The hardness of approximate optima in lattices, codes, and systems of linear equations
Journal of Computer and System Sciences - Special issue: papers from the 32nd and 34th annual symposia on foundations of computer science, Oct. 2–4, 1991 and Nov. 3–5, 1993
Approximating shortest lattice vectors is not harder than approximating closet lattice vectors
Information Processing Letters
On the limits of nonapproximability of lattice problems
Journal of Computer and System Sciences - 30th annual ACM symposium on theory of computing
A sieve algorithm for the shortest lattice vector problem
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
Complexity of Lattice Problems
Complexity of Lattice Problems
Hardness of Approximating the Minimum Distance of a Linear Code
FOCS '99 Proceedings of the 40th Annual Symposium on Foundations of Computer Science
STOC '83 Proceedings of the fifteenth annual ACM symposium on Theory of computing
International Journal of Metaheuristics
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Given a permutation group G ≤ Sn 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 2O(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 2O(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 logO(1) n unless P = NP. In contrast, we show that SDP for approximation factor more than n/ log n 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.