We consider some generalizations of the Asymmetric Traveling Salesman Path problem. Suppose we have an asymmetric metric G = (V,A) with two distinguished nodes s,t. We are also given a positive integer k. The goal is to find k paths of minimum total cost from s to t whose union spans all nodes. We call this the k-Person Asymmetric Traveling Salesmen Path problem (k-ATSPP). Our main result for k-ATSPP is a bicriteria approximation that, for some parameter b >= 1 we may choose, finds between k and k + k b paths of total length O(b log |V|) times the optimum value of an LP relaxation based on the Held-Karp relaxation for the Traveling Salesman problem. On one extreme this is an O(log |V|)-approximation that uses up to 2k paths and on the other it is an O(k log |V|)-approximation that uses exactly k paths. Next, we consider the case where we have k pairs of nodes (s_1,t_1), ..., (s_k,t_k). The goal is to find an s_i-t_i path for every pair such that each node of G lies on at least one of these paths. Simple approximation algorithms are presented for the special cases where the metric is symmetric or where s_i = t_i for each i. We also show that the problem can be approximated within a factor O(log n) when k=2. On the other hand, we demonstrate that the general problem cannot be approximated within any bounded ratio unless P = NP.
The variant of finding Hamiltonian paths in asymmetric metrics, namely ATSPP, has only recently been studied from the perspective of approximation algorithms. The first approximation algorithm was an @math -approximation by Lam and Newman @cite_17 . Following this, Chekuri and Pal @cite_2 brought the ratio down to @math . Finally, Feige and Singh @cite_9 proved that an @math -approximation for ATSP implies a @math -approximation for ATSPP for any constant @math . Combining their result with the recent ATSP algorithm in @cite_15 yields an @math -approximation for ATSPP.
@cite_15 @cite_9 @cite_2 @cite_17
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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 present a deterministic (1+√5 2)-approximation algorithm for the s-t path TSP for an arbitrary metric. Given a symmetric metric cost on @math vertices including two prespecified endpoints, the problem is to find a shortest Hamiltonian path between the two endpoints; Hoogeveen showed that the natural variant of Christofides' algorithm is a 5 3-approximation algorithm for this problem, and this asymptotically tight bound in fact had been the best approximation ratio known until now. We modify this algorithm so that it chooses the initial spanning tree based on an optimal solution to the Held-Karp relaxation rather than a minimum spanning tree; we prove this simple but crucial modification leads to an improved approximation ratio, surpassing the 20-year-old barrier set by the natural Christofides' algorithm variant. Our algorithm also proves an upper bound of 1+√5 2 on the integrality gap of the path-variant Held-Karp relaxation. The techniques devised in this paper can be applied to other optimization problems as well: these applications include improved approximation algorithms and improved LP integrality gap upper bounds for the prize-collecting s-t path problem and the unit-weight graphical metric s-t path TSP. In metric asymmetric traveling salesperson problems the input is a complete directed graph in which edge weights satisfy the triangle inequality, and one is required to find a minimum weight walk that visits all vertices. In the asymmetric traveling salesperson problem (ATSP) the walk is required to be cyclic. In asymmetric traveling salesperson path problem (ATSPP), the walk is required to start at vertex sand to end at vertex t. We improve the approximation ratio for ATSP from @math to @math . This improvement is based on a modification of the algorithm of [JACM 05] that achieved the previous best approximation ratio. We also show a reduction from ATSPP to ATSP that loses a factor of at most 2 + i¾?in the approximation ratio, where i¾?> 0 can be chosen to be arbitrarily small, and the running time of the reduction is polynomial for every fixed i¾?. Combined with our improved approximation ratio for ATSP, this establishes an approximation ratio of @math for ATSPP, improving over the previous best ratio of 4log e ni¾? 2.76log 2 nof Chekuri and Pal [Approx 2006]. We analyze approximation algorithms for several variants of the traveling salesman problem with multiple objective functions. First, we consider the symmetric TSP (STSP) with γ-triangle inequality. For this problem, we present a deterministic polynomial-time algorithm that achieves an approximation ratio of @math and a randomized approximation algorithm that achieves a ratio of @math . In particular, we obtain a 2+e approximation for multi-criteria metric STSP. Then we show that multi-criteria cycle cover problems admit fully polynomial-time randomized approximation schemes. Based on these schemes, we present randomized approximation algorithms for STSP with γ-triangle inequality (ratio @math ), asymmetric TSP (ATSP) with γ-triangle inequality (ratio @math ), STSP with weights one and two (ratio 4 3) and ATSP with weights one and two (ratio 3 2).