TY - JOUR

T1 - Approximation algorithms for asymmetric TSP by decomposing directed regular multigraphs

AU - Kaplan, Haim

AU - Lewenstein, Moshe

AU - Shafrir, Nira

AU - Sviridenko, Maxim

PY - 2005

Y1 - 2005

N2 - 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 [dβ2] 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 log2 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).

AB - 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 [dβ2] 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 log2 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).

KW - Approximation algorithms

UR - http://www.scopus.com/inward/record.url?scp=30544433248&partnerID=8YFLogxK

U2 - 10.1145/1082036.1082041

DO - 10.1145/1082036.1082041

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AN - SCOPUS:30544433248

SN - 0004-5411

VL - 52

SP - 602

EP - 626

JO - Journal of the ACM

JF - Journal of the ACM

IS - 4

ER -