Algorithmic Problems for Metrics on Permutation Groups

Abstract

Given a permutation group G ≤ S _n by a generating set, we explore MWP (the minimum weight problem) and SDP (the subgroup distance problem) for some natural metrics on permutations. These problems are know to be NP-hard. We study both exact and approximation versions of these problems. We summarize our main results:

• For our upper bound results we focus on the Hamming and the l _∞ permutation metrics. For the l _∞ metric, we give a randomized 2^O(n) time algorithm for finding an optimal solution to MWP. Interestingly, this algorithm adapts ideas from the Ajtai-Kumar-Sivakumar algorithm for the shortest vector problem in lattices [AKS01]. For the Hamming metric, we again give a 2^O(n) time algorithm for finding an optimal solution to MWP. This algorithm is based on the classical Schrier-Sims algorithm for finding pointwise stabilizer subgroups of permutation groups.

• It is known that SDP is NP-hard([BCW06]) and it easily follows that SDP is hard to approximate within a factor of log^O(1) n unless P = NP. In contrast, we show that SDP for approximation factor more than n/logn is not NP-hard unless there is an unlikely containment of complexity classes.

• For several permutation metrics, we show that the minimum weight problem is polynomial-time reducible to the subgroup distance problem for solvable permutation groups.