Average time analysis of simplified Davis-Putnam procedures

Goldberg, Allen J.; Purdom, Paul W.; Brown, Cynthia A.

doi:10.1016/0020-0190(82)90110-7

Cited by 80 publications

(27 citation statements)

References 3 publications

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“…If q=q(n) approaches C } n for a constant C the following results are known about Form n (q, 3): The average size of the whole backtracking tree of a simplified Davis Putnam procedure is exponential in n [2,9]. (There are different families of input spaces where this size is polynomial [10].) For almost all unsatisfiable formulas from Form n (q, 3) the length of the shortest resolution proof is exponential, i.e, (1+=) n for a fixed =>0 [5].…”

Section: Introductionmentioning

confidence: 98%

A Threshold for Unsatisfiability

Goerdt

1996

Journal of Computer and System Sciences

197

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A propositional formula is in 2-CNF (2-conjunctive normalform) iff it is the conjunction of clauses each of which has exactly two literals. We show: If C=1+=, where =>0 is fixed and q(n) C } n, then almost all formulas in 2-CNF with q(n) different clauses, where n is the number of variables, are unsatisfiable. If C=1&= and q(n) C } n, then almost all formulas with q(n) clauses are satisfiable. By``almost all'' we mean that the probability of the set of unsatisfiable or satisfiable formulas among all formulas with q(n) clauses approaches 1 as n Ä . So C=1 gives us a threshold separating satisfiability and unsatisfiability of formulas in 2-CNF in a probabilistic, asymptotic sense. To prove our result we translate the satisfiability problem for formulas in 2-CNF into a graph theoretical question. Then we apply techniques from the theory of random graphs. ]

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Section: Introductionmentioning

confidence: 98%

A Threshold for Unsatisfiability

Goerdt

1996

Journal of Computer and System Sciences

197

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“…A historically important example is the Boolean satisfiability where each clause is generated by selecting literals with some fixed probability. Goldberg introduced this random ensemble and showed that the average running time of the Davis-Putnam algorithm [DP60,DLL62] is polynomial for almost all choices of parameter settings [Gol79,GPB82]. Thus in the eighties some computer scientist tended to think that all the NP-complete problems are in fact on average easy and it is hard to find the evil instances which makes them NP-complete.…”

Section: The Average Case Hardnessmentioning

confidence: 99%

Statistical physics of hard optimization problems

Zdeborová¹

2009

Acta Physica Slovaca. Reviews and Tutorials

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Statistical physics of hard optimization problemsOptimization is fundamental in many areas of science, from computer science and information theory to engineering and statistical physics, as well as to biology or social sciences. It typically involves a large number of variables and a cost function depending on these variables. Optimization problems in the non-deterministic polynomial (NP)-complete class are particularly difficult, it is believed that the number of operations required to minimize the cost function is in the most difficult cases exponential in the system size. However, even in an NP-complete problem the practically arising instances might, in fact, be easy to solve. The principal question we address in this article is: How to recognize if an NP-complete constraint satisfaction problem is typically hard and what are the main reasons for this? We adopt approaches from the statistical physics of disordered systems, in particular the cavity method developed originally to describe glassy systems. We describe new properties of the space of solutions in two of the most studied constraint satisfaction problems - random satisfiability and random graph coloring. We suggest a relation between the existence of the so-called frozen variables and the algorithmic hardness of a problem. Based on these insights, we introduce a new class of problems which we named "locked" constraint satisfaction, where the statistical description is easily solvable, but from the algorithmic point of view they are even more challenging than the canonical satisfiability.

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“…This includes the "unbiased" sample space when p = 1 3 and all expressions are equally likely. Later work [49] showed the same average-case complexity even if pure literals are handled as well. The scientific and engineering communities are often interested in proofs of unsatisfiability, but Goldberg made no mention of the frequency of unsatisfiable random CNF expressions over the parameter space of that distribution.…”

Section: A History Of Probabilistic Resultsmentioning

confidence: 71%