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A General Stochastic Approach to Solving Problems with Hard and Soft Constraints
 The Satisfiability Problem: Theory and Applications
, 1996
"... . Many AI problems can be conveniently encoded as discrete constraint satisfaction problems. It is often the case that not all solutions to a CSP are equally desirable  in general, one is interested in a set of "preferred" solutions (for example, solutions that minimize some cost function) . Pref ..."
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Cited by 46 (1 self)
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. Many AI problems can be conveniently encoded as discrete constraint satisfaction problems. It is often the case that not all solutions to a CSP are equally desirable  in general, one is interested in a set of "preferred" solutions (for example, solutions that minimize some cost function) . Preferences can be encoded by incorporating "soft" constraints in the problem instance. We show how both hard and soft constraints can be handled by encoding problems as instances of weighted MAXSAT (finding a model that maximizes the sum of the weights of the satisfied clauses that make up a problem instance). We generalize a localsearch algorithm for satisfiability to handle weighted MAXSAT. To demonstrate the effectiveness of our approach, we present experimental results on encodings of a set of wellstudied network Steinertree problems. This approach turns out to be competitive with some of the best current specialized algorithms developed in operations research. 1. Introduction Traditi...
Solving Problems with Hard and Soft Constraints Using a Stochastic Algorithm for MAXSAT
, 1995
"... Stochastic local search is an effective technique for solving certain classes of large, hard propositional satisfiability problems, including propositional encodings of problems such as circuit synthesis and graph coloring (Selman, Levesque, and Mitchell 1992; Selman, Kautz, and Cohen 1994). Many pr ..."
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Cited by 42 (3 self)
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Stochastic local search is an effective technique for solving certain classes of large, hard propositional satisfiability problems, including propositional encodings of problems such as circuit synthesis and graph coloring (Selman, Levesque, and Mitchell 1992; Selman, Kautz, and Cohen 1994). Many problems of interest to AI and operations research cannot be conveniently encoded as simple satisfiability, because they involve both hard and soft constraints  that is, any solution may have to violate some of the less important constraints. We show how both kinds of constraints can be handled by encoding problems as instances of weighted MAXSAT (finding a model that maximizes the sum of the weights of the satisfied clauses that make up a problem instance). We generalize our localsearch algorithm for satisfiability (GSAT) to handle weighted MAXSAT, and present experimental results on encodings of the Steiner tree problem, which is a wellstudied hard combinatorial search problem. On many...
A Hybrid GRASP with Perturbations for the Steiner Problem in Graphs
 INFORMS Journal on Computing
, 2001
"... t We propose and describe a hybrid GRASP with weight perturbations and adaptive pathrelinking heuristic (HGP+PR) for the Steiner problem in graphs. In this... ..."
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Cited by 34 (16 self)
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t We propose and describe a hybrid GRASP with weight perturbations and adaptive pathrelinking heuristic (HGP+PR) for the Steiner problem in graphs. In this...
A Parallel GRASP For The Steiner Tree Problem In Graphs Using A Hybrid Local Search Strategy
 Journal of Global Optimization
, 1999
"... In this paper, we present a parallel greedy randomized adaptive search procedure (GRASP) for the Steiner problem in graphs. GRASP is a two phase metaheuristic. In the first phase, solutions are constructed using a greedy randomized procedure. Local search is applied in the second phase, leading to a ..."
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Cited by 23 (12 self)
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In this paper, we present a parallel greedy randomized adaptive search procedure (GRASP) for the Steiner problem in graphs. GRASP is a two phase metaheuristic. In the first phase, solutions are constructed using a greedy randomized procedure. Local search is applied in the second phase, leading to a local minimum with respect to a specified neighborhood. In the Steiner problem in graphs, feasible solutions can be characterized by their nonterminal nodes (Steiner nodes) or by their keypaths. According to this characterization, two GRASP procedures are described using different local search strategies. Both use an identical construction procedure. The first uses a nodebased neighborhood for local search, while the second uses a pathbased neighborhood. Computational results comparing the two procedures show that while the nodebased variant produces better quality solutions, the pathbased variant is about twice as fast. A hybrid GRASP procedure combining the two neighbo...
The pilot method: A strategy for heuristic repetition with application to the Steiner problem in graphs
 Networks
, 1999
"... Abstract: As a metaheuristic to obtain solutions of enhanced quality, we formulate the socalled pilot method. It is a tempered greedy method that is to avoid the greedy trap by looking ahead for each possible choice (memorizing the best result). Repeatedly, a socalled master solution is modified, ..."
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Cited by 20 (8 self)
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Abstract: As a metaheuristic to obtain solutions of enhanced quality, we formulate the socalled pilot method. It is a tempered greedy method that is to avoid the greedy trap by looking ahead for each possible choice (memorizing the best result). Repeatedly, a socalled master solution is modified, each time in a minimal fashion to account for the “best ” choice, where all choices have been judged by means of a separate heuristic result, the “pilot ” solution. We apply the method to the wellknown Steiner problem in a weighted graph, that is, the problem is to determine a subgraph of minimum total weight spanning a set of given vertices. The pilot method may be seen as a system for heuristic repetition. As a higher time complexity order is usually associated with repetition, we propose policies to reduce the running times, while retaining an enhanced solution quality. Where possible, to encourage application of the pilot method to other combinatorial problems, we formulate in general terms. © 1999 John Wiley & Sons, Inc. Networks 34: 181–191, 1999 1.
Computing NearOptimal Solutions to the Steiner Problem in a Graph Using a Genetic Algorithm
, 1995
"... A new Genetic Algorithm (GA) for the Steiner Problem in a Graph (SPG) is presented. The algorithm is based on a bitstring encoding. A bitstring specifies selected Steiner vertices and the corresponding Steiner tree is computed using the Distance Network Heuristic. This scheme ensures that every bit ..."
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Cited by 19 (0 self)
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A new Genetic Algorithm (GA) for the Steiner Problem in a Graph (SPG) is presented. The algorithm is based on a bitstring encoding. A bitstring specifies selected Steiner vertices and the corresponding Steiner tree is computed using the Distance Network Heuristic. This scheme ensures that every bitstring correspond to a valid Steiner tree and thus eliminate the need for penalty terms in the cost function. The GA is tested on all SPG instances from the ORLibrary of which the largest graphs have 2,500 vertices and 62,500 edges. When executed 10 times on each of 58 graph examples, the GA finds the global optimum at least once for 55 graphs and every time for 43 graphs. In total the GA finds the global optimum in 77 % of all program executions and is within 1 % from the global optimum in more than 92 % of all executions. The performance is compared to that of two branchandcut algorithms and one of the very best deterministic heuristics, an iterated version of the Shortest Path Heuristi...
Multicast Routing Under Optical Layer Constraints
 In IEEE INFOCOM
, 2004
"... It has been widely recognized that physical layer impairments, including power losses, must be taken into account when routing optical connections in transparent networks. In this paper we study the problem of constructing lighttrees under optical layer power budget constraints, with a focus on alg ..."
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Cited by 16 (2 self)
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It has been widely recognized that physical layer impairments, including power losses, must be taken into account when routing optical connections in transparent networks. In this paper we study the problem of constructing lighttrees under optical layer power budget constraints, with a focus on algorithms which can guarantee a certain level of quality for the signals received by the destination nodes. We define a new constrained lighttree routing problem by introducing a set of constraints on the sourcedestination paths to account for the power losses at the optical layer. We investigate a number of variants of this problem, we characterize their complexity, and we develop a suite of corresponding routing algorithms
A Parallel GRASP for the Steiner Problem in Graphs
 Proceedings of IRREGULAR’98 – 5th International Symposium on Solving Irregularly Structured Problems in Parallel, volume 1457 of Lecture Notes in Computer Science
, 1998
"... A greedy randomized adaptive search procedure (GRASP) is a metaheuristic for combinatorial optimization. Given an undirected graph with weights associated with its nodes, the Steiner tree problem consists in finding a minimum weight subgraph spanning a given subset of (terminal) nodes of the origina ..."
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Cited by 15 (8 self)
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A greedy randomized adaptive search procedure (GRASP) is a metaheuristic for combinatorial optimization. Given an undirected graph with weights associated with its nodes, the Steiner tree problem consists in finding a minimum weight subgraph spanning a given subset of (terminal) nodes of the original graph. In this paper, we describe a parallel GRASP for the Steiner problem in graphs. We review basic concepts of GRASP: construction and local search algorithms. The implementation of a sequential GRASP for the Steiner problem in graphs is described in detail. Feasible solutions are characterized by their nonterminal nodes. A randomized version of Kruskal's algorithm for the minimum spanning tree problem is used in the construction phase. Local search is based on insertions and eliminations of nodes to/from the current solution. Parallelization is done through the distribution of the GRASP iterations among the processors on a demanddriven basis, in order to improve load balancing. The ...
Reactive Tabu Search With Path Relinking For The Steiner Problem In Graphs
 In Proceedings of the Third Metaheuristics International Conference
, 1999
"... . Given an undirected graph with weights associated with its edges, the Steiner tree problem consists in finding a minimum weight subgraph spanning a given subset of nodes (terminals) of the original graph. We describe a reactive tabu search with path relinking algorithm for the Steiner problem in g ..."
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Cited by 14 (5 self)
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. Given an undirected graph with weights associated with its edges, the Steiner tree problem consists in finding a minimum weight subgraph spanning a given subset of nodes (terminals) of the original graph. We describe a reactive tabu search with path relinking algorithm for the Steiner problem in graphs, based on the extension of a previously developed tabu search algorithm using a neighborhood defined by insertions and eliminations of Steiner nodes. Computational experiments on benchmark problems are reported, comparing the reactive tabu search with other metaheuristic implementations. The reactive tabu search algorithm outperforms other algorithms, obtaining better or comparably good solutions. We also describe a robust parallel implementation based on an independent multiple path strategy and report improved computational results on a 32processor cluster running under Linux. Key words. Combinatorial optimization, Steiner problem, graphs, metaheuristics, tabu search, reactive tabu...
A Branch and Cut Algorithm for the Steiner Problem in Graphs
 Networks
, 1998
"... Abstract: In this paper, we consider the Steiner problem in graphs, which is the problem of connecting together, at minimum cost, a number of vertices in an undirected graph with nonnegative edge costs. We use the formulation of this problem as a shortest spanning tree (SST) problem with additional ..."
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Cited by 9 (2 self)
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Abstract: In this paper, we consider the Steiner problem in graphs, which is the problem of connecting together, at minimum cost, a number of vertices in an undirected graph with nonnegative edge costs. We use the formulation of this problem as a shortest spanning tree (SST) problem with additional constraints given previously in the literature. We strengthen this SST formulation and present a branch and cut algorithm to solve the problem to optimality. This algorithm incorporates reduction tests and is used to solve a number of problems drawn from the literature. A number of general issues relating to branch and cut algorithms are also highlighted. � 1998 John Wiley & Sons, Inc. Networks 31: 39–59, 1998