TY - GEN

T1 - An improved integrality gap for asymmetric TSP paths

AU - Friggstad, Zachary

AU - Gupta, Anupam

AU - Singh, Mohit

PY - 2013

Y1 - 2013

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84875516616&partnerID=8YFLogxK

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U2 - 10.1007/978-3-642-36694-9_16

DO - 10.1007/978-3-642-36694-9_16

M3 - Conference contribution

AN - SCOPUS:84875516616

SN - 9783642366932

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 181

EP - 192

BT - Integer Programming and Combinatorial Optimization - 16th International Conference, IPCO 2013, Proceedings

T2 - 16th Conference on Integer Programming and Combinatorial Optimization, IPCO 2013

Y2 - 18 March 2013 through 20 March 2013

ER -