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.
We give a 7.18-approximation algorithm for the minimum latency problem that uses only @math calls to the prize-collecting Steiner tree (PCST) subroutine of Goemans and Williamson. This improves the previous best algorithms in both performance guarantee and running time. A previous algorithm of Goemans and Kleinberg for the minimum latency problem requires an approximation algorithm for the @math -minimum spanning tree ( @math -MST) problem which is called as a black box for each value of @math . Their algorithm can achieve an approximation factor of 10.77 while making @math PCST calls, a factor of 8.98 using @math PCST calls, or a factor of @math using @math PCST calls, via the @math -MST algorithms of Garg, Arya and Ramesh, and Arora and Karakostas, respectively. Here @math denotes the number of nodes in the instance, and @math is the largest edge cost in the input. In all cases, the running time is dominated by the PCST calls. Since the PCST subroutine can be implemented to run in @math time, the overall running time of our algorithm is @math . We also give a faster randomized version of our algorithm that achieves the same approximation guarantee in expectation, but uses only @math PCST calls, and derandomize it to obtain a deterministic algorithm with factor @math , using @math PCST calls. The basic idea for our improvement is that we do not treat the @math -MST algorithm as a black box. This allows us to take advantage of some special situations in which the PCST subroutine delivers a 2-approximate @math -MST. We are able to obtain the same approximation ratio that would be given by Goemans and Kleinberg if we had access to 2-approximate @math -MSTs for all values of @math , even though we have them only for some values of @math that we are not able to specify in advance. We also extend our algorithm to a weighted version of the minimum latency problem. We study the directed minimum latency problem: given an n-vertex asymmetric metric (V,d) with a root vertex ri¾? V, find a spanning path originating at rthat minimizes the sum of latencies at all vertices (the latency of any vertex vi¾? Vis the distance from rto valong the path). This problem has been well-studied on symmetric metrics, and the best known approximation guarantee is 3.59 [3]. For any @math O( n^ ^3 ) @math =O( n )$, which implies (for any fixed i¾?> 0) a polynomial time O(n1 2 + i¾?)-approximation algorithm for directed latency. In the special case of metrics induced by shortest-paths in an unweighted directed graph, we give an O(log2n) approximation algorithm. As a consequence, we also obtain an O(log2n) approximation algorithm for minimizing the weighted completion time in no-wait permutation flowshop scheduling. We note that even in unweighted directed graphs, the directed latency problem is at least as hard to approximate as the well-studied asymmetric traveling salesman problem, for which the best known approximation guarantee is O(logn). Given a tour visiting n points in a metric space, the latency of one of these points p is the distance traveled in the tour before reaching p. The minimum latency problem (MLP) asks for a tour passing through n given points for which the total latency of the n points is minimum; in effect, we are seeking the tour with minimum average arrival time. This problem has been studied in the operations research literature, where it has also been termed the delivery-man problem and the traveling repairman problem. The approximability of the MLP was first considered by Sahni and Gonzalez in 1976; however, unlike the classical traveling salesman problem (TSP), it is not easy to give any constant-factor approximation algorithm for the MLP. Recently, (A. Blum, P. Chalasani, D. Coppersimith, W. Pulleyblank, P. Raghavan, M. Sudan, Proceedings of the 26th ACM Symposium on the Theory of Computing, 1994, pp. 163-171) gave the first such algorithm, obtaining an approximation ratio of 144. In this work, we develop an algorithm which improves this ratio to 21.55; moreover, combining our algorithm with a recent result of Garg (N. Garg, Proceedings of the 37th IEEE Symposium on Foundations of Computer Science, 1996, pp. 302-309) provides an approximation ratio of 10.78. The development of our algorithm involves a number of techniques that seem to be of interest from the perspective of the TSP and its variants more generally. We are given a set of points @math and a symmetric distance matrix @math giving the distance between @math and @math . We wish to construct a tour that minimizes @math , where @math is the latency of @math , defined to be the distance traveled before first visiting @math . This problem is also known in the literature as the deliveryman problem or the traveling repairman problem . It arises in a number of applications including disk-head scheduling, and turns out to be surprisingly different from the traveling salesman problem in character. We give exact and approximate solutions to a number of cases, including a constant-factor approximation algorithm whenever the distance matrix satisfies the triangle inequality. We give improved approximation algorithms for a variety of latency minimization problems. In particular, we give a 3.59-approximation to the minimum latency problem, improving on previous algorithms by a multiplicative factor of 2. Our techniques also give similar improvements for related problems like k-traveling repairmen and its multiple depot variant. We also observe that standard techniques can be used to speed up the previous and this algorithm by a factor of O sup spl tilde (n).