Typical case complexity of Satisfiability Algorithms and the threshold phenomenon

Franco, John

doi:10.1016/j.dam.2005.05.008

“…We believe that Friegut (Friegut 1999) was the first to prove the existence of a phase transition in boolean formulas. More details have been found experimentally, and surveys of this work have been given by Franco in Franco (2001) and Franco (2005). Our results do say something about boolean formulas, since a boolean formula in conjunctive normal form can be easily transformed into a quadratic system over GF(2) (see, for example, Håstad et al (1993)).…”

Section: Related Workmentioning

confidence: 69%

Phase transition of multivariate polynomial systems

Fusco¹,

Bach²

2009

Math. Struct. Comp. Sci.

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A random multivariate polynomial system with more equations than variables is likely to be unsolvable. On the other hand if there are more variables than equations, the system has at least one solution with high probability. In this paper we study in detail the phase transition between these two regimes, which occurs when the number of equations equals the number of variables. In particular the limiting probability for no solution is 1/e at the phase transition, over a prime field.We also study the probability of having exactly s solutions, with s ≥ 1. In particular, the probability of a unique solution is asymptotically 1/e if the number of equations equals the number of variables. The probability decreases very rapidly if the number of equations increases or decreases.Our motivation is that many cryptographic systems can be expressed as large multivariate polynomial systems (usually quadratic) over a finite field. Since decoding is unique, the solution of the system must also be unique. Knowing the probability of having exactly one solution may help us to understand more about these cryptographic systems. For example, whether attacks should be evaluated by trying them against random systems depends very much on the likelihood of a unique solution.

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“…The satisfiability problem is known to be NP-complete in general and for many restricted cases, for example see [5,6,7,8,11,19]. Finding the strongest possible restrictions under which the satisfiability problem remains NP-complete is important since this can make it easier to prove the NP-completeness of new problems by allowing easier reductions.…”

Section: Introductionmentioning

confidence: 99%

$(2/2/3)$-SAT problem and its applications in dominating set problems

Ahadi,

Dehghan

2016

Preprint

0

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The satisfiability problem is known to be NP-complete in general and for many restricted cases. One way to restrict instances of k-SAT is to limit the number of times a variable can be occurred. It was shown that for an instance of 4-SAT with the property that every variable appears in exactly 4 clauses (2 times negated and 2 times not negated), determining whether there is an assignment for variables such that every clause contains exactly two true variables and two false variables is NP-complete. In this work, we show that deciding the satisfiability of 3-SAT with the property that every variable appears in exactly four clauses (two times negated and two times not negated), and each clause contains at least two distinct variables is NP-complete. We call this problem (2/2/3)-SAT. For an r-regular graph G = (V, E) with r ≥ 3, it was asked in [Discrete Appl. Math., 160(15):2142-2146 to determine whether for a given independent set T there is an independent dominating set D that dominates T such that T ∩ D = ∅? As an application of (2/2/3)-SAT problem we show that for every r ≥ 3, this problem is NP-complete. Among other results, we study the relationship between 1-perfect codes and the incidence coloring of graphs and as another application of our complexity results, we prove that for a given cubic graph G deciding whether G is 4-incidence colorable is NP-complete.

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Phase Transition of Multivariate Polynomial Systems

Fusco

¹

,

Bach

²

Lecture Notes in Computer Science

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A random multivariate polynomial system with more equations than variables is likely to be unsolvable. On the other hand if there are more variables than equations, the system has at least one solution with high probability. In this paper we study in detail the phase transition between these two regimes, which occurs when the number of equations equals the number of variables. In particular the limiting probability for no solution is 1/e at the phase transition, over a prime field.We also study the probability of having exactly s solutions, with s ≥ 1. In particular, the probability of a unique solution is asymptotically 1/e if the number of equations equals the number of variables. The probability decreases very rapidly if the number of equations increases or decreases.Our motivation is that many cryptographic systems can be expressed as large multivariate polynomial systems (usually quadratic) over a finite field. Since decoding is unique, the solution of the system must also be unique. Knowing the probability of having exactly one solution may help us to understand more about these cryptographic systems. For example, whether attacks should be evaluated by trying them against random systems depends very much on the likelihood of a unique solution.

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Typical case complexity of Satisfiability Algorithms and the threshold phenomenon

Abstract: This is a written record of a survey talk on the topic of typical case complexity of Satisfiability algorithms which was presented at the LICS workshop on this subject in June, 2003, Ottawa, Canada.

Cited by 4 publications

References 67 publications

Phase transition of multivariate polynomial systems

Phase transition of multivariate polynomial systems

$(2/2/3)$-SAT problem and its applications in dominating set problems

Phase Transition of Multivariate Polynomial Systems

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