Asymmetric Traveling Salesman Path and Directed Latency Problems

TL;DR

This paper introduces improved approximation algorithms for the asymmetric traveling salesman path and directed latency problems, achieving an O(log n) ratio and bounding their LP relaxation integrality gaps, thus advancing understanding of these NP-hard problems.

Contribution

It presents a new O(log n) approximation algorithm for ATSPP and directed latency, and establishes bounds on the LP relaxation integrality gap, solving an open problem.

Findings

Approximation ratio for ATSPP improved to O(log n)

Bound on LP relaxation integrality gap for ATSPP established as O(log n)

New algorithms for k-person ATSPP and directed latency developed

Abstract

We study integrality gaps and approximability of two closely related problems on directed graphs. Given a set V of n nodes in an underlying asymmetric metric and two specified nodes s and t, both problems ask to find an s-t path visiting all other nodes. In the asymmetric traveling salesman path problem (ATSPP), the objective is to minimize the total cost of this path. In the directed latency problem, the objective is to minimize the sum of distances on this path from s to each node. Both of these problems are NP-hard. The best known approximation algorithms for ATSPP had ratio O(log n) until the very recent result that improves it to O(log n/ log log n). However, only a bound of O(sqrt(n)) for the integrality gap of its linear programming relaxation has been known. For directed latency, the best previously known approximation algorithm has a guarantee of O(n^(1/2+eps)), for any constant eps > 0. We present a new algorithm for the ATSPP problem that has an approximation ratio of O(log n), but whose analysis also bounds the integrality gap of the standard LP relaxation of ATSPP by the same factor. This solves an open problem posed by Chekuri and Pal [2007]. We then pursue a deeper study of this linear program and its variations, which leads to an algorithm for the k-person ATSPP (where k s-t paths of minimum total length are sought) and an O(log n)-approximation for the directed latency problem.