Abstract
We give a constant factor approximation algorithm for the Asymmetric Traveling Salesman Problem on shortest path metrics of directed graphs with two different edge weights. For the case of unit edge weights, the first constant factor approximation was given recently in [17]. This was accomplished by introducing an easier problem called Local-Connectivity ATSP and showing that a good solution to this problem can be used to obtain a constant factor approximation for ATSP. In this paper, we solve Local-Connectivity ATSP for two different edge weights. The solution is based on a flow decomposition theorem for solutions of the Held-Karp relaxation, which may be of independent interest.
Please refer to the full version (http://arxiv.org/abs/1511.07038) for proofs and more detailed explanations (with figures).
O. Svensson—Supported by ERC Starting Grant 335288-OptApprox.
L.A. Végh—Supported by EPSRC First Grant EP/M02797X/1.
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Notes
- 1.
An estimation algorithm is a polynomial-time algorithm for approximating/estimating the optimal value without necessarily finding a solution to the problem.
- 2.
For ATSP, we can think of a node-weighted graph as an edge-weighted graph where the weight of an edge (u, v) equals the node weight of u.
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Svensson, O., Tarnawski, J., Végh, L.A. (2016). Constant Factor Approximation for ATSP with Two Edge Weights. In: Louveaux, Q., Skutella, M. (eds) Integer Programming and Combinatorial Optimization. IPCO 2016. Lecture Notes in Computer Science(), vol 9682. Springer, Cham. https://doi.org/10.1007/978-3-319-33461-5_19
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