# On the Hardness of Approximating NP Witnesses

## 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.)

## Keywords

HAMILTONIAN Cycle Vertex Cover Maximum Clique Hardness Result Satisfying Assignment## Preview

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