2-Matchings, the Traveling Salesman Problem, and the Subtour LP: A Proof of the Boyd-Carr Conjecture

Abstract

, ankevanzuylen.com Determining the precise integrality gap for the subtour linear programming (LP) relaxation of the traveling salesman problem is a significant open question, with little progress made in thirty years in the general case of symmetric costs that obey triangle inequality. Boyd and Carr [Boyd S, Carr R (2011) Finding low cost TSP and 2-matching solutions using certain half-integer subtour vertices. Discrete Optim. 8:525-539. Prior version accessed June 27, 2011, http://www.site.uottawa .ca/~sylvia/recentpapers/halftri.pdf.] observe that we do not even know the worst-case upper bound on the ratio of the optimal 2-matching to the subtour LP; they conjecture the ratio is at most 10/9. In this paper, we prove the Boyd-Carr conjecture. In the case that the support of a fractional 2-matching has no cut edge, we can further prove that an optimal 2-matching has cost at most 10/9 times the cost of the fractional 2-matching.

On the integrality gap of the subtour LP for the 1,2-TSP

“…We prove an upper bound on the integrality gap for the subtour LP of 5 4 , which is the first bound on the integrality gap with value less than 4 3 for a natural class of TSP instances. Under an analog of a conjecture of Schalekamp et al [20], we show that the integrality gap is at most 7 6 , and with an additional assumption on the structure of the solution, we can improve this bound to 10 9 . We describe these results in more detail below.…”

“…Conjecture 1 (Schalekamp et al [20]) Let α n be the integrality gap of the subtour LP on n vertices. Then there exists an instance which has an optimal subtour LP solution that is an F2M and for which the optimal tour has cost at least α n times the subtour LP cost.…”

“…The integrality gap of the subtour LP for the 1,2-TSP is 10 9 . Schalekamp, Williamson, and van Zuylen [20] have conjectured that to determine the integrality gap for the subtour LP, we can restrict ourselves to considering instances, which have an optimal solution that is an extreme point of the F2M polytope.…”

Improved integrality gap upper bounds for traveling salesperson problems with distances one and two

“…For relevant special cases, the conjecture is known to be true [7,18]. It is also known that SER has a close relation to 2-matchings, as was pointed out for instance by Schalekamp et al [22] in the context of perfect 2-matchings.…”

The salesman’s improved tours for fundamental classes

“…Note that the 1-edges of 1 2 -integer points form a set of disjoint paths that we call 1-paths of x, and the 1 2 -edges form a set of edge-disjoint cycles we call the 1 2 -cycles of x. For Conjecture 1, it seems that 1 2 -integer vertices play an important role (see [1], [7], [20]). In fact it has been conjectured by Schalekamp, Williamson and van Zuylen [20] that a subclass of these 1 2 -integer vertices are the ones that give the largest gap.…”