A hybrid algorithm is a collection of heuristics, paired with a polynomial time selector S that runs on the input to decide which heuristic should be executed to solve the problem. Hybrid algorithms are interesting in scenarios where the selector must decide between heuristics that are "good" with respect to different complexity measures.

In this paper, we focus on hybrid algorithms with a "hardness-defying" property: for a problem Pi, there is a set of complexity measures {m_i} whereby Pi is known or conjectured to be hard (or unsolvable) for each m_i , but for each heuristic h_i of the hybrid algorithm, one can give a complexity guarantee for h_i on the instances of Pi that S selects for h_i that is strictly better than m_i. For example, we show that for NP-hard problems such as Max-Ek-Lin-p, Longest Path and Minimum Bandwidth, a given instance can either be solved exactly in ``sub-exponential'' (2^o(n)) time, or be approximated in polynomial time with an approximation ratio exceeding that of the known or conjectured inapproximability of the problem, assuming P != NP. We also prove some inherent limitations to the design of hybrid algorithms that arise under the assumption that NP requires exponential time algorithms.

35 pages

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