A probabilistic algorithm for k-SAT and constraint satisfaction problems (original) (raw)
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Probabilistic performance of a heuristic for the satisfiability problem
Discrete Applied Mathematics, 1988
An algorithm for the sattsftabtbty problem (SAT) IS presented and us probabmsttc behavior 1s analyzed when combined with two other algorithms studied earber The analysis IS based on an instance drstrrbutron which IS parametenzed to simulate a variety of sample charactertsttcs. The algorithm dynamically asstgns values to hterals appearing m a given mstance until a satrsfymg assignment IS found or the algorithm "gives up" without determining whether or not a solutron exists. It 1s shown that rf n clauses are constructed independently from r Boolean variables, where the probability that a variable appears m a clause as a posmve literal 1s p and as a negative literal IS p, then almost all randomly generated instances of SAT are solved in polynomtal trme If p<04ln(n)/r or p>ln(n)/r or p=cln(n)/r, 0.4<c<l and limR,+olnl-C/rl-EO. It 1s also shown that tf p=cln(n)/r, 0.4<c< 1 and l~m,r,,,,nt-c/r=w then almost all randomly generated mstances of SAT have no solutton Thus the combined algorithm 1s very effective m the probabtbstic sense on mstances of SAT that have soluttons. The combined algorithm 1s effective m some limited sense in verifying unsattsfiablhty.
Information Sciences, 1990
Two algorithms for the k-satisfiability problem are presented, and a probabilistic analysis is performed. The analysis is based on an instance distribution which is parametrized to simulate a variety of sample characteristics. The algorithms assign values to literals appearing in a given instance of k-satisfiability, one at a time, until a solution is found or it is discovered that further assignments cannot lead to finding a solution. One algorithm chooses the next literal from a unit clause if one exists and randomly from the set of remaining literals otherwise. The other algorithm uses a generalization of the unit-clause rule as a heuristic for selecting the next literal: at each step a literal is chosen randomly from a clause containing the least number of literals. The algorithms run in polynomial time, and it is shown that they find a solution to a random instance of k-satisfiability with probability bounded below by a constant greater than zero for two different ranges of parameter values. It is also shown that the second algorithm mentioned finds a solution with probability approaching one for a wide range of parameter values.
A deterministic (2−2/(k+1))n algorithm for k-SAT based on local search
Theoretical Computer Science, 2002
Local search is widely used for solving the propositional satisÿability problem. showed that randomized local search solves 2-SAT in polynomial time. Recently, Sch oning (1999) proved that a close algorithm for k-SAT takes time (2 − 2=k) n up to a polynomial factor. This is the best known worst-case upper bound for randomized 3-SAT algorithms (cf. also recent preprint by Schuler et al.).
The Probabilistic Analysis of a Greedy Satisfiability Algorithm
Lecture Notes in Computer Science, 2002
On input a random 3-CNF formula of clauses-to-variables ratio r3 apply repeatedly the following simple heuristic: Set to True a literal that appears in the maximum number of clauses, irrespectively of their size and the number of occurrences of the negation of the literal (ties are broken randomly; 1-clauses when they appear get priority). We prove that for r3 < 3.42 this heuristic succeeds with probability asymptotically bounded away from zero. Previously, heuristics of increasing sophistication were shown to succeed for r3 < 3.26. We improve up to r3 < 3.52 by further exploiting the degree of the negation of the evaluated to True literal.
On the Complexity of Random Satisfiability Problems with Planted Solutions
Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing - STOC '15, 2015
Random satisfiability problems with planted solutions exhibit an intriguing complexity gap: for problems on n variables with k variables per constraint, after O(n log n) random clauses the planted assignment becomes the unique solution but the best-known algorithms need at least max{n r/2 , n log n} to efficiently identify it (or even one that is correlated with it), for clause distributions that are (r − 1)-wise independent (thus r can be as high as k). We show a nearly tight unconditional lower bound ofΩ(max{n r/2 , n log n}) clauses for any statistical algorithma restricted class of algorithms introduced in [41, 29] that covers most algorithmic approaches commonly used in theory and practice. We complement this with a nearly matching upper bound: a simple, iterative, statistical algorithm that usesÕ(n r/2 ) clauses and time linear in this to find the planted assignment with high probability. As known approaches for planted satisfiability problems (spectral, MCMC, gradient-based, etc.) all have statistical analogues, this provides a rigorous explanation of the large gap between the identifiability and algorithmic identifiability thresholds for random satisfiability problems with planted solutions.
1992
Given an arbitrary boolean expression, satisfiability (SAT) refers to the task of finding a truth assignment to the boolean variables that makes the expression true. For example, the boolean expression a & b is true if and only if the boolean variables a and b are true. Satisfiability is of interest to the logic, operations research, and computational complexity communities. Due to the emphasis of the logic community, satisfiability algorithms tend to be sound and complete. However, a current trend is to relax some of these requirements.
Analytic and algorithmic solution of random satisfiability problems
2002
Abstract We study the satisfiability of random Boolean expressions built from many clauses with K variables per clause (K-satisfiability). Expressions with a ratio α of clauses to variables less than a threshold α c are almost always satisfiable, whereas those with a ratio above this threshold are almost always unsatisfiable. We show the existence of an intermediate phase below α c, where the proliferation of metastable states is responsible for the onset of complexity in search algorithms.