Paper by Erik D. Demaine

Reference:

Erik D. Demaine, Uriel Feige, MohammadTaghi Hajiaghayi, and Mohammad R. Salavatipour, “Combination Can Be Hard: Approximability of the Unique Coverage Problem”, SIAM Journal on Computing, volume 38, number 4, September 2008, pages 1464–1483.

BibTeX

@Article{UniqueCoverage_SICOMP,
  AUTHOR        = {Erik D. Demaine and Uriel Feige and MohammadTaghi Hajiaghayi
                   and Mohammad R. Salavatipour},
  TITLE         = {Combination Can Be Hard: Approximability of the Unique
                   Coverage Problem},
  JOURNAL       = {SIAM Journal on Computing},
  journalurl    = {http://www.siam.org/journals/sicomp/sicomp.htm},
  VOLUME        = 38,
  NUMBER        = 4,
  PAGES         = {1464--1483},
  MONTH         = {September},
  YEAR          = 2008,

  withstudent   = 1,
  papers        = {UniqueCoverage_SODA2006},
  replaces      = {UniqueCoverage_SODA2006},
  doi           = {https://dx.doi.org/10.1137/060656048},
  dblp          = {https://dblp.org/rec/journals/siamcomp/DemaineFHS08},
  comments      = {This paper is also available from <A HREF="http://dx.doi.org/10.1137/060656048">SIAM</A>.},
}

Abstract:

We prove semi-logarithmic inapproximability for a maximization problem called unique coverage: given a collection of sets, find a subcollection that maximizes the number of elements covered exactly once. Specifically, assuming that NP ⊈ BPTIME(2^nε) for an arbitrary ε > 0, we prove O(1/log^σ n) inapproximability for some constant σ = σ(ε). We also prove O(1/log^1/3−ε n) inapproximability, for any ε > 0, assuming that refuting random instances of 3SAT is hard on average; and prove O(1/log n) inapproximability under a plausible hypothesis concerning the hardness of another problem, balanced bipartite independent set. We establish an Ω(1/log n)-approximation algorithm, even for a more general (budgeted) setting, and obtain an Ω(1/log B)-approximation algorithm when every set has at most B elements. We also show that our inapproximability results extend to envy-free pricing, an important problem in computational economics. We describe how the (budgeted) unique coverage problem, motivated by real-world applications, has close connections to other theoretical problems including max cut, maximum coverage, and radio broadcasting.

Comments:

This paper is also available from SIAM.

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Related papers:

UniqueCoverage_SODA2006 (Combination Can Be Hard: Approximability of the Unique Coverage Problem)