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.

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 show that nodes of high degree tend to occur infrequently in random graphs but frequently in a wide variety of graphs associated with real world search problems. We then study some alternative models for randomly generating graphs which have been proposed to give more realistic topologies. For example, we show that Watts and Strogatz 's small world model has a narrow distribution of node degree. On the other hand, Barabasi and Albert's power law model, gives graphs with both nodes of high degree and a small world topology. These graphs may therefore be useful for benchmarking. We then measure the impact of nodes of high degree and a small world topology on the cost of coloring graphs. The long tail in search costs observed with small world graphs disappears when these graphs are also constructed to contain nodes of high degree. We conjecture that this is a result of the small size of their "backbone", pairs of edges that are frozen to be the same color.

We consider the resolution proof complexity of propositional formulas which encode random instances of graph k-colorability. We obtain a tradeoff between the graph density and the resolution proof complexity. For random graphs with linearly many edges we obtain linear-exponential lower bounds on the length of resolution refutations. For any # > 0, we obtain sub-exponential lower bounds of the form for some # > 0 for non-k-colorability proofs of graphs with n vertices and O(n -# ) edges. We obtain sharper lower bounds for Davis-Putnam-DPLL proofs and for proofs in a system considered by McDiarmid.

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.

Distributed Constraint Satisfaction (DCSP) has long been considered an important problem in multi-agent systems research. This is because many real-world problems can be represented as constraint satisfaction and these problems often present themselves in a distributed form. In this article, we present a new complete, distributed algorithm called asynchronous partial overlay (APO) for solving DCSPs that is based on a cooperative mediation process. The primary ideas behind this algorithm are that agents, when acting as a mediator, centralize small, relevant portions of the DCSP, that these centralized subproblems overlap, and that agents increase the size of their subproblems along critical paths within the DCSP as the problem solving unfolds. We present empirical evidence that shows that APO outperforms other known, complete DCSP techniques. 1.

In this paper, we analyze the decision version of the NK landscape model from the perspective of threshold phenomena and phase transitions under two random distributions, the uniform probability model and the fixed ratio model. For the uniform probability model, we prove that the phase transition is easy in the sense that there is a polynomial algorithm that can solve a random instance of the problem with the probability asymptotic to 1 as the problem size tends to infinity. For the fixed ratio model, we establish several upper bounds for the solubility threshold, and prove that random instances with parameters above these upper bounds can be solved polynomially. This, together with our empirical study for random instances generated below and in the phase transition region, suggests that the phase transition of the fixed ratio model is also easy.

It is now fairly well understood that a vast number of AI problems can be formulated as Constraint Satisfaction Problems (CSPs) and striking improvements have been made in solving them using both centralized and distributed methods. However, many real world problems change over time and very little work has been done in developing methods, particularly distributed ones, for solving problems which exhibit this behavior. This paper presents two new protocols for solving dynamic, distributed constraint satisfaction problems which are based on the classic Distributed Breakout Algorithm (DBA) and the Asynchronous Partial Overlay (APO) algorithm. These two new algorithms are compared on a broad class of problems varying the problems ’ overall difficulty as well as the rate at which they change over time. The results indicate that neither of the algorithms complete dominates the other on all problem types, but that depending on environmental conditions and the needs of the user, one method may be preferable over the other.

In recent years, there has been much interest in phase transitions of combinatorial problems.

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