Average-case completeness of a word problem for groups

This paper presents a bounded word problem for groups whose random instances are hard on average. The problem is to decide, when given a finitely presented group G, words ~, y, z, and an integer k, whether (Z–lyZ)Z can be obtained from Z(x–l yz) in G using elementary Tietze transformations for at most k times. We show that the problem is average-case NP-complete when instances are chosen at random under a uniform distribution. NP-completeness of more general bounded word problems for groups is shown as well.

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