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 inequality, we then show how to obtain from this pair of cycle covers a tour whose weight is at most 0.842 log 2   n  larger than optimal. This improves upon a previous approximation algorithm with approximation guarantee of 0.999 log 2   n . 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).
['Haim Kaplan', 'Moshe Lewenstein', 'Nira Shafrir', 'Maxim Sviridenko']
Approximation algorithms for asymmetric TSP by decomposing directed regular multigraphs