This article briefly summarizes the work that has been done and presents the current standing of neural networks for combinatorial optimization by considering each of the major classes of combinatorial optimization problems. Areas which have not yet been studied are identified for future research.

The past few years have seen an enormous progress in the performance of Boolean satisfiability (SAT) solvers. Despite the worst-case exponential run time of all known algorithms, satisfiability solvers are increasingly leaving their mark as a generalpurpose tool in areas as diverse as software and hardware verification [29–31, 228], automatic test pattern generation [138, 221], planning [129, 197], scheduling [103], and even challenging problems from algebra [238]. Annual SAT competitions have led to the development of dozens of clever implementations of such solvers [e.g. 13,

A parallel distributed framework to solve constraint satisfaction problems based on connectionist ideas of distributed information processing is presented. In this approach, each variable of a given problem is associated with a simple agent continuously applying a variable manipulation rule in the spirit of local conflict minimization to satisfy all constraints this variable is involved in. All agents are working simultaneously forming together a recurrent dynamical system which should selforganize after some iterations to a feasible problem solution. We investigate whether and how manipulation rules can be chosen such that the whole process converges without coordinating agent activities. Keywords: constraint satisfaction problems, local search, parallel distributed processing 1 Introduction Constraint satisfaction is a problem which appears in many real-life situations and has received a great deal of attention in recent years. Many AI problems of theoretical and practical interest...

Heuristic repair algorithms, a class of local search methods, demonstrate impressive efficiency in solving some large-scale and hard instances of constraint satisfaction problems (CSP's). In this paper, we draw a surprising connection between heuristic repair techniques and the discrete Lagrange multiplier methods by transforming CSP's into zero-one constrained optimization problems. A Lagrangian-based search scheme LSDL is proposed. We show how GENET, a representative heuristic repair algorithm, can be reconstructed from LSDL. The dual viewpoint of GENET as heuristic repair method and Lagrange multiplier method allows us to investigate variants of GENET from both perspectives. Benchmarking results confirm that first, our reconstructed GENET has the same fast convergence behavior as other GENET implementations reported in the literature, competing favourably with other state-of-the-art methods on a set of hard graph colouring problems. Second, our best variant, which combines technique...

Many real-life problems belong to the class of constraint satisfaction problems (CSP's), which are NP-complete, and some NP-hard, in general. When the problem size grows, it becomes difficult to program solutions and to execute the solution in a timely manner. In this paper, we present a general framework for integrating artificial neural networks and logic programming to provide an efficient and yet easy-to-program environment for solving CSP's. To realize this framework, we propose a novel constraint logic programming language PROCLANN. Operationally, PROCLANN uses the standard goal reduction strategy as frontend to generate constraints and an efficient backend constraint-solver based on artificial neural networks. PROCLANN retains the simple and elegant declarative semantics of constraint logic programming. Its operational semantics is probabilistic in nature. We show that PROCLANN is sound and weakly complete. A novelty of PROCLANN is that while it is a committed-choice l...

GENET is a heuristic repair algorithm which demonstrates impressive efficiency in solving some large-scale and hard instances of constraint satisfaction problems (CSPs). In this paper, we draw a surprising connection between GENET and discrete Lagrange multiplier methods. Based on the work of Wah and Shang, we propose a discrete Lagrangian-based search scheme LSDL, defining a class of search algorithms for solving CSPs. We show how GENET can be reconstructed from LSDL.The dual viewpoint of GENET as a heuristic repair method and a discrete Lagrange multiplier method allows us to investigate variants of GENET from both perspectives. Benchmarking results confirm that first, our reconstructed GENET has the same fast convergence behavior as the original GENET implementation, and has competitive performance with other local search solvers DLM, WalkSAT, and WSAT(OIP), on a set of difficult benchmark problems. Second, our improved variant, which combines techniques from heuristic repair an...

Hard or large-scale constraint satisfaction and optimization problems, occur widely in artificial intelligence and operations research. These problems are often difficult to solve with global search methods, but many of them can be efficiently solved by local search methods. Evolutionary algorithms are local search methods which have considerable success in tackling difficult, or ill-defined optimization problems. In contrast they have not been so successful in tackling constraint satisfaction problems. Other local search methods, in particular GENET and EGENET are designed specifically for constraint satisfaction problems, and have demonstrated remarkable success in solving hard examples of these problems. In this paper we examine how we can transfer the mechanisms that were so successful in (E)GENET to evolutionary algorithms, in order to tackle constraint satisfaction algorithms efficiently. An empirical comparison of our evolutionary algorithm improved by mechanisms from EGENET and shows how it can markedly improve on the efficiency of EGENET in solving certain hard instances of constraint satisfaction problems.

Constraint Satisfaction Problems (CSPs) are at the heart of many AI applications. The currently existing constraint programming languages and systems are mostly based on heuristic search techniques. The major problems of these techniques are the limited parallelism available in the algorithms and the inadequacy of handling over-constrained CSPs. GENET is a competitive neural network model developed for solving CSPs with large size and tight constraints. It realizes a stochastic heuristic search in a fully parallelized processing fashion. This paper presents the VLSI design of a cascadable GENET module, which may be configured to suit the structure of the CSPs in application. A realistic estimation of the potential speed-up over the existing programming languages and systems is given, which is based on the result of software simulation of the GENET's behaviour in solving over several thousands of randomly generated CSPs with various tightness of constraints. It is concluded that GENET, ...

. Although graph drawing has been extensively studied, little attention has been paid to the problem of node overlapping. The problem arises because almost all existing graph layout algorithms assume that nodes are points. In practice, however, nodes may be labelled and these labels may overlap. We propose four dierent approaches for removing node overlapping, all of which are based on constrained optimization techniques. The rst is the simplest. It performs the minimal linear scaling which will remove node-overlapping. The second approach relies on formulating the node overlapping problem as a convex quadratic programming problem which can then be solved by any quadratic solver. The disadvantage is that since constraints must be linear the node overlapping constraints cannot be expressed directly but must be strengthened to obtain a linear constraint strong enough to ensure no node overlapping. The third and fourth approaches are based on local search methods. The third is an adapta...

In this paper we discuss the possibilities of applying genetic algorithms (GA) for solving constraint satisfaction problems (CSP). We point out how the greediness of deterministic classical CSP solving techniques can be counterbalanced by the random mechanisms of GAs. We tested our ideas by running experiments on four different CSPs: N-queens, graph 3-colouring, the traffic lights and the Zebra problem. Three of the problems have proven to be GA-easy, and even for the GA-hard one the performance of the GA could be boosted by techniques familiar in classical methods. Thus GAs are promising tools for solving CSPs. In the discussion, we address the issues of non-solvable CSPs and the generation of all the solutions. 1.1 Introduction In this paper we consider genetic algorithms (GA) for solving constraint satisfaction problems (CSP) with finite domains. The majority of CSP solving algorithms, which we will refer to as classical ones, are deterministic and constructive search algorithms....