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University of Rochester. Computer Science Department.
Abstract
Heuristic approaches often do so well that they seem to pretty much
always give the right answer. How close can heuristic algorithms get
to always giving the right answer, without inducing seismic
complexity-theoretic consequences? This article first discusses how a
series of results by Berman, Buhrman, Hartmanis, Homer, Longpr\'{e},
Ogiwara, Sch\"{o}ning, and Watanabe, from the early 1970s through the
early 1990s, explicitly or implicitly limited how well heuristic
algorithms can do on NP-hard problems. In particular, many desirable
levels of heuristic success cannot be obtained unless severe, highly
unlikely complexity class collapses occur. Second, we survey work
initiated by Goldreich and Wigderson, who showed how under plausible
assumptions deterministic heuristics for randomized computation can
achieve a very high frequency of correctness. Finally, we consider
formal ways in which theory can help explain the effectiveness of
heuristics that solve NP-hard problems in practice
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