A Local Search Algorithm for Large Maximum Weight Independent Set Problems

Motivated by a real-world vehicle routing application, we consider the maximum-weight independent set problem: Given a node-weighted graph, find a set of independent (mutually nonadjacent) nodes whose node-weight sum is maximum. Some of the graphs arising in the vehicle routing application are large, having hundreds of thousands of nodes and hundreds of millions of edges. To solve instances of this size, we develop a new local search algorithm, which is a metaheuristic based on the greedy randomized adaptive search (GRASP) framework. This algorithm, named METAMIS, uses a wider range of simple local search operations than previously described in the literature. We introduce data structures that make these operations efficient. A new variant of path-relinking is introduced to escape local optima and so is a new alternating augmenting-path local search move that improves algorithm performance. We compare an implementation of our algorithm with a state-of-the-art publicly available code on public benchmark sets, including some large instances. Our algorithm is, in general, competitive and outperforms this openly available code on large vehicle routing instances of the maximum weight independent set problem. We hope that our results will lead to even better maximum-weight independent set algorithms

Computing Approximate Solutions Of The Maximum Covering Problem With Grasp

. We consider the maximum covering problem, a combinatorial optimization problem that arises in many facility location problems. In this problem, a potential facility site covers a set of demand points. With each demand point, we associate a nonnegative weight. The task is to select a subset of p > 0 sites from the set of potential facility sites, such that the sum of weights of the covered demand points is maximized. We describe a greedy randomized adaptive search procedure (GRASP) for the maximum covering problem that finds good, though not necessarily optimum, placement configurations. We describe a well-known upper bound on the maximum coverage which can be computed by solving a linear program and show that on large instances, the GRASP can produce facility placements that are nearly optimal. 1. INTRODUCTION We consider the maximum covering problem (MCP) [11], a combinatorial optimization problem that has been applied to numerous facility location problems, including rural health c..

An Efficient Implementation of a Network Interior Point Method

. We describe dlnet, an implementation of the dual affine scaling algorithm for minimum cost capacitated network flow problems. The efficiency of this implementation is the result of three factors: the small number of iterations taken by interior point methods, efficient solution of the linear system that determines the ascent direction using a preconditioned conjugate gradient algorithm and strategies to produce an optimal primal integer solution. The combination of these ingredients results in a code that can solve minimum cost network flow problems having over 250,000 vertices in a few hours of running time on a workstation-class computer. We compare dlnet with network simplex code netflo and relaxation code relaxt-3 on an extensive range of minimum cost network flow problems, including minimum cost circulation, maximum flow and transshipment problems. The computational results show that dlnet offers more predictable running times than those of netflo and relaxt-3. Its performance,..

An Implementation Of The Dual Affine Scaling Algorithm For Minimum Cost Flow On Bipartite Uncapacitated Networks

. We describe an implementation of the dual affine scaling algorithm for linear programming specialized to solve minimum cost flow problems on bipartite uncapacitated networks. This implementation uses a preconditioned conjugate gradient algorithm to solve the system of linear equations that determines the search direction at each iteration of the interior point algorithm. Two preconditioners are considered: a diagonal preconditioner and a preconditioner based on the incidence matrix of an approximate maximum weighted spanning tree of the network. Under dual nondegeneracy, this spanning tree allows for early identification of the optimal solution. Applying an ffl-perturbation to the cost vector, an optimal extreme-point primal solution is produced in the presence of dual degeneracy. The implementation is tested by solving several large instances of randomly generated assignment problems, comparing solution times with the network simplex code netflo and the relaxation algorithm code re..

A Grasp For Satisfiability

A greedy randomized adaptive search procedure (Grasp) is a randomized heuristic that has been shown to quickly produce good quality solutions for a wide variety of combinatorial optimization problems. In this paper, we describe a Grasp for the satisfiability (SAT) problem. This algorithm can be also directly applied to both the weighted and unweighted versions of the maximum satisfiability (MAX-SAT) problem. We review basic concepts of Grasp: construction and local search algorithms. The implementation of Grasp for the SAT problem is described in detail. Computational experience on a large set of test problems is presented

Interior Point Algorithms For Network Flow Problems

. Computational algorithms for the solution of network flow problems are of great practical significance. In the last decade, a new class of computationally efficient algorithms, based on the interior point method, has been proposed and applied to solve large scale network flow problems. In this chapter, we review interior point approaches for network flows, with emphasis on computational issues. Key words. Network flow problems, interior point methods, computational testing, computer implementation. AMS(MOS) subject classifications. 90B10, 90C05, 90C06, 90C35, 65-05, 65F10, 65F50 1. Introduction. A large number of problems in transportation, communications, and manufacturing can be modeled as network flow problems. In these problems one seeks to find the most efficient, or optimal, way to move flow (e.g. materials, information, buses, electrical currents) on a network (e.g. postal network, computer network, transportation grid, power grid). Among these optimization problems, many a..