Polynomial time approximation schemes for dense instances of script N ℘-hard problems

Sanjeev Arora, David Karger, Marek Karpinski

Research output: Contribution to journalArticlepeer-review

148 Scopus citations


We present a unified framework for designing polynomial time approximation schemes (PTASs) for "dense" instances of many script N sign script P sign-hard optimization problems, including maximum cut, graph bisection, graph separation, minimum k-way cut with and without specified terminals, and maximum 3-satisfiability. By dense graphs we mean graphs with minimum degree Ω(n), although our algorithms solve most of these problems so long as the average degree is Ω(n). Denseness for non-graph problems is defined similarly. The unified framework begins with the idea of exhaustive sampling: picking a small random set of vertices, guessing where they go on the optimum solution, and then using their placement to determine the placement of everything else. The approach then develops into a PTAS for approximating certain smooth integer programs, where the objective function and the constraints are "dense" polynomials of constant degree.

Original languageEnglish (US)
Pages (from-to)193-210
Number of pages18
JournalJournal of Computer and System Sciences
Issue number1
StatePublished - Feb 1999

All Science Journal Classification (ASJC) codes

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


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