Abstract. We study the runtime distributions of backtrack procedures for propositional satisfiability and constraint satisfaction. Such procedures often exhibit a large variability in performance. Our study reveals some intriguing properties of such distributions: They are often characterized by very long tails or “heavy tails”. We will show that these distributions are best characterized by a general class of distributions that can have infinite moments (i.e., an infinite mean, variance, etc.). Such nonstandard distributions have recently been observed in areas as diverse as economics, statistical physics, and geophysics. They are closely related to fractal phenomena, whose study was introduced by Mandelbrot. We also show how random restarts can effectively eliminate heavy-tailed behavior. Furthermore, for harder problem instances, we observe long tails on the left-hand side of the distribution, which is indicative of a non-negligible fraction of relatively short, successful runs. A rapid restart strategy eliminates heavy-tailed behavior and takes advantage of short runs, significantly reducing expected solution time. We demonstrate speedups of up to two orders of magnitude on SAT and CSP encodings of hard problems in planning, scheduling, and circuit synthesis. Key words: satisfiability, constraint satisfaction, heavy tails, backtracking 1.

A good decoding algorithm is critical

There has been significant recent progress in reasoning and constraint processing methods. In areas such as planning and finite model-checking, current solution techniques can handle combinatorial problems with up to a million variables and five million constraints. The good scaling behavior of these methods appears to defy what one would expect based on a worst-case complexity analysis. In order to bridge this gap between theory and practice, we propose a new framework for studying the complexity of these techniques on practical problem instances. In particular, our approach incorporates general structural properties observed in practical problem instances into the formal complexity

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.

Consider the following simple, greedy Davis-Putnam algorithm applied to a random 3CNF formula of fixed density (clauses to variables ratio): Arbitrarily select and set to True a literal that appears in as many clauses as possible, irrespective of their size (and irrespective of the number of occurrences of the negation of the literal). Delete these clauses from the formula, and also delete the negation of this literal from any clauses it appears. Repeat. If however unit clauses ever appear, then first repeatedly and in any order set the literals in them to True and delete and shrink clauses accordingly, until no unit clause remains. Also if at any step an empty clause appears, then do not backtrack, but just terminate the algorithm and report failure. A slight modification of this algorithm is probabilistically analyzed in this paper (rigorously). It is proved that for random formulas of n variables and density up to 3.42, it succeeds in producing a satisfying truth assignment with bounded away from zero probability, as n approaches infinity. Therefore the satisfiability threshold is at least 3.42.

Abstract not found

Of late, new insight into the study of random k-SAT formulae has been gained from the introduction of a concept inspired by models of physics, the `backbone ' of a SAT formula which corresponds to the variables having a fixed truth value in all assignments satisfying the maximum number of clauses.

We give the first example of a sharp threshold in proof complexity. More precisely, we show that for any sufficiently small � and � � �, random formulas consisting of 2-clauses and 3-clauses, which are known to be unsatisfiable almost certainly, almost certainly require resolution and Davis-Putnam proofs of unsatisfiability of exponential size, whereas it is easily seen that random formulas with 2-clauses (and 3-clauses) have linear size proofs of unsatisfiability almost certainly. A consequence of our result also yields the first proof that typical random 3-CNF formulas at ratios below the generally accepted range of the satisfiability threshold (and thus expected to be satisfiable almost certainly) cause natural Davis-Putnam algorithms to take exponential time to find satisfying assignments.

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...

We dene the `frozen development' of coloring random graphs. We identify two nodes in a graph as frozen if they are the same color in all legal colorings. This is analogous to studies of the development of a backbone or spine in SAT (the Satisability problem). We rst describe in detail the algorithmic techniques used to study frozen development. We present strong empirical evidence that freezing in 3-coloring is sudden. A single edge typically causes the size of the graph to collapse in size by 28%. We also use the frozen development to calculate unbiased estimates of probability of colorability in random graphs, even where this probability is as low as 10 300 . We investigate the links between frozen development and the solution cost of graph coloring. In SAT, a discontinuity in the order parameter has been correlated with the hardness of SAT instances, and our data for coloring is suggestive of an asymptotic discontinuity. The uncolorability threshold is known to give rise to har...