New approximation techniques for some linear ordering problems

Satish Rao, Andrea Richa

Research output: Contribution to journalArticle

30 Citations (Scopus)

Abstract

We describe logarithmic approximation algorithms for the NP-hard graph optimization problems of minimum linear arrangement, minimum containing interval graph, and minimum storage-time product. This improves upon the best previous approximation bounds of Even, Naor, Rao, and Schieber [J. ACM, 47 (2000), pp. 585-616] for these problems by a factor of Ω(log log n). We use the lower bound provided by the volume W of a spreading metric for each of the ordering problems above (as defined by Even et al.) in order to find a solution with cost at most a logarithmic factor times W for these problems. We develop a divide-and-conquer strategy where the cost of a solution to a problem at a recursive level is C plus the cost of a solution to the subproblems at this level, and where the spreading metric volume on the subproblems is less than the original volume by Ω(C/ log n), ensuring that the resulting solution has cost O(log n) times the original spreading metric volume. We note that this is an existentially tight bound on the relationship between the spreading metric volume and the true optimal values for these problems. For planar graphs, we combine a structural theorem of Klein, Plotkin, and Rao [Proceedings of the 25th ACM Symposium on Theory of Computing, 1993, pp. 682-690] with our new recursion technique to show that the spreading metric cost volumes are within an O(log log n) factor of the cost of an optimal solution for the minimum linear arrangement, and the minimum containing interval graph problems.

Original languageEnglish (US)
Pages (from-to)388-404
Number of pages17
JournalSIAM Journal on Computing
Volume34
Issue number2
DOIs
StatePublished - 2005

Fingerprint

Linear Ordering Problem
Costs
Metric
Approximation
Interval Graphs
Arrangement
Logarithmic
Divide and conquer
Approximation algorithms
Best Approximation
Recursion
Planar graph
Approximation Algorithms
NP-complete problem
Optimal Solution
Lower bound
Optimization Problem
Computing
Graph in graph theory
Theorem

Keywords

  • Approximation algorithms
  • Interval graph completion
  • Minimum linear arrangement
  • Spreading metrics
  • Storage - time product

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computational Theory and Mathematics
  • Applied Mathematics

Cite this

New approximation techniques for some linear ordering problems. / Rao, Satish; Richa, Andrea.

In: SIAM Journal on Computing, Vol. 34, No. 2, 2005, p. 388-404.

Research output: Contribution to journalArticle

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