On Two Segmentation Problems

Noga Alon, Benny Sudakov

Research output: Contribution to journalArticle

15 Scopus citations

Abstract

The hypercube segmentation problem is the following: Given a set S of m vertices of the discrete d-dimensional cube {0,1}d, find k vertices P1,..., Pk, Pi ∈ {0, 1}d and a partitions of S into k segments S1,..., Sk so as to maximize the sum ∑ki=1c∈Si Pi⊙c, where ⊙ is the overlap operator between two vertices of the d-dimensional cube, defined to be the number of positions they have in common. This problem (among other ones) was considered by Kleinberg, Papadimitriou, and Raghavan, where the authors designed a deterministic approximation algorithm that runs in polynomial time for every fixed k and produces a solution whose value is within 0.828 of the optimum, as well as a randomized algorithm that runs in linear time for every fixed k and produces a solution whose expected value is within 0.7 of the optimum. Here we design an improved approximation algorithm; for every fixed ε > 0 and every fixed k our algorithm produces in linear time a solution whose value is within (1 - ε) of the optimum. Therefore, for every fixed k, this is a polynomial approximation scheme for the problem. The algorithm is deterministic, but it is convenient to first describe it as a randomized algorithm and then to derandomize it using some properties of expander graphs. We also consider a segmented version of the minimum spanning tree problem, where we show that no approximation can be achieved in polynomial time, unless P = NP.

Original languageEnglish (US)
Pages (from-to)173-184
Number of pages12
JournalJournal of Algorithms
Volume33
Issue number1
DOIs
StatePublished - Oct 1999

All Science Journal Classification (ASJC) codes

  • Control and Optimization
  • Computational Mathematics
  • Computational Theory and Mathematics

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