AF: Small: Looking Under Rocks: A Search for a Provably Stronger TSP Relaxation

Cornell University, Ithaca NY

Investigators

Abstract

The traveling salesman problem is one of the most widely known problems of computational difficulty. The goal of the problem is to find the cheapest route for a salesman to visit a number of cities and return to home. It is not known whether there is a means of finding the best possible tour short of enumerating the exponentially large number of possible tours. Widely used algorithms for finding the best possible solution involve computing lower bounds on the length of the tour, and the closer the bound is to cost of the best possible tour, the quicker these algorithms run. This project will investigate alternative lower bounds for the traveling salesman problem to the ones widely used in practice, in the hopes of finding a provably better lower bound. A better understanding of lower bounds for the traveling salesman problem should help with solving other computationally difficult problems. The traveling salesman problem is one that is easy to explain to undergraduates and high school students. The PI and his graduate student plan to use their research as a means of outreach to such students, and to involve undergraduates in their project. Current methods for computing the optimal solution to the traveling salesman problem involve repeatedly solving a well-known linear programming relaxation of the problem. Although this bound is extremely good in practice (that is, it is very close to value of an optimal solution), its worst-case behavior is not well understood, despite decades of research. This project intends to study alternative lower bounds for the traveling salesman problem: it will start by studying semidefinite programming relaxations of the problem, and also consider other constraints that might be added to the linear program in order to be able to prove stronger statements about its worst-case behavior than are currently known. In addition, the project will consider some variants of the traveling salesman problem that have not been as well-studied, including the circulant TSP; for this variant, we are not even sure if the problem is NP-hard or has a polynomial-time algorithm. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.