Consensus Patterns (Probably) Has no EPTAS

Given n length-L strings S = {s(1), ... , s(n)} over a constant size alphabet S together with an integer l, where l <= L, the objective of Consensus Patterns is to find a length-l string s, a substring t(i) of each si in S such that Sigma(for all i) d(t(i), s) is minimized. Here d(x, y) denotes the Hamming distance between the two strings x and y. Consensus Patterns admits a PTAS [Li et al., JCSS 2002] is fixed parameter tractable when parameterized by the objective function value [Marx, SICOMP 2008], and although it is a well-studied problem, improvement of the PTAS to an EPTAS seemed elusive. We prove that Consensus Patterns does not admit an EPTAS unless FPT=W[1], answering an open problem from [Fellows et al., STACS 2002, Combinatorica 2006]. To the best of our knowledge, Consensus Patterns is the first problem that admits a PTAS, and is fixed parameter tractable when parameterized by the value of the objective function but does not admit an EPTAS under plausible complexity assumptions. The proof of our hardness of approximation result combines parameterized reductions and gap preserving reductions in a novel manner.