| Approximation Algorithms for Asymmetric TSP by Decomposing Directed Regular Multigraphs (2003) | |||||||||||||||
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| A directed multigraph is said to be d-regular if the indegree and outdegree of every vertex is exactly d. By Hall’s theorem one can represent such a multigraph as a combination of at most n 2 cycle covers each taken with an appropriate multiplicity. We prove that if the d-regular multigraph does not contain more than ⌊d/2 ⌋ copies of any 2-cycle then we can find a similar decomposition into n 2 pairs of cycle covers where each 2-cycle occurs in at most one component of each pair. Our proof is constructive and gives a polynomial algorithm to find such a decomposition. Since our applications only need one such a pair of cycle covers whose weight is at least the average weight of all pairs, we also give an alternative, simpler algorithm to extract a single such pair. This combinatorial theorem then comes handy in rounding a fractional solution of an LP relaxation of the maximum Traveling Salesman Problem (TSP) problem. The first stage of the rounding procedure obtains two cycle covers that do not share a 2-cycle with weight at least twice the weight of the optimal solution. Then we show how to extract a tour from the 2 cycle covers, whose weight is at least 2/3 of the weight of the longest tour. This improves upon the previous 5/8 approximation with a simpler algorithm. Utilizing a reduction from maximum TSP to the shortest superstring problem we obtain a 2.5-approximation algorithm for the latter problem which is again much simpler than the previous one. For minimum asymmetric TSP the same technique gives two cycle covers, not sharing a 2-cycle, with weight at most twice the weight of the optimum. Assuming triangle | |||||||||||||||
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