The Scaling Window of the 2-SAT Transition

We consider the random 2-satisfiability problem, in which each instance is a formula that is the conjunction of m clauses of the form (x or y), chosen uniformly at random from among all 2-clauses on n Boolean variables and their negations. As m and n tend to infinity in the ratio m/n --> alpha, the problem is known to have a phase transition at alpha_c = 1, below which the probability that the formula is satisfiable tends to one and above which it tends to zero. We determine the finite-size scaling about this transition, namely the scaling of the maximal window W(n,delta) = (alpha_-(n,delta),alpha_+(n,delta)) such that the probability of satisfiability is greater than 1-delta for alpha < alpha_- and is less than delta for alpha > alpha_+. We show that W(n,delta)=(1-Theta(n^{-1/3}),1+Theta(n^{-1/3})), where the constants implicit in Theta depend on delta. We also determine the rates at which the probability of satisfiability approaches one and zero at the boundaries of the window. Namely, for m=(1+epsilon)n, where epsilon may depend on n as long as |epsilon| is sufficiently small and |epsilon|*n^(1/3) is sufficiently large, we show that the probability of satisfiability decays like exp(-Theta(n*epsilon^3)) above the window, and goes to one like 1-Theta(1/(n*|epsilon|^3)) below the window. We prove these results by defining an order parameter for the transition and establishing its scaling behavior in n both inside and outside the window. Using this order parameter, we prove that the 2-SAT phase transition is continuous with an order parameter critical exponent of 1. We also determine the values of two other critical exponents, showing that the exponents of 2-SAT are identical to those of the random graph.Comment: 57 pages. This version updates some reference

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

Criticality and Universality in the Unit-Propagation Search Rule

The probability Psuccess(alpha, N) that stochastic greedy algorithms successfully solve the random SATisfiability problem is studied as a function of the ratio alpha of constraints per variable and the number N of variables. These algorithms assign variables according to the unit-propagation (UP) rule in presence of constraints involving a unique variable (1-clauses), to some heuristic (H) prescription otherwise. In the infinite N limit, Psuccess vanishes at some critical ratio alpha\_H which depends on the heuristic H. We show that the critical behaviour is determined by the UP rule only. In the case where only constraints with 2 and 3 variables are present, we give the phase diagram and identify two universality classes: the power law class, where Psuccess[alpha\_H (1+epsilon N^{-1/3}), N] ~ A(epsilon)/N^gamma; the stretched exponential class, where Psuccess[alpha\_H (1+epsilon N^{-1/3}), N] ~ exp[-N^{1/6} Phi(epsilon)]. Which class is selected depends on the characteristic parameters of input data. The critical exponent gamma is universal and calculated; the scaling functions A and Phi weakly depend on the heuristic H and are obtained from the solutions of reaction-diffusion equations for 1-clauses. Computation of some non-universal corrections allows us to match numerical results with good precision. The critical behaviour for constraints with >3 variables is given. Our results are interpreted in terms of dynamical graph percolation and we argue that they should apply to more general situations where UP is used.Comment: 30 pages, 13 figure

Extremal Optimization at the Phase Transition of the 3-Coloring Problem

We investigate the phase transition of the 3-coloring problem on random graphs, using the extremal optimization heuristic. 3-coloring is among the hardest combinatorial optimization problems and is closely related to a 3-state anti-ferromagnetic Potts model. Like many other such optimization problems, it has been shown to exhibit a phase transition in its ground state behavior under variation of a system parameter: the graph's mean vertex degree. This phase transition is often associated with the instances of highest complexity. We use extremal optimization to measure the ground state cost and the ``backbone'', an order parameter related to ground state overlap, averaged over a large number of instances near the transition for random graphs of size $n$ up to 512. For graphs up to this size, benchmarks show that extremal optimization reaches ground states and explores a sufficient number of them to give the correct backbone value after about $O(n^{3.5})$ update steps. Finite size scaling gives a critical mean degree value $\alpha_{\rm c}=4.703(28)$. Furthermore, the exploration of the degenerate ground states indicates that the backbone order parameter, measuring the constrainedness of the problem, exhibits a first-order phase transition.Comment: RevTex4, 8 pages, 4 postscript figures, related information available at http://www.physics.emory.edu/faculty/boettcher

Critical behaviour of combinatorial search algorithms, and the unitary-propagation universality class

The probability P(alpha, N) that search algorithms for random Satisfiability problems successfully find a solution is studied as a function of the ratio alpha of constraints per variable and the number N of variables. P is shown to be finite if alpha lies below an algorithm--dependent threshold alpha\_A, and exponentially small in N above. The critical behaviour is universal for all algorithms based on the widely-used unitary propagation rule: P[ (1 + epsilon) alpha\_A, N] ~ exp[-N^(1/6) Phi(epsilon N^(1/3)) ]. Exponents are related to the critical behaviour of random graphs, and the scaling function Phi is exactly calculated through a mapping onto a diffusion-and-death problem.Comment: 7 pages; 3 figure

Intermediate Phases, structural variance and network demixing in chalcogenides: the unusual case of group V sulfides

We review Intermediate Phases (IPs) in chalcogenide glasses and provide a structural interpretation of these phases. In binary group IV selenides, IPs reside in the 2.40 < r < 2.54 range, and in binary group V selenides they shift to a lower r, in the 2.29< r < 2.40 range. Here r represents the mean coordination number of glasses. In ternary alloys containing equal proportions of group IV and V selenides, IPs are wider and encompass ranges of respective binary glasses. These data suggest that the local structural variance contributing to IP widths largely derives from four isostatic local structures of varying connectivity r; two include group V based quasi-tetrahedral (r = 2.29) and pyramidal (r = 2.40) units, and the other two are group IV based corner-sharing (r = 2.40) and edge-sharing (r = 2.67) tetrahedral units. Remarkably, binary group V (P, As) sulfides exhibit IPs that are shifted to even a lower r than their selenide counterparts; a result that we trace to excess Sn chains either partially (As-S) or completely (P-S) demixing from network backbone, in contrast to excess Sen chains forming part of the backbone in corresponding selenide glasses. In ternary chalcogenides of Ge with the group V elements (As, P), IPs of the sulfides are similar to their selenide counterparts, suggesting that presence of Ge serves to reign in the excess Sn chain fragments back in the backbone as in their selenide counterparts

Finite size scaling for the core of large random hypergraphs

The (two) core of a hypergraph is the maximal collection of hyperedges within which no vertex appears only once. It is of importance in tasks such as efficiently solving a large linear system over GF[2], or iterative decoding of low-density parity-check codes used over the binary erasure channel. Similar structures emerge in a variety of NP-hard combinatorial optimization and decision problems, from vertex cover to satisfiability. For a uniformly chosen random hypergraph of $m=n\rho$ vertices and $n$ hyperedges, each consisting of the same fixed number $l\geq3$ of vertices, the size of the core exhibits for large $n$ a first-order phase transition, changing from $o(n)$ for $\rho>\rho _{\mathrm{c}}$ to a positive fraction of $n$ for $\rho<\rho_{\mathrm{c}}$, with a transition window size $\Theta(n^{-1/2})$ around $\rho_{\mathrm{c}}>0$. Analyzing the corresponding ``leaf removal'' algorithm, we determine the associated finite-size scaling behavior. In particular, if $\rho$ is inside the scaling window (more precisely, $\rho=\rho_{\mathrm{c}}+rn^{-1/2}$), the probability of having a core of size $\Theta(n)$ has a limit strictly between 0 and 1, and a leading correction of order $\Theta(n^{-1/6})$. The correction admits a sharp characterization in terms of the distribution of a Brownian motion with quadratic shift, from which it inherits the scaling with $n$. This behavior is expected to be universal for a wide collection of combinatorial problems.Comment: Published in at http://dx.doi.org/10.1214/07-AAP514 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org