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Towards Improving Christofides Algorithm for Half-Integer TSP

Abstract : We study the traveling salesman problem (TSP) in the case when the objective function of the subtour linear programming relaxation is minimized by a half-cycle point: xe∈{0,1/2,1} where the half-edges form a 2-factor and the 1-edges form a perfect matching. Such points are sufficient to resolve half-integer TSP in general and they have been conjectured to demonstrate the largest integrality gap for the subtour relaxation. For half-cycle points, the best-known approximation guarantee is 3/2 due to Christofides famous algorithm. Proving an integrality gap of α for the subtour relaxation is equivalent to showing that αx can be written as a convex combination of tours, where x is any feasible solution for this relaxation. To beat Christofides bound, our goal is to show that (2−ϵ)x can be written as a convex combination of tours for some positive constant ϵ. Let ye=2−ϵ when xe=1 and ye=3/4 when xe=1/2. As a first step towards this goal, our main result is to show that y can be written as a convex combination of tours. In other words, we show that we can save on 1-edges, which has several applications. Among them, it gives an alternative algorithm for the recently studied uniform cover problem. Our main new technique is a procedure to glue tours over proper 3-edge cuts that are tight with respect to x , thus reducing the problem to a base case in which such cuts do not occur.
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https://hal.archives-ouvertes.fr/hal-03049527
Contributor : Alantha Newman <>
Submitted on : Wednesday, December 9, 2020 - 9:34:47 PM
Last modification on : Thursday, January 7, 2021 - 12:51:21 PM

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  • HAL Id : hal-03049527, version 1

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Arash Haddadan, Alantha Newman. Towards Improving Christofides Algorithm for Half-Integer TSP. European Symposium on Algorithms, Sep 2019, Munich, Germany. ⟨hal-03049527⟩

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