Improved Approximations for TSP with Simple Precedence Constraints

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In this paper, we consider variants of the traveling salesman problem with precedence constraints. We characterize hard input instances for Christofides’ algorithm and Hoogeveen’s algorithm by relating the two underlying problems, i. e., the traveling salesman problem and the problem of finding a minimum-weight Hamiltonian path between two prespecified vertices. We show that the sets of metric worst-case instances for both algorithms are disjoint in the following sense. There is an algorithm that, for any input instance, either finds a Hamiltonian tour that is significantly better than 1.5-approximative or a set of Hamiltonian paths between all pairs of endpoints, all of which are significantly better than 5/3-approximative.

In the second part of the paper, we give improved algorithms for the ordered TSP, i. e., the TSP, where the precedence constraints are such that a given subset of vertices has to be visited in some prescribed linear order. For the metric case, we present an algorithm that guarantees an approximation ratio of 2.5 − 2/k, where k is the number of ordered vertices. For near-metric input instances satisfying a β-relaxed triangle inequality, we improve the best previously known ratio to \(k\beta^{\log_2 (3k-3)}\) .

The research was partially funded by the ANR-projects “ALADDIN” and “IDEA”, the INRIA project “CEPAGE”, and by SNF grant 200021-109252.