### Approximation algorithms for asymmetric TSP by decomposing directed regular multigraphsApproximation algorithms for asymmetric TSP by decomposing directed regular multigraphs

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 Author Kaplan, Haim ♦ Lewenstein, Moshe ♦ Shafrir, Nira ♦ Sviridenko, Maxim Source ACM Digital Library Content type Text Publisher Association for Computing Machinery (ACM) File Format PDF Copyright Year ©2005 Language English
 Subject Domain (in DDC) Computer science, information & general works ♦ Data processing & computer science Subject Keyword Approximation algorithms Abstract A directed multigraph is said to be $\textit{d}-regular$ if the indegree and outdegree of every vertex is exactly $\textit{d}.$ 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 $\textit{d}-regular$ multigraph does not contain more than $⌊\textit{d/2}⌋$ 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 log2 $\textit{n}$ larger than optimal. This improves upon a previous approximation algorithm with approximation guarantee of 0.999 log2 $\textit{n}.$ 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). ISSN 00045411 Age Range 18 to 22 years ♦ above 22 year Educational Use Research Education Level UG and PG Learning Resource Type Article Publisher Date 2005-07-01 Publisher Place New York e-ISSN 1557735X Journal Journal of the ACM (JACM) Volume Number 52 Issue Number 4 Page Count 25 Starting Page 602 Ending Page 626

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Source: ACM Digital Library