Asymmetric Traveling Salesman Path and Directed Latency Problems

📝 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.

💡 Analysis

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.

📄 Content

arXiv:0907.0726v2 [cs.DS] 1 Jun 2010 Asymmetric Traveling Salesman Path and Directed Latency Problems∗ Zachary Friggstad† Mohammad R. Salavatipour‡ Zoya Svitkina§ Department of Computing Science University of Alberta Edmonton, Alberta T6G 2E8, Canada June 1, 2010 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) [7,9] until the very recent result that improves it to O(log n/ log log n) [3, 9]. However, only a bound of O(√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(n1/2+ǫ), for any constant ǫ > 0 [25]. 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 in [7]. 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. 1 Introduction Let G = (V, E) be a complete directed graph on a set of n nodes and let d : E →R+ be a cost function satisfying the directed triangle inequality duw ≤duv + dvw for all u, v, w ∈V . However, d is not necessarily symmetric: it may be that duv ̸= dvu for some nodes u, v ∈V . In the metric Asymmetric Traveling Salesman Path Problem (ATSPP), we are also given two distinct nodes s, t ∈V . The goal is to find a path s = v1, v2, . . . , vn = t that visits all the nodes in V while minimizing the sum Pn−1 j=1 dvjvj+1. ATSPP can be used to model scenarios such as minimizing the ∗A preliminary version of this paper appeared in the Proceedings of 21st Annual ACM-SIAM Symposium on Discrete Algorithms †Supported by NSERC and iCORE scholarships ‡Supported by NSERC and an Alberta Ingenuity New Faculty award §Supported by Alberta Ingenuity 1 total cost of travel for a person trying to visit a set of cities on the way from a starting point to a destination. This is a variant of the classical Asymmetric Traveling Salesman Problem (ATSP), where the goal is to find a minimum-cost cycle visiting all nodes. In the k-person ATSPP, given an integer k ≥1, the goal is to find k paths from s to t such that every node is contained in at least one path and the sum of path lengths is minimized. Related to ATSPP is the directed latency problem. On the same input, the goal is to find a path s = v1, v2, . . . , vn = t that minimizes the sum of latencies of the nodes. Here, the latency of node vi in the path is defined as Pi−1 j=1 dvjvj+1. The objective can be thought of as minimizing the total waiting time of clients or the average response time. There are possible variations in the problem definition, such as asking for a cycle instead of a path, or specifying only s but not t, but they easily reduce to the version that we consider. Other names used in the literature for this problem are the deliveryman problem [23] and the traveling repairman problem [1]. 1.1 Related work Both ATSPP and the directed latency problem are closely related to the classical Traveling Sales- man Problem (TSP), which asks to find the cheapest Hamiltonian cycle in a complete undirected graph with edge costs [15, 21]. In general weighted graphs, TSP is not approximable. However, in most practical settings it can be assumed that edge costs satisfy the triangle inequality (i.e. duw ≤duv +dvw). Though metric TSP is still NP-hard, the well-known algorithm of Christofides [8] has an approximation ratio of 3/2. Later the analysis in [27, 29] showed that this approximation algorithm actually bounds the integrality gap of a linear programming relaxation for TSP known as the Held-Karp LP. This integrality gap is also known to be at least 4/3. Furthermore, for all ǫ > 0, approximating TSP within a factor of 220/219 −ǫ is NP-hard [26]. Christofides’ heuristic was adapted to the problem of finding the cheapest Hamiltonian path in a metric graph with an approximation guarantee of 3/2 if at most one endpoint is specified or 5/3 if both endpoints are given [16]. In contrast to TSP, no constant-factor approximation for its asymmetric version is known. The current best approximation for ATSP is the very recent result of Asadpour et al. [3], which gives an O(log n/ log log n)-approximation algorithm. It also upper-bounds the integrali

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