We prove new integrality gap upper bounds for the traveling salesperson problem with distances one and two ((1,2)-TSP). We obtain these bounds by investigating the structure of the support graph of optimal solutions to the subtour elimination linear programming relaxation, with one additional cutting plane inequality.
For undirected (1,2)-TSP, our main results are as follows:
All instances have an integrality gap of at most 5/4.
Instances admitting half-integral solutions have integrality gap at most 7/6.
Instances admitting subcubic solutions Of cost at most the order of the instance have integrality gap at most 10/9, even without the cutting plane. This bound is tight, and holds in particular for basic solutions in the fractional 2-matching polytope with cost at most the order of the instance.
For directed (1,2)-TSP instances we show an integrality gap upper bound of 3/2 for general instances, and of 5/4 for instances admitting half-integral solutions.
We prove these bounds by providing local search algorithms that, in polynomial time, find 2-matchings with few components in the support of the solution. We show that the run times of our algorithms cannot be considerably improved under standard complexity-theoretic assumptions: we show that finding improved TSP solutions via local search is intractable for edge change parameterized by the size of the neighborhoods even for instances with distances one and two; this strengthens a result of Daniel Marx. (C) 2017 Elsevier B.V. All rights reserved.
- Traveling salesman
- Integrality gap
- Subtour elimination
- SALESMAN PROBLEM
- APPROXIMATION ALGORITHM
- WEIGHTS ZERO
- SUBTOUR LP