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

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## Abstract

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 30 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 et al., 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.

## Keywords

Traveling salesman problem Subtour elimination Linear programming Integrality gap## Mathematics Subject Classification

90C05 90C27 05C70## Notes

### Acknowledgments

We thank Sylvia Boyd for useful and encouraging discussions. We thank two anonymous referees for helpful comments and suggestions.

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