Nomadic applications create replicas of shared objects that evolve independently while they are disconnected. When reconnecting, the system has to reconcile the divergent replicas. Log-based reconciliation is a novel approach in which the input is a common initial state and logs of actions that were performed on each replica. The output is a consistent global schedule that maximises the number of accepted actions. The reconciler merges the logs according to the schedule, and replays the operations in the merged log against the initial state, yielding to a reconciled common nal state. In this paper, we show the NP-hardness of the log-based reconciliation problem, and study a constraint logic program that uses integer constraints for expressing precedences between actions, and boolean constraints for expressing dependencies between actions. Interestingly, we demonstrate the existence of a single computational complexity peak on randomly generated problems, around densities 7 for precedence constraints and 0 for dependency constraints between actions. Around this peak we observe phase transitions in the two dimensions of the density, where the mean running time of the program shifts from polynomial in the order to exponential. On realistic benchmarks, rst evaluation results show that the program nds nearly optimal solutions up to a thousands of actions and proves optimality up to a hundred of actions. 1

There are several algorithms for producing the canonical DFA from a given NFA. While the theoretical complexities of these algorithms are known, there has not been a systematic empirical comparison between them. In this work we propose a probabilistic framework for testing the performance of automatatheoretic algorithms. We conduct a direct experimental comparison between Hopcroft 's and Brzozowski's algorithms. We show that while Hopcroft's algorithm has better overall performance, Brzozowski's algorithm performs better for "highdensity " NFA. We also consider the universality problem, which is traditionally solved explicitly via the subset construction. We propose an encoding that allows this problem to be solved symbolically via a model-checker. We compare the performance of this approach to that of the standard explicit algorithm, and show that the explicit approach performs significantly better.

We consider three distributed configuration tasks that arise in the setup and operation of multihop wireless networks: partition into coordinating cliques, Hamiltonian cycle formation and conflict-free channel allocation. We show that the probabilities of accomplishing these tasks undergo zero-one phase transitions with respect to the transmission range of individual nodes. We model these tasks as distributed constraint satisfaction problems (DCSPs) and show that, even though they are NP-hard in general, these problems can be solved efficiently on average when the network is operated sufficiently far from the transition region. Phase transition analysis is shown to be a useful mechanism for quantifying the critical range of energy and bandwidth resources needed for the scalable performance of self-configuring wireless networks. Keywords: self-configuration, wireless networks, distributed constraint satisfaction

The complexity class PP consists of all decision problems solvable by polynomial-time probabilistic Turing machines. It is well known that PP is a highly intractable complexity class and that PP-complete problems are in all likelihood harder than NP-complete problems. We investigate the existence of phase transitions for a family of PPcomplete Boolean satisfiability problems under the fixed clauses-to-variables ratio model. A typical member of this family is the decision problem #3SAT( 2 n=2 ): given a 3CNF-formula, is it satisfied by at least the square-root of the total number of possible truth assignments? We provide evidence to the effect that there is a critical ratio r 3

An analysis for the phase transition in a random NK landscape model, NK(n,k,z), is given. This model is motivated from population genetics and the solubility problem for the model is equivalent to a random (k + 1)-SAT problem. Gao and Culberson [Y. Gao, J. Culberson, An analysis of phase transition in NK landscapes, Journal of Artificial Intelligence Research 17 (2002) 309–332] showed that a random instance generated by NK(n, 2,z) with z>z0 = 27−7√5 4 is asymptotically insoluble. Based on empirical results, they conjectured that the phase transition occurs around the value z = z0. We prove that an instance generated by NK(n, 2,z)with z<z0 is soluble with positive probability by providing a polynomial time algorithm. Using branching process arguments, we prove again that an instance generated by NK(n, 2,z)with z>z0 is asymptotically insoluble. The results show the phase transition around z = z0 for NK(n, 2,z). In the course of the analysis, we introduce a generalized random 2-SAT formula, which is of self interest, and show its phase transition phenomenon.

We consider random instances I of a constraint satisfaction problem generalizing k-SAT: given n boolean variables, m ordered k-tuples of literals, and q "bad" clause assignments, find an assignment which does not set any of the k-tuples to a bad clause assignment. We consider the case where k =#328 n), and generate instance I by including every k-tuple of literals independently with probability p. Appropriate choice of the bad clause assignments results in random instances of k-SAT and notall -equal k-SAT. For constant q, a second moment method calculation yields the sharp threshold Pr[I is satisfiable] = k-1 .

The unsatisfiability threshold conjecture: techniques behind upper bound improvements

3-SAT is a canonical NP-complete problem: satisfiable and unsatisfiable instances cannot generally be distinguished in polynomial time. However, random 3-SAT formulas show a phase transition: for any large number of variables n, sparse random formulas (with m &le; 3.145n clauses) are almost always satisfiable, dense ones (with m &ge; 4.596n clauses) are almost always unsatisfiable, and the transition occurs sharply when m=n crosses some threshold. It is believed that the limiting threshold is around 4.2, but it is not even known that a limit exists. Proofs of the satisfiability...

The technique of using di#erential equations to approximate the mean path of Markov chains has proved very useful in the average-case analysis of algorithms. Here, we significantly expand the range of this technique, by showing that it can be used to handle algorithms that favor high-degree vertices. In particular, we consider the problem of 3-coloring sparse random graphs and analyze a "smoothed" version of the Brelaz heuristic. This allows us to prove that i) almost all graphs with average degree d, i.e. G(n, p = d/n), are 3-colorable for d 4.03, and that ii) almost all 4-regular graphs are 3-colorable. This improves over the previous lower bound of 3.847 for the G(n, p) 3-colorability threshold and gives the first non-trivial result on the 3-colorability of random regular graphs.