Improved integrality gap upper bounds for TSP with distances one and two

We study the structure of solutions to linear programming formulations for the traveling salesperson problem (TSP). We perform a detailed analysis of the support of the subtour elimination linear programming relaxation, which leads to algorithms that find 2-matchings with few components in polynomial time. The number of components directly leads to integrality gap upper bounds for the TSP with distances one and two, for both undirected and directed graphs. Our main results concern the subtour elimination relaxation with one additional cutting plane inequality: - For undirected instances we obtain an integrality gap upper bound of 5/4 without any further restrictions, of 7/6 if the optimal LP solution is half-integral. - For instances of order n where the fractional LP value has a cost of n, we obtain a tight integrality gap upper bound of 10/9 if there is an optimal solution with subcubic support graph. The latter property that the graph is subcubic is implied if the solution is a basic solution in the fractional 2-matching polytope. - For directed instances we obtain an integrality gap upper bound of 3/2, and of 4/3 if given an optimal 1/2-integral solution. In the case of undirected graphs, we can avoid to add the cutting plane inequality if we accept slightly increased values. For the tight result, the cutting plane is not required. Additionally, we show that relying on the structure of the support is not an artefact of our algorithm, but is necessary under standard complexity-theoretic assumptions: we show that finding improved solutions via local search is W[1]-hard for k-edge change neighborhoods even for the TSP with distances one and two, which strengthens a result of D\'aniel Marx.Comment: 36 pages, 12 figure

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.Comment: Changes wrt previous version: upper bound on integrality gap improved to 5/4 (using the same techniques as in the previous version