Constraint Composite Graph-Based Weighted CSP Solvers: An Empirical Study

Test-cost-sensitive rough set based approach for minimum weight vertex cover problem

An efficient local search framework for the minimum weighted vertex cover problem

Minimum weighted vertex cover (MWVC) problem of weighted networks is one of the most important combinatorial optimization problems in network optimization. This paper models the MWVC problem as an asymmetric game on weighted networks. Then, an asynchronous game-based algorithm (AGB) is proposed to solve the MWVC problem, which can guarantee to obtain a vertex cover (VC). Numerical illustrations verify the performance of the proposed game solution on the representative weighted networks.

The minimum weight vertex cover problem (MWVCP) is one of the classic combinatorial optimisation problems. In this paper, we have proposed a modified jumping particle swarm optimisation approach, n...

Toward better approximation for the minimum-weighted vertex cover (MWVC) problem in multiagent systems, we present a distributed algorithm from the perspective of learning in games. For self-organized coordination and optimization, we see each vertex as a potential game player who makes decisions using local information of its own and the immediate neighbors. The resulting Nash equilibrium is classified into two categories, i.e., the inferior Nash equilibrium (INE) and the dominant Nash equilibrium (DNE). We show that the optimal solution must be a DNE. To achieve better approximation ratios, local rules of perturbation and weighted memory are designed, with the former destroying the stability of an INE and the latter facilitating the refinement of a DNE. By showing the existence of an improvement path from any INE to a DNE, we prove that when the memory length is larger than 1, our algorithm converges in finite time to DNEs, which could not be improved by exchanging the action of a selected node with all its unselected neighbors. Moreover, additional freedom for solution efficiency refinement is provided by increasing the memory length. Finally, intensive comparison experiments demonstrate the superiority of the presented methodology to the state of the art, both in solution efficiency and computation speed.

The generalized vertex cover problem and some variations

The exponential-time hypothesis (ETH) states that 3-SAT is not solvable in subexponential time, i.e. not solvable in O(c^n) time for arbitrary c > 1, where n denotes the number of variables. Problems like k-SAT can be viewed as special cases of the constraint satisfaction problem (CSP), which is the problem of determining whether a set of constraints is satisfiable. In this paper we study the worst-case time complexity of NP-complete CSPs. Our main interest is in the CSP problem parameterized by a constraint language Gamma (CSP(Gamma)), and how the choice of Gamma affects the time complexity. It is believed that CSP(Gamma) is either tractable or NP-complete, and the algebraic CSP dichotomy conjecture gives a sharp delineation of these two classes based on algebraic properties of constraint languages. Under this conjecture and the ETH, we first rule out the existence of subexponential algorithms for finite domain NP-complete CSP(Gamma) problems. This result also extends to certain infinite-domain CSPs and structurally restricted CSP(Gamma) problems. We then begin a study of the complexity of NP-complete CSPs where one is allowed to arbitrarily restrict the values of individual variables, which is a very well-studied subclass of CSPs. For such CSPs with finite domain D, we identify a relation SD such that (1) CSP({SD}) is NP-complete and (2) if CSP(Gamma) over D is NP-complete and solvable in O(c^n) time, then CSP({SD}) is solvable in O(c^n) time, too. Hence, the time complexity of CSP({SD}) is a lower bound for all CSPs of this particular kind. We also prove that the complexity of CSP({SD}) is decreasing when |D| increases, unless the ETH is false. This implies, for instance, that for every c>1 there exists a finite-domain Gamma such that CSP(Gamma) is NP complete and solvable in O(c^n) time.

We present a deterministic distributed 2-approximation algorithm for the Minimum Weight Vertex Cover problem in the CONGEST model whose round complexity is \(O(\log n\log \varDelta / \log ^2\log \varDelta )\). This improves over the currently best known deterministic 2-approximation implied by [KVY94]. Our solution generalizes the \((2+\epsilon )\)-approximation algorithm of [BCS17], improving the dependency on \(\epsilon ^{-1}\) from linear to logarithmic. In addition, for every \(\epsilon =(\log \varDelta )^{-c}\), where \(c\ge 1\) is a constant, our algorithm computes a \(\left( 2+\epsilon \right) \)-approximation in \(O(\log {\varDelta }/\log \log {\varDelta })\) rounds (which is asymptotically optimal).

The Constraint Satisfaction Problem (CSP) is a central and generic computational problem which provides a common framework for many theoretical and practical applications. A central line of research is concerned with the identification of classes of instances for which CSP can be solved in polynomial time; such classes are often called “islands of tractability.” A prominent way of defining islands of tractability for CSP is to restrict the relations that may occur in the constraints to a fixed set, called a constraint language , whereas a constraint language is conservative if it contains all unary relations. Schaefer’s famous Dichotomy Theorem (STOC 1978) identifies all islands of tractability in terms of tractable constraint languages over a Boolean domain of values. Since then, many extensions and generalizations of this result have been obtained. Recently, Bulatov (TOCL 2011, JACM 2013) gave a full characterization of all islands of tractability for CSP and the counting version #CSP that are defined in terms of conservative constraint languages. This article addresses the general limit of the mentioned tractability results for CSP and #CSP, that they only apply to instances where all constraints belong to a single tractable language (in general, the union of two tractable languages is not tractable). We show that we can overcome this limitation as long as we keep some control of how constraints over the various considered tractable languages interact with each other. For this purpose, we utilize the notion of a strong backdoor of a CSP instance, as introduced by Williams et al. (IJCAI 2003), which is a set of variables that when instantiated, moves the instance to an island of tractability, that is, to a tractable class of instances. We consider strong backdoors into scattered classes , consisting of CSP instances where each connected component belongs entirely to some class from a list of tractable classes. Figuratively speaking, a scattered class constitutes an archipelago of tractability . The main difficulty lies in finding a strong backdoor of given size k ; once it is found, we can try all possible instantiations of the backdoor variables and apply the polynomial time algorithms associated with the islands of tractability on the list component-wise. Our main result is an algorithm that, given a CSP instance with n variables, finds in time f ( k ) n O (1) a strong backdoor into a scattered class (associated with a list of finite conservative constraint languages) of size k or correctly decides that there is not such a backdoor. This also gives the running time for solving (#)CSP, provided that (#)CSP is polynomial-time tractable for the considered constraint languages. Our result makes significant progress towards the main goal of the backdoor-based approach to CSPs—the identification of maximal base classes for which small backdoors can be detected efficiently.

Handbook of Constraint Programming

We discuss a coevolutionary genetic algorithm for constraint satisfaction. Our basic idea is to explore effective genetic information in the population, i.e., schemata, and to exploit the genetic information in order to guide the population to better solutions. Our coevolutionary genetic algorithm (CGA) consists of two GA populations; the first GA, called searches for the solutions in a given environment (problem), and the second GA, called P-GA, searches for effective genetic information involved in the H-GA, namely, good schemata. Thus, each individual in P-GA consists of alleles in H-GA or don't care symbol representing a schema in the H-GA. These GA populations separately evolve in each genetic space at different abstraction levels and affect with each other by two genetic operators: superposition and transcription. We then applied our CGA to constraint satisfaction problems (CSPs) incorporating a new stochastic repair operator for P-GA to raise the consistency of schemata with the (local) constraint conditions in CSPs. We carried out two experiments: First, we examined the performance of CGA on various general CSPs that are generated randomly for a wide variety of density and tightness of constraint conditions in the CSPs that are the basic measures of characterizing CSPs. Next, we examined structured CSPs involving latent cluster structures among the variables in the CSPs. For these experiments, computer simulations confirmed us the effectiveness of our CGA.

We use the Constraint Satisfaction Problem (CSP) framework to model and solve the problem of designing substitution functions for substitutionpermutation (SP) networks as proposed by Shannon for the architecture of ciphers. Many ciphers are designed using the SP pattern, and differ mainly by two parametrized functions: substitution and permutation. The most difficult of the two is the substitution function, which has to be nonlinear (a requirement that was difficult to define and quantify). Over time, researchers such as Nyberg, Pieprzyk and Matsui have proposed various metrics of nonlinearity that make the function robust to modern attacks. Before us, people have attempted various ways to design functions that respect these metrics. In the past people hand-picked substitution tables (S-boxes) by trying various values. Recently they use difficult to analyze constructs (such as Bent functions, spectral inversion, inverses in Galois fields) whose outputs are tested for nonlinearity. While efficient, such techniques are neither exhaustive (optimal), nor did they manage to generate better substitutions than the ones hand-picked in the past. We show that Matsui's nonlinearity requirement can be naturally modelled using CSPs. Based on a combination of existing CSP techniques and some new filtering operators that we designed specially for the new types of constraints, we manage to obtain better S-boxes than any previously published ones. The simplicity of the CSP framework and availability of general CSP solvers like ours, makes it easy for more people to design their own ciphers with easy to understand security parameters. Here we report on this new application of CSPs.

An instance of the Constraint Satisfaction Problem (CSP) is given by a family of constraints on overlapping sets of variables, and the goal is to assign values from a xed domain to the variables so that all constraints are satised. In the optimization version, the goal is to maximize the number of satised constraints. An approximation algorithm for CSP is called robust if it outputs an assignment satisfying an (1????g("))-fraction of constraints on any (1????")-satisable instance, where the loss function g is such that g(") ! 0 as " ! 0. We study how the robust approximability of CSPs depends on the set of constraint relations allowed in instances, the so-called constraint language. All constraint languages admitting a robust polynomial-time algorithm (with some g) have been characterised by Barto and Kozik, with the general bound on the loss g being doubly exponential, specically g(") = O((log log(1="))= log(1=")). It is natural to ask when a better loss can be achieved: in particular, polynomial loss g(") = O("1=k) for some constant k. In this paper, we consider CSPs with a constraint language having a nearunanimity polymorphism. This general condition almost matches a known necessary condition for having a robust algorithm with polynomial loss. We give two randomized robust algorithms with polynomial loss for such CSPs: one works for any near-unanimity polymorphism and the parameter k in the loss depends on the size of the domain and the arity of the relations in ????, while the other works for a special ternary near-unanimity operation called dual discriminator with k = 2 for any domain size. In the latter case, the CSP is a common generalisation of Unique Games with a xed domain and 2-Sat. In the former case, we use the algebraic approach to the CSP. Both cases use the standard semidenite programming relaxation for CSP.

An Iterative local-search framework for solving constraint satisfaction problem

We perform a comprehensive study of mappings between constraint satisfaction problems (CSPs) and propositional satisfiability (SAT). We analyse four different mappings of SAT problems into CSPs, and two of CSPs into SAT problems. For each mapping, we compare the impact of achieving arc-consistency on the CSP with unit propagation on the SAT problem. We then extend these results to CSP algorithms that maintain (some level of) arc-consistency during search like FC and MAC, and to the Davis-Putnam procedure (which performs unit propagation at each search node). Because of differences in the branching structure of their search, a result showing the dominance of achieving arc-consistency on the CSP over unit propagation on the SAT problem does not necessarily translate to the dominance of MAC over the Davis-Putnam procedure. These results provide insight into the relationship between propositional satisfiability and constraint satisfaction.