The Measure Hypothesis and Efficiency of Polynomial Time Approximation Schemes

A polyomial time approximation scheme for an optimization problem X is an algorithm A such that for each instance x of X and each ǫ > 0, A computes a (1 + ǫ)-approximate solution to instance x of X in time is O(|x|f(1/ǫ)) for some function f . If the running time of A is instead bounded by g(1/ǫ) · |x|O(1) for some function g, A is called an efficient polynomial time approximation scheme. PTAS denotes the class of all NP optimization problems for which a polytime approximation scheme exists, and EPTAS is the class of all such problems for which an efficient polytime approximation scheme exists. It is an open question whether P 6= NP implies the strictness of the inclusion EPTAS ⊆ PTAS. Bazgan [2] and independently Cesati and Trevisan [5] gave a separation under the stronger assumption FPT 6= W [P ]. In this paper we prove EPTAS ( PTAS under some different assumption, namely existence of NP search problems ΠR with a superpolynomial lower bound for the deterministic time complexity. This assumption is weaker than the NP Machine Hypothesis [15] and hence is implied by the Measure Hypothesis μp(NP ) 6= 0. Furthermore, using a sophisticated combinatorial counting argument we construct a recursive oracle under which our assumption holds but that of Cesati and Trevisan does not hold, implying that using relativizing proof techniques one cannot show that our assumption implies FPT 6= W [P ].