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.)
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Feige, U., Langberg, M., Nissim, K. (2000). On the Hardness of Approximating NP Witnesses. In: Jansen, K., Khuller, S. (eds) Approximation Algorithms for Combinatorial Optimization. APPROX 2000. Lecture Notes in Computer Science, vol 1913. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44436-X_13
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DOI: https://doi.org/10.1007/3-540-44436-X_13
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