boolean constraint satisfaction problems, such as random $k$-SAT, we study how the number of locally maximal solutions evolves when constraints are added. We give the exponential order of the expected number of these distinguished solutions and prove it depends on the sensitivity of the allowed constraint functions only. As a by-product we provide a general tool for computing an upper bound of the satisfiability threshold for any problem of a large class of random Boolean CSPs.

Classification of Bipartite Boolean Constraint Satisfaction through Delta-Matroid Intersection

Matroid intersection has a known polynomial time algorithm using an oracle. We generalize this result to delta-matroids that do not have equality as a restriction and give a polynomial time algorithm for delta-matroid intersection on delta-matroids without equality using an oracle. We note that when equality is present, delta-matroid intersection is as general as delta-matroid parity. We also obtain algorithms using an oracle for delta-matroid parity on delta-matroids without inequality, and for delta-matroid intersection where one delta-matroid does not contain either equality or inequality, and the second delta-matroid is arbitrary. Both these results also generalize matroid intersection. The results imply a dichotomy for bipartite Boolean constraint satisfaction problems using an oracle when one of the two sides does not contain equality, leaving open cases of delta-matroid parity when both sides have equality; the results also imply a full dichotomy for $k$-partite Boolean constraint satisfaction problems for $k\geq 3$. We then discuss polynomial cases of Boolean constraint satisfaction problems with two occurrences per variable through delta-matroid parity that cannot be obtained using the oracle approach.

Threshold properties of random boolean constraint satisfaction problems

We study threshold properties of random constraint satisfaction problems under a probabilistic model due to Molloy [Models for random constraint satisfaction problems, in: Proceedings of the 32nd ACM Symposium on Theory of Computing, 2002]. We give a sufficient condition for the existence of a sharp threshold. In the boolean case, it gives an independent proof for the more difficult half of a classification result conjectured by Creignou and Daudé [Generalized satisfiability problems: minimal elements and phase transitions. Theor. Comput. Sci. 302(1–3) (2003) 417–430], proved in a restricted case by the same authors [Combinatorial sharpness criterion and phase transition classification for random CSPs, Inform. Comput. 190(2) (2004) 220–238], and established by them [Coarse and sharp thresholds for random generalized satisfiability problems, in: M. Drmota, P. Flajolet, D. Gardy, B. Gittenberger (Eds.), Mathematics and Computer Science III: Algorithms, Trees, Combinatorics and Probabilities, Birkhauser, Basel, September 2004, pp. 507–517] while this paper was in the refereeing process.

Bases for Boolean co-clones

The complexity of various problems in connection with Boolean constraints, like, for example, quantified Boolean constraint satisfaction, have been studied recently. Depending on what types of constraints may be used, the complexity of such problems varies. A very interesting observation of the recent past has been that the thus derived classification of constraints can be explained with the help of universal algebra. More precisely, the difficulty of such a constraint problem often depends on the co-clone the constraints are from. A co-clone is a set of Boolean relations that is closed under very natural closure operations. Nearly all these co-clones can be generated by said operators out of a finite set of relations, a so-called base. Knowing a, preferably simple, base for each co-clone can therefore be of great value when studying the complexity of Boolean constraint problems, since this knowledge reduces the infinitely many cases of equivalent problems to a single one—the constraint satisfaction problem for this base. In this paper we give a finite and simple base for every Boolean co-clone, where this is possible. We give evidence that the presented bases are as easy as possible.

The Complexity of Boolean Constraint Satisfaction Local Search Problems

The Complexity of Boolean Constraint Satisfaction Local Search Problems

The complexity of Boolean constraint satisfaction local search problems

The class of generalized satisfiability problems, first introduced by Schaefer in 1978, presents a uniform way of studying the complexity of Boolean constraint satisfaction problems with respect to...

SIGACT news complexity theory column 34

This issue's guest columnists are angels. The column was faced with a last-minute cancellation, and they prepared on very short notice a wonderful column. And I'm sure that many readers of the column will want to learn even more by, for example, reading their monograph Complexity Classifications of Boolean Constraint Satisfaction Problems [7].Future complexity columns include Holzer/McKenzie on a familiar complexity class turning up in a surprising context, Gasarch (and a host of contributors) on the future of NP, Schaeffer/Umans on completeness for higher levels of the polynomial hierarchy, Nickelsen/Tantau on partial information classes, and Paturi on the complexity of k -SAT,Regarding the Schaeffer/Umans column, their 2-part column will be in part a kind of Garey and Johnson for completeness results for ∑ i p and π i p , i ≥ 2. You can help their project by, if you have proven natural sets complete for NP NP , coNP NP , NP NP NP , coNP NP NP , etc., letting them (MSchaefer@cti.depaul.edu and umans@microsoft.com) know of your result and its best citation location. Thanks!

The Approximability of Constraint Satisfaction Problems

We study optimization problems that may be expressed as constraint satisfaction problems. An instance of a Boolean constraint satisfaction problem is given by m constraints applied to n Boolean variables. Different computational problems arise from constraint satisfaction problems depending on the nature of the underlying constraints as well as on the goal of the optimization task. Here we consider four possible goals: Max CSP (Min CSP) is the class of problems where the goal is to find an assignment maximizing the number of satisfied constraints (minimizing the number of unsatisfied constraints). Max Ones (Min Ones) is the class of optimization problems where the goal is to find an assignment satisfying all constraints with maximum (minimum) number of variables set to 1. Each class consists of infinitely many problems and a problem within a class is specified by a finite collection of finite Boolean functions that describe the possible constraints that may be used. Tight bounds on the of every problem in Max CSP were obtained by Creignou [ J. Comput. System Sci., 51 (1995), pp. 511--522]. In this work we determine tight bounds on the approximability (i.e., the ratio to within which each problem may be approximated in polynomial time) of every problem in Max Ones, Min CSP, and Min Ones. Combined with the result of Creignou, this completely classifies all optimization problems derived from Boolean constraint satisfaction. Our results capture a diverse collection of optimization problems such as MAX 3-SAT, Max Cut, Max Clique, Min Cut, Nearest Codeword, etc. Our results unify recent results on the (in-)approximability of these optimization problems and yield a compact presentation of most known results. Moreover, these results provide a formal basis to many statements on the behavior of natural optimization problems that have so far been observed only empirically.

Boolean constraint satisfaction: complexity results for optimization problems with arbitrary weights

A boolean constraint satisfaction problem consists of some finite set of constraints (i.e., functions from 0 /1-vectors to {0,1}) and an instance of such a problem is a set of constraints applied to specified subsets of n boolean variables. The goal is to find an assignment to the variables which satisfy all constraint applications. The computational complexity of optimization problems in connection with such problems has been studied extensively but the results have relied on the assumption that the weights are non-negative. The goal of this article is to study variants of these optimization problems where arbitrary weights are allowed. For the four problems that we consider, we give necessary and sufficient conditions for when the problems can be solved in polynomial time. In addition, we show that the problems are NP-equivalent in all other cases.