We study the impact of backbones in optimization and approximation problems. We show that some optimization problems like graph coloring resemble decision problems, with problem hardness positively correlated with backbone size. For other optimization problems like blocks world planning and traveling salesperson problems, problem hardness is weakly and negatively correlated with backbone size, while the cost of finding optimal and approximate solutions is positively correlated with backbone size. A third class of optimization problems like number partitioning have regions of both types of behavior. We find that to observe the impact of backbone size on problem hardness, it is necessary to eliminate some symmetries, perform trivial reductions and factor out the effective problem size.

Tabu search algorithms are among the most effective approaches for solving the job-shop scheduling problem (JSP). Yet, we have little understanding of why these algorithms work so well, and under what conditions. We develop a model of problem difficulty for tabu search in the JSP, borrowing from similar models developed for SAT and other NP - complete problems. We show that the mean distance between random local optima and the nearest optimal solution is highly correlated with the cost of locating optimal solutions to typical, random JSPs. Additionally, this model accounts for the cost of locating suboptimal solutions, and provides an explanation for differences in the relative difficulty of square versus rectangular JSPs. We also identify two important limitations of our model. First, model accuracy is inversely correlated with problem difficulty, and is exceptionally poor for rare, very high-cost problem instances. Second, the model is significantly less accurate for structured, non-random JSPs. Our results are also likely to be useful in future research on difficulty models of local search in SAT, as local search cost in both SAT and the JSP is largely dictated by the same search space features. Similarly, our research represents the first attempt to quantitatively model the cost of tabu search for any NP -complete problem, and may possibly be leveraged in an effort to understand tabu search in problems other than job-shop scheduling.

Many real-world problems involve constraints that cannot be all satisfied. Solving an overconstrained problem then means to find solutions minimizing the number of constraints violated, which is an optimization problem. In this research, we study the behavior of the phase transitions and backbones of constraint optimization problems. We rst investigate the relationship between the phase transitions of Boolean satisfiability, or precisely 3-SAT (a well-studied NP-complete decision problem), and the phase transitions of MAX 3-SAT (an NP-hard optimization problem). To bridge the gap between the easy-hard-easy phase transitions of 3-SAT and the easy-hard transitions of MAX 3-SAT, we analyze bounded 3-SAT, in which solutions of bounded quality, e.g., solutions with at most a constant number of constraints violated, are sufficient.

We study the problem of (efficiently) deleting such clauses from conjunctive normal forms (clause-sets) which can not contribute to any proof of unsatisfiability. For that purpose we introduce the notion of an autarky system, associated with a canonical normal form for every clause-set by deleting superfluous clauses. Clause-sets where no clauses can be deleted are called lean, a natural generalization of minimally unsatisfiable clause-sets, opening the possibility for combinatorial approaches (and including also satisfiable instances). Three special examples for autarky systems are considered: general autarkies, linear autarkies (based on linear programming) and matching autarkies (based on matching theory). We give new characterizations of lean and linearly lean clause-sets by "universal linear programming problems," while matching lean clause-sets are characterized in terms of "deficiency, " the difference between the number of clauses and the number of variables, and ...

. Random 2+p-SAT interpolates between the polynomialtime problem Random 2-SAT when p = 0 and the NP-complete problem Random 3-SAT when p = 1. At some value p = p0 0:41, a dramatic change in the structural nature of instances is predicted by statistical mechanics methods. This is reflected by a change in the typical cost scaling for a complete search method TABLEAU, seen experimentally. We show empirically the same change of of behaviour in the local search algorithm NOVELTY + , a recent variant of WSAT. Between p = 0:3 and p = 0:5 we see typical cost scaling of NOVELTY + at the 50% satisfiability point apparently change from slow polynomial growth to superpolynomial. That this behaviour is seen in two such different algorithms lends credibility to the hypothesis that there is change of typical-case complexity around p0 . Previous work linked the emergence of a backbone of fully constrained variables to the cost peak seen in Random k-SAT. Initial experiments suggest that for those...

The best performing algorithms for a particular oversubscribed scheduling application, Air Force Satellite Control Network (AFSCN) scheduling, appear to have little in common. Yet, through careful experimentation and modeling of performance in real problem instances, we can relate characteristics of the best algorithms to characteristics of the application. In particular, we find that plateaus dominate the search spaces (thus favoring algorithms that make larger changes to solutions) and that some randomization in exploration is critical to good performance (due to the lack of gradient information on the plateaus). Based on our explanations of algorithm performance, we develop a new algorithm that combines characteristics of the best performers; the new algorithm’s performance is better than the previous best. We show how hypothesis driven experimentation and search modeling can both explain algorithm performance and motivate the design of a new algorithm. 1.

Stochastic Local Search (SLS) algorithms are amongst the most effective approaches for solving hard and large propositional satisfiability (SAT) problems. Prominent and successful SLS algorithms for SAT, including many members of the WalkSAT and GSAT families of algorithms, tend to show highly regular behaviour when applied to hard SAT instances: The run-time distributions (RTDs) of these algorithms are closely approximated by exponential distributions.

Software management oracles often contain numerous subjective features. At each subjective point, a range of behaviors is possible. Stochastic simulation samples a subset of the possible behaviors. After many such stochastic simulations, the TAR2 treatment learner can find control actions that have (usually) the same impact despite the subjectivity of the oracle. 1.

We introduce our work on the backdoor key, a concept that shows promise for characterizing problem hardness in backtracking search algorithms. The general notion of backdoors was recently introduced to explain the source of heavy-tailed behaviors in backtracking algorithms (Williams, Gomes, & Selman 2003a; 2003b). We describe empirical studies that show that the key faction, i.e., the ratio of the key size to the corresponding backdoor size, is a good predictor of problem hardness of ensembles and individual instances within an ensemble for structure domains with large key fraction.

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