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.

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

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

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