Approximation Algorithms for Asymmetric TSP by Decomposing Directed Regular Multigraphs

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 n2 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 n2 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 cov...

• Type: Conference
• Year: 2003
• Where: FOCS
• Authors: Haim Kaplan, Moshe Lewenstein, Nira Shafrir, Maxim Sviridenko
• Keywords: 2-cycle Occurs, Cycle Covers, FOCS 2003, Simpler Algorithm, Theoretical Computer Science