On the Integrality Gap of the Subtour LP for the 1,2-TSP

In this paper, we study the integrality gap of the subtour LP relaxation for the traveling salesman problem in the special case when all edge costs are either 1 or 2. For the general case of symmetric costs that obey triangle inequality, a famous conjecture is that the integrality gap is 4/3. Little progress towards resolving this conjecture has been made in thirty years. We conjecture that when all edge costs $c_{ij}\in \{1,2\}$, the integrality gap is $10/9$. We show that this conjecture is true when the optimal subtour LP solution has a certain structure. Under a weaker assumption, which is an analog of a recent conjecture by Schalekamp, Williamson and van Zuylen, we show that the integrality gap is at most $7/6$. When we do not make any assumptions on the structure of the optimal subtour LP solution, we can show that integrality gap is at most $5/4$; this is the first bound on the integrality gap of the subtour LP strictly less than $4/3$ known for an interesting special case of the TSP. We show computationally that the integrality gap is at most $10/9$ for all instances with at most 12 cities.