9,905 research outputs found
An Atypical Survey of Typical-Case Heuristic Algorithms
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}ening, 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.Comment: This article is currently scheduled to appear in the December 2012
issue of SIGACT New
Faster exponential-time algorithms in graphs of bounded average degree
We first show that the Traveling Salesman Problem in an n-vertex graph with
average degree bounded by d can be solved in O*(2^{(1-\eps_d)n}) time and
exponential space for a constant \eps_d depending only on d, where the
O*-notation suppresses factors polynomial in the input size. Thus, we
generalize the recent results of Bjorklund et al. [TALG 2012] on graphs of
bounded degree.
Then, we move to the problem of counting perfect matchings in a graph. We
first present a simple algorithm for counting perfect matchings in an n-vertex
graph in O*(2^{n/2}) time and polynomial space; our algorithm matches the
complexity bounds of the algorithm of Bjorklund [SODA 2012], but relies on
inclusion-exclusion principle instead of algebraic transformations. Building
upon this result, we show that the number of perfect matchings in an n-vertex
graph with average degree bounded by d can be computed in
O*(2^{(1-\eps_{2d})n/2}) time and exponential space, where \eps_{2d} is the
constant obtained by us for the Traveling Salesman Problem in graphs of average
degree at most 2d.
Moreover we obtain a simple algorithm that counts the number of perfect
matchings in an n-vertex bipartite graph of average degree at most d in
O*(2^{(1-1/(3.55d))n/2}) time, improving and simplifying the recent result of
Izumi and Wadayama [FOCS 2012].Comment: 10 page
Narrow proofs may be maximally long
We prove that there are 3-CNF formulas over n variables that can be refuted in resolution in width w but require resolution proofs of size n(Omega(w)). This shows that the simple counting argument that any formula refutable in width w must have a proof in size n(O(w)) is essentially tight. Moreover, our lower bound generalizes to polynomial calculus resolution and Sherali-Adams, implying that the corresponding size upper bounds in terms of degree and rank are tight as well. The lower bound does not extend all the way to Lasserre, however, since we show that there the formulas we study have proofs of constant rank and size polynomial in both n and w.Peer ReviewedPostprint (author's final draft
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