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 0(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 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 a 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 and minimum TSP problems. For maximum TSP, we obtain a tour whose weight is at least 2/3 of the weight of the longest tour, improving a previous 5/8 approximation. For minimum TSP we obtain a tour whose weight is at most 0.842log₂ n times the optimal, improving a previous 0.999log₂ n approximation. 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. Other applications of the rounding procedure are approximation algorithms for maximum 3-cycle cover (factor 2/3, previously 3/5) and maximum asymmetric TSP with triangle inequality (factor 10/13, previously 3/4 ).