On the hardness of approximating N P witnesses

  • Authors:

  • Uriel Feige;Michael Langberg;Kobbi Nissim

  • Affiliations:

  • -;-;-

  • Venue:
  • APPROX '00 Proceedings of the Third International Workshop on Approximation Algorithms for Combinatorial Optimization
  • Year:
  • 2000

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Abstract

The search version for NP-complete combinatorial optimization problems asks for finding a solution of optimal value. Such a solution is called a witness. We follow a recent paper by Kumar and Sivakumar, and study a relatively new notion of approximate solutions that ignores the value of a solution and instead considers its syntactic representation (under some standard encoding scheme). The results that we present are of a negative nature. We show that for many of the well known NP-complete problems (such as 3-SAT, CLIQUE, 3-COLORING, SET COVER) it is NP-hard to produce a solution whose Hamming distance from an optimal solution is substantially closer than what one would obtain by just taking a random solution. In fact, we have been able to show similar results for most of Karp's 21 original NP-complete problems. (At the moment, our results are not tight only for UNDIRECTED HAMILTONIAN CYCLE and FEEDBACK EDGE SET).