An Improved Integrality Gap for Asymmetric TSP Paths

The Asymmetric Traveling Salesperson Path (ATSPP) problem is one where, given an asymmetric metric space (V,d) with specified vertices s and t, the goal is to find an s-t path of minimum length that visits all the vertices in V. This problem is closely related to the Asymmetric TSP (ATSP) problem, which seeks to find a tour (instead of an s-t path) visiting all the nodes: for ATSP, a ρ-approximation guarantee implies an O(ρ)-approximation for ATSPP. However, no such connection is known for the integrality gaps of the linear programming relxations for these problems: the current-best approximation algorithm for ATSPP is O(logn/loglogn), whereas the best bound on the integrality gap of the natural LP relaxation (the subtour elmination LP) for ATSPP is O(logn). In this paper, we close this gap, and improve the current best bound on the integrality gap from O(logn) to O(logn/loglogn). The resulting algorithm uses the structure of narrow s-t cuts in the LP solution to construct a (random) tree witnessing this integrality gap. We also give a simpler family of instances showing the integrality gap of this LP is at least 2.

[1]  Alexander Schrijver,et al.  Combinatorial optimization. Polyhedra and efficiency. , 2003 .

[2]  Jan Vondrák,et al.  Dependent Randomized Rounding via Exchange Properties of Combinatorial Structures , 2010, 2010 IEEE 51st Annual Symposium on Foundations of Computer Science.

[3]  David B. Shmoys,et al.  Improving christofides' algorithm for the s-t path TSP , 2011, STOC '12.

[4]  Michel X. Goemans,et al.  On the Integrality Ratio for the Asymmetric Traveling Salesman Problem , 2006, Math. Oper. Res..

[5]  Jens Vygen,et al.  Shorter tours by nicer ears: 7/5-approximation for the graph-TSP, 3/2 for the path version, and 4/3 for two-edge-connected subgraphs , 2012, ArXiv.

[6]  David R. Karger,et al.  A new approach to the minimum cut problem , 1996, JACM.

[7]  M. R. Rao,et al.  Combinatorial Optimization , 1992, NATO ASI Series.

[8]  Aravind Srinivasan,et al.  Randomized Distributed Edge Coloring via an Extension of the Chernoff-Hoeffding Bounds , 1997, SIAM J. Comput..

[9]  R. Ravi,et al.  The Directed Minimum Latency Problem , 2008, APPROX-RANDOM.

[10]  Elliot Anshelevich,et al.  Path decomposition under a new cost measure with applications to optical network design , 2004, TALG.

[11]  Alantha Newman,et al.  Traveling salesman path problems , 2008, Math. Program..

[12]  Alan M. Frieze,et al.  On the worst-case performance of some algorithms for the asymmetric traveling salesman problem , 1982, Networks.

[13]  Amin Saberi,et al.  An O(log n/ log log n)-approximation algorithm for the asymmetric traveling salesman problem , 2010, SODA '10.

[14]  R. Ravi,et al.  Poly-logarithmic Approximation Algorithms for Directed Vehicle Routing Problems , 2007, APPROX-RANDOM.

[15]  Mohit Singh,et al.  Improved Approximation Ratios for Traveling Salesperson Tours and Paths in Directed Graphs , 2007, APPROX-RANDOM.

[16]  András Sebő.  Eight-Fifth approximation for the path TSP , 2013, IPCO 2013.

[17]  Moshe Lewenstein,et al.  Approximation algorithms for asymmetric TSP by decomposing directed regular multigraphs , 2005, JACM.

[18]  Amin Saberi,et al.  The asymmetric traveling salesman problem on graphs with bounded genus , 2009, SODA '11.

[19]  Chandra Chekuri,et al.  An O(log n) Approximation Ratio for the Asymmetric Traveling Salesman Path Problem , 2007, Theory Comput..

[20]  Jack Edmonds,et al.  Submodular Functions, Matroids, and Certain Polyhedra , 2001, Combinatorial Optimization.

[21]  Julia Chuzhoy,et al.  Approximation Algorithms for the Directed k-Tour and k-Stroll Problems , 2010, APPROX-RANDOM.

[22]  Chandra Chekuri,et al.  An O(logn) Approximation Ratio for the Asymmetric Traveling Salesman Path Problem , 2006, APPROX-RANDOM.

[23]  András Sebö,et al.  Eight-Fifth Approximation for the Path TSP , 2012, IPCO.

[24]  Markus Bläser,et al.  A new approximation algorithm for the asymmetric TSP with triangle inequality , 2003, TALG.

[25]  William J. Cook,et al.  Combinatorial optimization , 1997 .

[26]  Mohammad R. Salavatipour,et al.  Asymmetric traveling salesman path and directed latency problems , 2009, SODA '10.