Approximation algorithms for asymmetric TSP by decomposing directed regular multigraphs

H. Kaplan, M. Lewenstein, N. Shafrir, M. Sviridenko

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

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 ).

Original languageEnglish
Title of host publicationProceedings - 44th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2003
PublisherIEEE Computer Society
Pages56-65
Number of pages10
ISBN (Electronic)0769520405
DOIs
StatePublished - 2003
Event44th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2003 - Cambridge, United States
Duration: 11 Oct 200314 Oct 2003

Publication series

NameProceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
Volume2003-January
ISSN (Print)0272-5428

Conference

Conference44th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2003
Country/TerritoryUnited States
CityCambridge
Period11/10/0314/10/03

Keywords

  • Algorithm design and analysis
  • Application software
  • Approximation algorithms
  • Biology computing
  • Computational biology
  • Computer applications
  • Computer science
  • Polynomials
  • Traveling salesman problems

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