... quickly across a wide range of hard SAT problems than any other SAT tester in the literature on comparable platforms. On a Sun SPARCStation 10 running SunOS 4.1.3 U1, POSIT can solve hard random 400-variable 3-SAT problems in about 2 hours on the average. In general, it can solve hard n-variable random 3-SAT problems with search trees of size O(2 n=18:7 ). In addition to justifying these claims, this dissertation describes the most significant achievements of other researchers in this area, and discusses all of the widely known general techniques for speeding up SAT search algorithms. It should be useful to anyone interested in NP-complete problems or combinatorial optimization in general, and it should be particularly useful to researchers in either Artificial Intelligence or Operations Research.

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.

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 describe an experimental investigation of the satisfiability phase transition for several different classes of randomly generated problems. We show that the "conventional" picture of easy-hard-easy problem difficulty is inadequate. In particular, there is a region of very variable problem difficulty where problems are typically underconstrained and satisfiable. Within this region, problems can be orders of magnitude harder than problems in the middle of the satisfiability phase transition. These extraordinary hard problems appear to be associated with a constraint gap, a minimum in the amount of constraint propagation compared to the amount of search. We show that the position and shape of this constraint gap are very consistent with problem size. Unlike hard problems in the middle of satisfiability phase transition, hard problems in the variable region are not critically constrained between satisfiability and unsatisfiability. Indeed, hard problems in the variable region often cont...

Phase transitions in constraint satisfaction problems (CSP's) are the subject of intense study. We identify an order parameter for random binary CSP's. There is a rapid transition in the probability of a CSP having a solution at a critical value of this parameter. The order parameter allows different phase transition behaviour to be compared in an uniform manner, for example CSP's generated under different regimes. We then show that within classes, the scaling of behaviour can be modelled by a tehnique called "finite size scaling". This applies not only to probability of solubility, as has been observed before in other NP-problems, but also to search cost, the first time this has been observed. Furthermore, the technique applies with equal validity to several different methods of varying problem size. As well as contributing to the understanding of phase transitions, we contribute by allowing much finer grained comparison of algorithms, and for accurate empirical extrapolations of beha...

This paper presents an experimental investigation of the following questions: how does the average-case complexity of random 3-SAT, understood as a function of the order (number of variables) for xed density (ratio of number of clauses to order) instances, depend on the density? Is there a phase transition in which the complexity shifts from polynomial to exponential in the order? Is the transition dependent or independent of the solver? Our experiment design uses three complete SAT solvers embodying dierent algorithms: GRASP, CPLEX, and CUDD. We observe new phase transitions for all three solvers, where the median running time shifts from polynomial in the order to exponential. The location of the phase transition appears to be solver-dependent. While GRASP and CUDD shift from polynomial to exponential complexity at a density of about 3.8, CUDD exhibits this transition between densities of 0.1 and 0.5. This experimental result underscores the dependence between the solver and the complexity phase transition, and challenges the widely held belief that random 3-SAT exhibits a phase transition in computational complexity very close to the crossover point.

We consider the use of random CNF formulas in evaluating the performance of SAT testing algorithms, and in particular the role that the phase transition phenomenon plays in this use. Examples from the literature illustrate the importance of understanding the properties of formula distributions prior to designing an experiment. We expect this to be of increasing importance in the field. 1 Introduction Satisfiability testing lies at the core of many computational problems and because of its close relationship to various reasoning tasks, this is especially so in Artificial Intelligence. Randomly generated CNF formulas are a popular class of test problems for evaluating the performance of SAT testing programs. Not surprisingly, the choice of formula distribution is crucial to the validity of any investigation using random formulas. In [23], we argued that some families of distributions were more useful sources of test material than others, and suggested choosing formulas from the "hard reg...

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In order to effectively respond to changing market conditions, business partners must be able to rapidly form supply chains. This thesis approaches the problem of automating supply chain formation—the process of determining the participants in a supply chain, who will exchange what with whom, and the terms of the exchanges—within an economic framework. In this thesis, supply chain formation is formalized as task dependency networks. This model captures subtask decomposition in the presence of resource contention—two important and challenging aspects of supply chain formation. In order to form supply chains in a decentralized fashion, price systems provide an economic framework for guiding the decisions of self-interested agents. In competitive price equilibrium, agents choose optimal allocations with respect to prices, and outcomes are optimal overall. Approximate competitive equilibria yield approximately optimal allocations. Different market protocols are proposed for agents to negotiate the allocation of resources to form supply chains. In the presence of resource contention, these protocols produce better solutions than the greedy protocols common in the artificial intelligence

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...