Tutte Colloquium - David P. Williamson

Title: Semidefinite Programming Relaxations of the Traveling Salesman Problem

Abstract:

Finding a polynomial-time solvable relaxation of the traveling salesman problem whose integrality gap better matches what is seen in practice has been an outstanding open problem in combinatorial optimization for some time.  We study several semidefinite programming relaxations of the traveling salesman problem proposed in the literature and show that all known relaxations have an unbounded integrality gap. To obtain our results, we search for feasible solutions within a highly structured class of matrices; the problem of finding such solutions reduces to finding feasible solutions for a related linear program, which we do analytically. The solutions we find imply the unbounded integrality gap. Further, they imply several corollaries that help us better understand the semidefinite program and its relationship to other TSP relaxations.

These results are joint work with Sam Gutekunst.