Average-case non-approximability of optimisation problems
FCT'05 Proceedings of the 15th international conference on Fundamentals of Computation Theory
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We examine classes of distributional problemsdefined in terms of polynomial-time decision algorithms with bounded error probability. The class AvgP [5] has been characterized in [4] using polynomial-time algorithm schemes with benign faults. The class HP [4] extends AvgP by allowing malign faults instead of benign faults. The class AvgHP in turn extends HP by allowing running times to be polynomial on average instead of bounded by a polynomial. Polynomial-time algorithms that decide membership for words of length n with an error probability less than F(n) for some function F lead to the classes F(n)-ErrP [9]. The class Dist-Nearly-P = \cap _{k \geqslant 1} ({1 \mathord{\left/ {\vphantom {1 {n^k }}} \right. \kern-\nulldelimiterspace} {n^k }})-ErrP is an instance of the ýNearlyý-classes as introduced in [11]. We call a distribution µ fair if µ(n) 驴1/p(n) for some polynomial p(n). We prove: 1. The inclusion AvgP 驴 HP is strict. 2. AvgHP equals HP. 3. Problems from HP with fair distributions are in Dist-Nearly-P. One cannot substantially improve this result. There are problems in AvgP with fair distributions which are not in (1/g(n))-ErrP for every super-polynomial function g(n). And there are problems in AvgP with µ(n) = 1/g(n) for a super-polynomial function g(n) which are not in Dist-Nearly-P.