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Title: A 4/3-Approximation Algorithm for Half-Integral Cycle Cut Instances of the TSP
A long-standing conjecture for the traveling salesman problem (TSP) states that the integrality gap of the standard linear programming relaxation of the TSP (sometimes called the Subtour LP or the Held-Karp bound) is at most 4/3 for symmetric instances of the TSP obeying the triangle inequality. In this paper we consider the half-integral case, in which a feasible solution to the LP has solution values in {0,1/2,1} . Karlin, Klein, and Oveis Gharan [9], in a breakthrough result, were able to show that in the half-integral case, the integrality gap is at most 1.49993; Gupta et al. [6] showed a slight improvement of this result to 1.4983. Both of these papers consider a hierarchy of critical tight sets in the support graph of the LP solution, in which some of the sets correspond to cycle cuts and the others to degree cuts. Here we show that if all the sets in the hierarchy correspond to cycle cuts, then we can find a distribution of tours whose expected cost is at most 4/3 times the value of the half-integral LP solution; sampling from the distribution gives us a randomized 4/3-approximation algorithm. We note that known bad cases for the integrality gap have a gap of 4/3 and have a half-integral LP solution in which all the critical tight sets in the hierarchy are cycle cuts; thus our result is tight.  more » « less
Award ID(s):
2007009
PAR ID:
10420897
Author(s) / Creator(s):
; ;
Editor(s):
Del Pia, Alberto; Kaibel, Volker
Date Published:
Journal Name:
Lecture notes in computer science
Volume:
13904
ISSN:
0302-9743
Page Range / eLocation ID:
217-230
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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