The Inapproximability of Non NP-hard Optimization Problems

Abstract

The inapproximability of non NP-hard optimization problems is investigated. Based on self-reducibility and approximation preserving reductions, it is shown that problems Log Dominating Set, Tournament Dominating Set and Rich Hypergraph Vertex Cover cannot be approximated to a constant ratio in polynomial time unless the corresponding NP-hard versions are also approximable in deterministic subexponential time. A direct connection is established between non NP-hard problems and a PCP characterization of NP. Reductions from the PCP characterization show that Log Clique is not approximable in polynomial time and Max Sparse SAT does not have a PTAS under the assumption that SAT cannot be solved in deterministic \( 2^{O(log n \sqrt n )} \)time and that NP \( \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \not\subset } \) DTIME\( (2^{O(n)} ) \).KeywordsPolynomial TimeConstant RatioVertex CoverMinimum Vertex CoverInapproximability ResultThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.