A major difficulty in evaluating incomplete local search style algorithms for constraint satisfaction problems is the need for a source of hard problem instances that are guaranteed to be satisfiable. A standard approach to evaluate incomplete search methods has been to use a general problem generator and a complete search method to filter out the unsatisfiable instances. Unfortunately, this approach cannot be used to create problem instances that are beyond the reach of complete search methods. So far, it has proven to be surprisingly difficult to develop a direct generator for satisfiable instances only. In this paper, we propose a generator that only outputs satisfiable problem instances. We also show how one can finely control the hardness of the satisfiable instances by establishing a connection between problem hardness and a new kind of phase transition phenomenon in the space of problem instances. Finally, we use our problem distribution to show the easy-hard-easy pattern in search complexity for local search procedures, analogous to the previously reported pattern for complete search methods.

In this paper we present a new randomized algorithm for SAT, i.e., the satisfiability problem for Boolean formulas in conjunctive normal form. Despite its simplicity, this algorithm performs well on many common benchmarks ranging from graph coloring problems to microprocessor verification.

A primary concern when using local search methods for CNF satisfiability is how to get rid of local minimas. Among many other heuristics, Weighting by Morris (1993) and Selman and Kautz (1993) works overwhelmingly better than others (Cha and Iwama 1995). Weighting increases the weight of each clause which is unsatisfied at a local minima. This paper introduces a more sophisticated weighting strategy, i.e., adding new clauses (ANC) that are unsatisfied at the local minima. As those new clauses, we choose resolvents of the clauses unsatisfied at the local minima and randomly selected neighboring clauses. The idea is that ANC is to make the slope of search space more smooth than the simple weighting. Experimental data show that ANC is faster than simple weighting: (i) When the number of variables is 200 or more, ANC is roughly four to ten times as fast as weighting in terms of the number of search steps. (ii) It might be more important that the divergence of computation time for each try ...

The local search algorithm WSat is one of the most successful algorithms for solving the satisfiability (SAT) problem. It is notably e#ective at solving hard Random 3-SAT instances near the so-called `satisfiability threshold', but still shows a peak in search cost near the threshold and large variations in cost over di#erent instances. We make a number of significant contributions to the analysis of WSat on high-cost random instances, using the recently-introduced concept of the backbone of a SAT instance. The backbone is the set of literals which are entailed by an instance. We find that the number of solutions predicts the cost well for small-backbone instances but is much less relevant for the large-backbone instances which appear near the threshold and dominate in the overconstrained region. We show a very strong correlation between search cost and the Hamming distance to the nearest solution early in WSat's search. This pattern leads us to introduce a measure of the ba...

Certifying the correctness of a SAT solver is straightforward for satisfiable instances of SAT.

The easy-hard-easy pattern in the difficulty of combinatorial search problems as constraints are added has been explained as due to a competition between the decrease in number of solutions and increased pruning. We test the generality of this explanation by examining one of its predictions: if the number of solutions is held fixed by the choice of problems, then increased pruning should lead to a monotonic decrease in search cost. Instead, we find the easy-hard-easy pattern in median search cost even when the number of solutions is held constant, for some search methods. This generalizes previous observations of this pattern and shows that the existing theory does not explain the full range of the peak in search cost. In these cases the pattern appears to be due to changes in the size of the minimal unsolvable subproblems, rather than changing numbers of solutions. 1. Introduction Recently, many authors have shown that the solution cost for various kinds of combinatorial search probl...

Abstract. We introduce a highly structured family of hard satisfiable 3-SAT formulas corresponding to an ordered spin-glass model from statistical physics. This model has provably “glassy ” behavior; that is, it has many local optima with large energy barriers between them, so that local search algorithms get stuck and have difficulty finding the true ground state, i.e., the unique satisfying assignment. We test the hardness of our formulas with two Davis-Putnam solvers, Satz and zChaff, the recently introduced Survey Propagation (SP), and two local search algorithms, WalkSAT and Record-to-Record Travel (RRT). We compare our formulas to random 3-XOR-SAT formulas and to two other generators of hard satisfiable instances, the minimum disagreement parity formulas of Crawford et al., and Hirsch’s hgen2. For the complete solvers the running time of our formulas grows exponentially in √ n, and exceeds that of random 3-XOR-SAT formulas for small problem sizes. SP is unable to solve our formulas with as few as 25 variables. For WalkSAT, our formulas appear to be harder than any other known generator of satisfiable instances. Finally, our formulas can be solved efficiently by RRT but only if the parameter d is tuned to the height of the barriers between local minima, and we use this parameter to measure the barrier heights in random 3-XOR-SAT formulas as well. 1

The evaluation of incomplete satisfiability solvers depends critically on the availability of hard satisfiable instances. A plausible source of such instances consists of random k-SAT formulas whose clauses are chosen uniformly from among all clauses satisfying some randomly chosen truth assignment A. Unfortunately, instances generated in this manner tend to be relatively easy and can be solved efficiently by practical heuristics. Roughly speaking, as the formula’s density increases, for a number of different algorithms, A acts as a stronger and stronger attractor. Motivated by recent results on the geometry of the space of satisfying truth assignments of random k-SAT and NAE-k-SAT formulas, we introduce a simple twist on this basic model, which appears to dramatically increase its hardness. Namely, in addition to forbidding the clauses violated by the hidden assignment A, we also forbid the clauses violated by its complement, so that both A and A are satisfying. It appears that under this “symmetrization ” the effects of the two attractors largely cancel out, making it much harder for algorithms to find any truth assignment. We give theoretical and experimental evidence supporting this assertion. 1

Abstract. Many current local search algorithms for SAT fall into one of two classes. Random walk algorithms such as Walksat/SKC, Novelty+ and HWSAT are very successful but can be trapped for long periods in deep local minima. Clause weighting algorithms such as DLM, GLS, ESG and SAPS are good at escaping local minima but require expensive smoothing phases in which all weights are updated. We show that Walksat performance can be greatly enhanced by weighting variables instead of clauses, giving the best known results on some benchmarks. The new algorithm uses an efficient weight smoothing technique with no smoothing phase. 1

. This paper explores the performance of a new complete nonsystematic search algorithm learn-SAT on two types of 3-SAT problems, (i) an extended range of AIM problems [1] and (ii) structured unsolvable problems [2]. These are thought to present a difficult challenge for non-systematic search algorithms. They have been extensively used to study powerful special purpose SAT algorithms. We consider two of these, viz. the tableau-based algorithm of Bayardo & Schrag [2] and relsat. We compare their performance with that of learn-SAT, which is based on restart-repair and learning no-goods. Surprisingly, learn-SAT does very well. Sometimes it does much better than the other two algorithms; at other times they are broadly equivalent; and then there are some "anomalies". One thing at least is clear, learn-SAT solves problems which many would predict are beyond its scope. The relative performance of the three algorithms generates several interesting questions. We point to some of them with a vi...