Trajectories in phase diagrams, growth processes and computational complexity: how search algorithms solve the 3-Satisfiability problem

Most decision and optimization problems encountered in practice fall into one of two categories with respect to any particular solving method or algorithm: either the problem is solved quickly (easy) or else demands an impractically long computational effort (hard). Recent investigations on model classes of problems have shown that some global parameters, such as the ratio between the constraints to be satisfied and the adjustable variables, are good predictors of problem hardness and, moreover, have an effect analogous to thermodynamical parameters, e.g. temperature, in predicting phases in condensed matter physics [Monasson et al., Nature 400 (1999) 133-137]. Here we show that changes in the values of such parameters can be tracked during a run of the algorithm defining a trajectory through the parameter space. Focusing on 3-Satisfiability, a recognized representative of hard problems, we analyze trajectories generated by search algorithms using growth processes statistical physics. These trajectories can cross well defined phases, corresponding to domains of easy or hard instances, and allow to successfully predict the times of resolution.Comment: Revtex file + 4 eps figure

Tricritical Points in Random Combinatorics: the (2+p)-SAT case

The (2+p)-Satisfiability (SAT) problem interpolates between different classes of complexity theory and is believed to be of basic interest in understanding the onset of typical case complexity in random combinatorics. In this paper, a tricritical point in the phase diagram of the random $2+p$-SAT problem is analytically computed using the replica approach and found to lie in the range $2/5 \le p_0 \le 0.416$. These bounds on $p_0$ are in agreement with previous numerical simulations and rigorous results.Comment: 7 pages, 1 figure, RevTeX, to appear in J.Phys.

Phase coexistence and finite-size scaling in random combinatorial problems

We study an exactly solvable version of the famous random Boolean satisfiability problem, the so called random XOR-SAT problem. Rare events are shown to affect the combinatorial ``phase diagram'' leading to a coexistence of solvable and unsolvable instances of the combinatorial problem in a certain region of the parameters characterizing the model. Such instances differ by a non-extensive quantity in the ground state energy of the associated diluted spin-glass model. We also show that the critical exponent $\nu$, controlling the size of the critical window where the probability of having solutions vanishes, depends on the model parameters, shedding light on the link between random hyper-graph topology and universality classes. In the case of random satisfiability, a similar behavior was conjectured to be connected to the onset of computational intractability.Comment: 10 pages, 5 figures, to appear in J. Phys. A. v2: link to the XOR-SAT probelm adde

Exponentially hard problems are sometimes polynomial, a large deviation analysis of search algorithms for the random Satisfiability problem, and its application to stop-and-restart resolutions

A large deviation analysis of the solving complexity of random 3-Satisfiability instances slightly below threshold is presented. While finding a solution for such instances demands an exponential effort with high probability, we show that an exponentially small fraction of resolutions require a computation scaling linearly in the size of the instance only. This exponentially small probability of easy resolutions is analytically calculated, and the corresponding exponent shown to be smaller (in absolute value) than the growth exponent of the typical resolution time. Our study therefore gives some theoretical basis to heuristic stop-and-restart solving procedures, and suggests a natural cut-off (the size of the instance) for the restart.Comment: Revtex file, 4 figure

Energy Proportionality and Performance in Data Parallel Computing Clusters

Energy consumption in datacenters has recently become a major concern due to the rising operational costs andscalability issues. Recent solutions to this problem propose the principle of energy proportionality, i.e., the amount of energy consumedby the server nodes must be proportional to the amount of work performed. For data parallelism and fault tolerancepurposes, most common file systems used in MapReduce-type clusters maintain a set of replicas for each data block. A coveringset is a group of nodes that together contain at least one replica of the data blocks needed for performing computing tasks. In thiswork, we develop and analyze algorithms to maintain energy proportionality by discovering a covering set that minimizesenergy consumption while placing the remaining nodes in lowpower standby mode. Our algorithms can also discover coveringsets in heterogeneous computing environments. In order to allow more data parallelism, we generalize our algorithms so that itcan discover k-covering sets, i.e., a set of nodes that contain at least k replicas of the data blocks. Our experimental results showthat we can achieve substantial energy saving without significant performance loss in diverse cluster configurations and workingenvironments

Simplest random K-satisfiability problem

We study a simple and exactly solvable model for the generation of random satisfiability problems. These consist of $\gamma N$ random boolean constraints which are to be satisfied simultaneously by $N$ logical variables. In statistical-mechanics language, the considered model can be seen as a diluted p-spin model at zero temperature. While such problems become extraordinarily hard to solve by local search methods in a large region of the parameter space, still at least one solution may be superimposed by construction. The statistical properties of the model can be studied exactly by the replica method and each single instance can be analyzed in polynomial time by a simple global solution method. The geometrical/topological structures responsible for dynamic and static phase transitions as well as for the onset of computational complexity in local search method are thoroughly analyzed. Numerical analysis on very large samples allows for a precise characterization of the critical scaling behaviour.Comment: 14 pages, 5 figures, to appear in Phys. Rev. E (Feb 2001). v2: minor errors and references correcte

Optimisation problems and replica symmetry breaking in finite connectivity spin-glasses

A formalism capable of handling the first step of hierarchical replica symmetry breaking in finite-connectivity models is introduced. The emerging order parameter is claimed to be a probability distribution over the space of field distributions (or, equivalently magnetisation distributions) inside the cluster of states. The approach is shown to coincide with the previous works in the replica symmetric case and in the two limit cases m=0,1 where m is Parisi's break-point. As an application to the study of optimization problems, the ground-state properties of the random 3-Satisfiability problem are investigated and we present a first RSB solution improving replica symmetric results.Comment: 16 pages Revtex file, 1 figure; amended version with two new appendices; to be published in J.Phys.

Cluster expansions in dilute systems: applications to satisfiability problems and spin glasses

We develop a systematic cluster expansion for dilute systems in the highly dilute phase. We first apply it to the calculation of the entropy of the K-satisfiability problem in the satisfiable phase. We derive a series expansion in the control parameter, the average connectivity, that is identical to the one obtained by using the replica approach with a replica symmetric ({\sc rs}) {\it Ansatz}, when the order parameter is calculated via a perturbative expansion in the control parameter. As a second application we compute the free-energy of the Viana-Bray model in the paramagnetic phase. The cluster expansion allows one to compute finite-size corrections in a simple manner and these are particularly important in optimization problems. Importantly enough, these calculations prove the exactness of the {\sc rs} {\it Ansatz} below the percolation threshold and might require its revision between this and the easy-to-hard transition.Comment: 21 pages, 7 figs, to appear in Phys. Rev.