We consider the asymmetric traveling salesman problem for which the triangular inequality is satisfied. For various heuristics we construct examples to show that the worst-case ratio of length of tour found to minimum length tour is ~2 (n) for n city problems. We also provide a new 0 ([log2 nl ) heuristic.

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In this work we present a random pseudo-polynomial algorithm for the problem of finding a base of specified value in a weighted represented matroid, subject to parity conditions. We also describe a specialized version of the algorithm suitable for finding a base of specified value in the intersection of two matroids. This result generalizes an existing pseudo-polynomial algorithm for computing exact arborescences in weighted graphs. Another (simpler) specialized version of our algorithms is also presented for computing perfect matchings of specified value in weighted graphs. 6

In this paper we try to describe the main characters of Heuristics ‘derived’ from Nature, a border area between Operations Research and Artificial Intelligence, with applications to graph optimization problems. These algorithms take inspiration from physics, biology, social sciences, and use a certain amount of repeated trials, given by one or more ‘agents’ operating with a mechanism of competition‐cooperation. Two introductory sections, devoted respectively to a presentation of some general concepts and to a tentative classification of Heuristics from Nature open the work. The paper is then composed of six review sections: each of them concerns a heuristic and its application to an NP‐hard combinatorial optimization problem. We consider the following topics: genetic algorithms with timetable problems, simulated annealing with dial‐a‐ride problems, sampling and clustering with communication spanning tree problems, tabu search with job‐shop‐scheduling problems, neural nets with location problems, ant system with travelling salesman and quadratic assignment problems.

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We study the problem of designing at minimum cost a two-connected network such that the shortest cycle to which each edge belongs (a “mesh”) does not exceed a given length K. This problem arises in the design of fiber-optic-based backbone telecommunication networks. A Branch-and-Cut approach to this problem is presented for which we introduce several families of valid inequalities and discuss the corresponding separation algorithms. Because the size of the problems solvable to optimality by this approach is too small, we also develop some heuristics. The computational performances of these exact and approximate methods are then thoroughly assessed both on randomly generated instances as well as instances suggested by real applications.

We consider the k-CARD TREE problem, i.e., the problem of finding in a given undirected graph G a subtree with kedges, having minimum weight. Applications of this problem arise in oil-field leasing and facility layout. Although the general problem is shown to be strongly NP hard, it can be solved in polynomial time if G is itself a tree. We give an integer programming formulation of k-CARD TREE and an efficient exact separation routine for a set of generalized subtour elimination constraints. The polyhedral structure of the convex hull of the integer solutions is studied. 0 7994 by

The concept of reload cost, that is of a cost incurred when two consecutive arcs along a path are of different types, naturally arises in a variety of applications related to transportation, telecommunication, and energy networks. Previous work on reload costs is devoted to the problem of finding a spanning tree of minimum reload cost diameter (with no arc costs) or of minimum reload cost. In this article, we investigate the complexity and approximability of the problems of finding optimum paths, tours, and flows under a general cost model including reload costs as well as regular arc costs. Some of these problems, such as shortest paths and minimum cost flows, turn out to be polynomially solvable while others, such as minimum shortest path tree and minimum unsplittable multicommodity flows, are NP-hard to approximate within any polynomial-time computable function