Long Tours and Short Superstrings

This paper considers weight-maximizing variants of the classical symmetric and asymmetric travelingsalesman problems. Like their weight-minimizing counterparts, these variants are MAX SNP-hard. We present the first nontrivial approximation algorithms for these problems. Our algorithm for directed graphs finds a tour whose weight is at least 38 63 ≈ 0.603 times the weight of a maximum-weight tour, and our algorithm for undirected graphs finds a tour whose weight is at least 5 7 ≈ 0.714 times optimal. These bounds compare favorably with the 1 2 and 2 3 bounds that can be obtained for undirected and directed graphs, respectively, by simply deleting the minimum-weight edge from each cycle of a maximum-weight cycle cover. Our algorithm for directed graphs can be used to improve several recent approximation results for the shortestsuperstring problem.