#

### Volume 2014

#### Article 1

Published by the Department of Computer Science, The University of Chicago.

####
Lift-and-Project Integrality Gaps for the Traveling Salesperson Problem

Thomas Watson

University of Toronto, Toronto ON

Canada

`thomasw AT cs DOT toronto DOT edu`

*March 28, 2014*
#### Abstract

We study the lift-and-project procedures of Lovasz-Schrijver and
Sherali-Adams applied to the standard linear programming relaxation of the
traveling salesperson problem with triangle inequality.
For the asymmetric TSP tour problem, Charikar, Goemans, and Karloff (FOCS 2004)
proved that the integrality gap of the standard relaxation is at least $2$.
We prove that after one round of the Lovasz-Schrijver or Sherali-Adams
procedures, the integrality gap of the asymmetric TSP tour problem is at least
$3/2$, with a caveat on which version of the standard relaxation is used. For
the symmetric TSP tour problem, the integrality gap of the standard relaxation
is known to be at least $4/3$, and Cheung (SIOPT 2005) proved that it remains
at least $4/3$ after $o(n)$ rounds of the Lovasz-Schrijver procedure,
where $n$ is the number of nodes. For the symmetric TSP path problem, the
integrality gap of the standard relaxation is known to be at least $3/2$,
and we prove that it remains at least $3/2$ after $o(n)$ rounds of the
Lovasz-Schrijver procedure, by a simple reduction to Cheung's result.

- The article:
**PDF** (474 KB)
- Source material: ZIP (317 KB)
**BibTeX** entry for this
article (286 bytes)

Submitted December 30, 2013, revised February 28, 2014, received in final form March 14, 2014, published March 28, 2014.

Volume 2013, Article 14
Article 2

Volume 2014
Published articles