, Volume 56, Issue 4, pp 577–604 | Cite as

2 2 Spreading Metrics for Vertex Ordering Problems

  • Moses Charikar
  • Mohammad Taghi Hajiaghayi
  • Howard KarloffEmail author
  • Satish Rao


We design approximation algorithms for the vertex ordering problems Minimum Linear Arrangement, Minimum Containing Interval Graph, and Minimum Storage-Time Product, achieving approximation factors of \(O(\sqrt{\log n}\log\log n)\) , \(O(\sqrt{\log n}\log\log n)\) , and \(O(\sqrt{\log T}\log\log T)\) , respectively, the last running in time polynomial in T (T being the sum of execution times). The technical contribution of our paper is to introduce “ 2 2 spreading metrics” (that can be computed by semidefinite programming) as relaxations for both undirected and directed “permutation metrics,” which are induced by permutations of {1,2,…,n}. The techniques introduced in the recent work of Arora, Rao and Vazirani (Proc. of 36th STOC, pp. 222–231, 2004) can be adapted to exploit the geometry of such 2 2 spreading metrics, giving a powerful tool for the design of divide-and-conquer algorithms. In addition to their applications to approximation algorithms, the study of such 2 2 spreading metrics as relaxations of permutation metrics is interesting in its own right. We show how our results imply that, in a certain sense we make precise, 2 2 spreading metrics approximate permutation metrics on n points to a factor of \(O(\sqrt{\log n}\log\log n)\) .


Semidefinite programming Minimum linear arrangement Minimum containing interval graph Minimum storage-time product Approximation algorithm Vertex ordering problem 


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Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • Moses Charikar
    • 1
  • Mohammad Taghi Hajiaghayi
    • 2
  • Howard Karloff
    • 2
    Email author
  • Satish Rao
    • 3
  1. 1.Dept. of Computer Science, Princeton UniversityPrincetonUSA
  2. 2.Algorithms and OptimizationAT&T Labs–ResearchFlorham ParkUSA
  3. 3.UC BerkeleyBerkeleyUSA

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