The asymmetric traveling salesman path LP has constant integrality ratio

Abstract

We show that the classical LP relaxation of the asymmetric traveling salesman path problem (ATSPP) has constant integrality ratio. If ρ ATSP and ρ ATSPP denote the integrality ratios for the asymmetric TSP and its path version, then ρ ATSPP ≤ 4ρ ATSP − 3.We prove an even better bound for node-weighted instances: if the integrality ratio for ATSP on node-weighted instances is ρ NW ATSP , then the integrality ratio for ATSPP on node-weighted instances is at most 2ρ NW ATSP − 1. Moreover, we show that for ATSP node-weighted instances and unweighted digraph instances are almost equivalent. From this we deduce a lower bound of 2 on the integrality ratio of unweighted digraph instances.

A Constant-factor Approximation Algorithm for the Asymmetric Traveling Salesman Problem

“…We remark that we can obtain a tighter upper bound of 319 for the integrality gap of the Held-Karp relaxation, and that our results also imply a constant-factor approximation algorithm for the Asymmetric Traveling Salesman Path Problem via black-box reductions, given by Feige and Singh for the approximation guarantee [FS07] and recently by Köhne, Traub and Vygen [KTV19] for the integrality gap-see Section 11.…”

“…Note that the approximation ratio here is not bounded in terms of the Held-Karp lower bound 12 . Köhne, Traub and Vygen [KTV19] give a similar reduction as Feige and Singh, but for integrality gaps, with a loss of β → 4β − 3. Together with Theorem 11.1, this implies:…”

A constant-factor approximation algorithm for the asymmetric traveling salesman problem

“…Note that the approximation ratio here is not bounded in terms of the Held-Karp lower bound 11 . Köhne, Traub and Vygen [KTV18] give a similar reduction as Feige and Singh, but for integrality gaps, with a loss of β → 4β − 3. Together with Theorem 11.1, this implies:…”

A full description of the isolated calmness of semidefinite programming problem

“…For example, the inequality problem of bilinear matrix, optimal structure optimization, truss design problem, optimal robust control, and optimization problem in robust feedback control design in control system can be reduced to semidefinite programming problem. The references [1], [1][2], [3], [4], [5] give some applications of problem (1). Here we only provide two famous applications.…”