We describe four versions of a Greedy Randomized Adaptive Search Procedure (GRASP) for finding approximate solutions of general instances of the Steiner Problem in Graphs. Di#erent construction and local search algorithms are presented. Preliminary computational results with one of the versions on a variety of test problems are reported. On the majority of instances from the OR-Library, a set of standard test problems, the GRASP produced optimal solutions. On those that optimal solutions were not found, the GRASP found good quality approximate solutions.

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 MAX-SAT (finding a model that maximizes the sum of the weights of the satisfied clauses that make up a problem instance). We generalize our local-search algorithm for satisfiability (GSAT) to handle weighted MAX-SAT, and present experimental results on encodings of the Steiner tree problem, which is a well-studied hard combinatorial search problem. On many...

. 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 MAX-SAT (finding a model that maximizes the sum of the weights of the satisfied clauses that make up a problem instance). We generalize a local-search algorithm for satisfiability to handle weighted MAX-SAT. To demonstrate the effectiveness of our approach, we present experimental results on encodings of a set of well-studied network Steiner-tree problems. This approach turns out to be competitive with some of the best current specialized algorithms developed in operations research. 1. Introduction Traditi...

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Abstract: In this paper, we present the implementation of a branch-and-cut algorithm for solving Steiner tree problems in graphs. Our algorithm is based on an integer programming formulation for directed graphs and comprises preprocessing, separation algorithms, and primal heuristics. We are able to solve nearly all problem instances discussed in the literature to optimality, including one problem that—to our knowledge—has not yet been solved. We also report on our computational experiences with some very large Steiner tree problems arising from the design of electronic circuits. All test problems are gathered in a newly introduced library called SteinLib that is accessible via the World Wide Web. � 1998 John

Cutting plane algorithms have turned out to be practically successful computational tools in combinatorial optimization, in particular, when they are embedded in a branch and bound framework. Implementations of such "branch and cut" algorithms are rather complicated in comparison to many purely combinatorial algorithms. The purpose of this article is to give an introduction to cutting plane algorithms from an implementor's point of view. Special emphasis is given to control and data structures used in practically successful implementations of branch and cut algorithms. We also address the issue of parallelization. Finally, we point out that in important applications branch and cut algorithms are not only able to produce optimal solutions but also approximations to the optimum with certified good quality in moderate computation times. We close with an overview of successful practical applications in the literature.

A new Genetic Algorithm (GA) for the Steiner Problem in a Graph (SPG) is presented. The algorithm is based on a bit-string 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 OR-Library 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 branch-and-cut algorithms and one of the very best deterministic heuristics, an iterated version of the Shortest Path Heuristi...

Cutting plane methods require the solution of a sequence of linear programs, where the solution to one provides a warm start to the next. A cutting plane algorithm for solving the linear ordering problem is described. This algorithm uses the primal-dual interior point method to solve the linear programming relaxations. A point which isagoodwarm start for a simplex-based cutting plane algorithm is generally not a good starting point for an interior point method. Techniques used to improve the warm start include attempting to identify cutting planes early and storing an old feasible point, which is used to help recenter when cutting planes are added. Computational results are described for some real-world problems; the algorithm appears to be competitive with a simplex-based cutting plane algorithm.

. This paper presents a computational study of a network design problem arising in the telecommunication industry. The objective can be formulated as that of finding an optimal degree constrained Steiner tree in a graph whose nodes and edges are weighted by costs. We develop a probabilistic Tabu Search heuristic for this problem, addressing issues of move evaluations, error correction, tabu memories, candidate list strategies and intensification strategies based on elite solution recovery. Computational results for a test set of difficult problem instances show that the new heuristic yields optimal solutions for all problems that could be solved by exact algorithms, while requiring only a fraction of the solution time. In addition, for larger and more realistic sized problems, which the exact methods were unable to solve, computational results show our method also outperforms the best local search heuristic currently available. Key words. Steiner Tree-Star, Tabu Search, Telecommunicat...

Abstract. The Prize-Collecting Steiner Tree Problem (PCST) on a graph with edge costs and vertex profits asks for a subtree minimizing the sum of the total cost of all edges in the subtree plus the total profit of all vertices not contained in the subtree. PCST appears frequently in the design of utility networks where profit generating customers and the network connecting them have to be chosen in the most profitable way. Our main contribution is the formulation and implementation of a branch-and-cut algorithm based on a directed graph model where we combine several state-of-the-art methods previously used for the Steiner tree problem. Our method outperforms the previously published results on the standard benchmark set of problems. We can solve all benchmark instances from the literature to optimality, including some of them for which the optimum was not known. Compared to a recent algorithm by Lucena and Resende, our new method is faster by more than two orders of magnitude. We also introduce a new class of more challenging instances and present computational results for them. Finally, for a set of large-scale real-world instances arising in the design of fiber optic networks, we also obtain optimal solution values. Keywords: Branch-and-Cut – Steiner Arborescence – Prize Collecting – Network Design 1