Unknown solution length problems with no asymptotically optimal run time

Benjamin Doerr, Carola Doerr, Timo Kötzing

Abstract: We revisit the problem of optimizing a fitness function of unknown dimension; that is, we face a function defined over bit-strings of large length N, but only n ≪ N of them have an influence on the fitness. Neither the position of these relevant bits nor their number is known. In previous work, variants of the (1 + 1) evolutionary algorithm (EA) have been developed that solve, for arbitrary s ∈ ℕ, such OneMax and LeadingOnes instances, simultaneously for all n ∈ ℕ, in expected time O(n(log(n))2 log log(n) ... log(s−1)(n)(log(s)(n))1+ε) and O(n2 log(n) log log(n) ... log(s−1)(n)(log(s)(n))1+ε), respectively; that is, in almost the same time as if n and the relevant bit positions were known. In this work, we prove the first, almost matching, lower bounds for this setting. For LeadingOnes, we show that, for every s ∈ ℕ, the (1 + 1) EA with any mutation operator treating zeros and ones equally has an expected run time of ω(n2 log(n) log log(n) ... log(s)(n)) when facing problem size n. Aiming at closing the small remaining gap, we realize that, quite surprisingly, there is no asymptotically best performance. For any algorithm solving, for all n, all instances of size n in expected time at most T(n), there is an algorithm doing the same in time T'(n) with T' = o(T). For OneMax we show results of similar flavor.