The problem of estimating the proportion of satisfiable instances of a given CSP (constraint satisfaction problem) can be tackled through weighting. It consists in putting onto each solution a non-negative real value based on its neighborhood in a way that the total weight is at least 1 for each satisfiable instance. We define in this paper a general weighting scheme for the estimation of satisfiability of general CSPs. First we give some sufficient conditions for a weighting system to be correct. Then we show that this scheme allows for an improvement on the upper bound on the existence of non-trivial cores in 3-SAT obtained by Maneva and Sinclair (2008) to 4.419. Another more common way of estimating satisfiability is ordering. This consists in putting a total order on the domain, which induces an orientation between neighboring solutions in a way that prevents circuits from appearing, and then counting only minimal elements. We compare ordering and weighting under various conditions.
Deep Dive into Estimating Satisfiability.
The problem of estimating the proportion of satisfiable instances of a given CSP (constraint satisfaction problem) can be tackled through weighting. It consists in putting onto each solution a non-negative real value based on its neighborhood in a way that the total weight is at least 1 for each satisfiable instance. We define in this paper a general weighting scheme for the estimation of satisfiability of general CSPs. First we give some sufficient conditions for a weighting system to be correct. Then we show that this scheme allows for an improvement on the upper bound on the existence of non-trivial cores in 3-SAT obtained by Maneva and Sinclair (2008) to 4.419. Another more common way of estimating satisfiability is ordering. This consists in putting a total order on the domain, which induces an orientation between neighboring solutions in a way that prevents circuits from appearing, and then counting only minimal elements. We compare ordering and weighting under various conditions
Constraint satisfaction problems cover a large variety of problems that arise in many areas of combinatorial optimization. They are central in complexity theory because they are N P -complete and also because one particular case -satisfiability of boolean formulas -was the first problem to be identified in this class. In general, they consist in defining constraints on a set of variables taking their values in a given finite domain. Constraints specify which combinations of values assigned to subsets of variables are allowed (or dually are forbidden). A solution is a valuation (i.e. the assignment of a value to each variable) that does not violate any constraint. The satisfiability problem is the following: given an instance, decide the existence of a solution for it.
Besides the design of algorithms for solving these problems, the research of structural properties for these problems has attracted much attention in the recent years. In particular, the empirical evidence of the existence of a threshold (rigorously established in some particular cases) in the satisfiability of some classes of CSPs has opened a field of research: attempts are made to rigorously establish the existence and the location of this threshold. This involves estimating the proportion of satisfiable instances in a given set of instances. The N P -completeness of these problems in general makes it difficult to determine whether a given instance is satisfiable; that may explain why direct counting of satisfiable instances is currently unfeasible. However, precisely because these problems are in N P , it is easy to determine whether some instance is satisfied by a given valuation and then to count the formulas satisfied by this valuation. Thus counting couples (formulas, solutions) is only accessible starting from a solution; moreover, given a solution, it is not complicated to investigate also its immediate neighborhood. But even at a distance of 2, i.e. with neighbors of neighbors, calculations become quite complicated (see Kirousis et al. [2]). This fact imposes a strong restriction on the design of both estimation techniques studied: they can only make use of local information. We shall refer to this as the locality condition.
Using one of the most popular techniques in the probabilistic method (cf. Alon and Spencer [3]), namely the first moment method, it is possible to bound from above the probability of satisfiability. The implementation of the first moment method makes use of Markov’s inequality; one needs to define a non-negative random variable X that must be at least 1 for a satisfiable formula (we call that a correct random variable). Ideally, X should be as small as possible; in other words, it should be 0 for unsatisfiable instances and as close to 1 as possible for satisfiable ones (if X is 1 for every satisfiable instance and 0 for every unsatisfiable instance then we get the exact probability of satisfiability). The most straightforward candidate for X is simply the number of solutions. The method consists in counting for every valuation the number of instances that are satisfied by it and then summing up over all valuations. But since the number of solutions is generally too large, the method over-estimates the proportion of satisfiable instances.
Many techniques have been developed to overcome this difficulty in various types of CSPs, Satisfiability of CNF formulas (Kamath et al. [4], Dubois and Boufkhad [5], Kirousis et al. [2], Dubois et al. [6,7], Boufkhad et al. [8], Kaporis et al. [9], Díaz et al. [10], Boufkhad and Hugel [11]), 3-Coloring of graphs (Achlioptas and Molloy [12]), Binary CSPs (Achlioptas et al. [13,14]). . . Most of these methods share a common point: they count minimal elements under some partial order over solutions. We will refer to this method as solution selection through a partial ordering or for short ordering. Due to the locality condition, the partial order must be locally computable (i.e. must depend only on the immediate neighbors of the considered solution). Two solutions of some instance are neighbors if they disagree only on the value taken by one variable. Both solutions may be ordered using a predetermined order on the values for this particular variable in this particular instance. Finally we count only those solutions having minimal values for all their variables with respect to their neighbors.
Recently, Maneva et al. [15] introduced a novel approach for the boolean satisfiability problem consisting in weighting partial valuations and solutions depending on their neighborhood. While not originally intended to estimate the proportion of satisfiable instances (but rather to analyze some properties of Belief Propagation algorithms), it was though specifically used by Maneva and Sinclair [1] to estimate the probability of existence of non-trivial cores in random 3-SAT instances. The existence of non-trivial cores contains an important information on the structure of the space of so
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