Introduction to Phase Transitions in Random Optimization Problems
Notes of the lectures delivered in Les Houches during the Summer School on Complex Systems (July 2006).
cond-mat.stat-mech↗
arXiv subjects
Remi Monasson
Publications and source records attributed to Remi Monasson.
Inferring DNA sequences from mechanical unzipping data: the large-bandwidth case
The complementary strands of DNA molecules can be separated when stretched apart by a force; the unzipping signal is correlated to the base content of the sequence but is affected by thermal and instrumental noise. We consider here the ideal case where opening events are known to a very good time resolution (very large bandwidth), and study how the sequence can be reconstructed from the unzipping data. Our approach relies on the use of statistical Bayesian inference and of Viterbi decoding algorithm. Performances are studied numerically on Monte Carlo generated data, and analytically. We show how multiple unzippings of the same molecule may be exploited to improve the quality of the prediction, and calculate analytically the number of required unzippings as a function of the bandwidth, the sequence content, the elasticity parameters of the unzipped strands.
q-bio.BM↗
Reconstructing a Random Potential from its Random Walks
The problem of how many trajectories of a random walker in a potential are needed to reconstruct the values of this potential is studied. We show that this problem can be solved by calculating the probability of survival of an abstract random walker in a partially absorbing potential. The approach is illustrated on the discrete Sinai (random force) model with a drift. We determine the parameter (temperature, duration of each trajectory, ...) values making reconstruction as fast as possible.
cond-mat.stat-mech↗
Can rare SAT formulas be easily recognized? On the efficiency of message passing algorithms for K-SAT at large clause-to-variable ratios
For large clause-to-variable ratio, typical K-SAT instances drawn from the uniform distribution have no solution. We argue, based on statistical mechanics calculations using the replica and cavity methods, that rare satisfiable instances from the uniform distribution are very similar to typical instances drawn from the so-called planted distribution, where instances are chosen uniformly between the ones that admit a given solution. It then follows, from a recent article by Feige, Mossel and Vilenchik, that these rare instances can be easily recognized (in O(log N) time and with probability close to 1) by a simple message-passing algorithm.
cs.CC↗
An algorithm for counting circuits: application to real-world and random graphs
We introduce an algorithm which estimates the number of circuits in a graph as a function of their length. This approach provides analytical results for the typical entropy of circuits in sparse random graphs. When applied to real-world networks, it allows to estimate exponentially large numbers of circuits in polynomial time. We illustrate the method by studying a graph of the Internet structure.
cond-mat.stat-mech↗
A generating function method for the average-case analysis of DPLL
A method to calculate the average size of Davis-Putnam-Loveland-Logemann (DPLL) search trees for random computational problems is introduced, and applied to the satisfiability of random CNF formulas (SAT) and the coloring of random graph (COL) problems. We establish recursion relations for the generating functions of the average numbers of (variable or color) assignments at a given height in the search tree, which allow us to derive the asymptotics of the expected DPLL tree size, 2^{N w + o(N)}, where N is the instance size. w is calculated as a function of the input distribution parameters (ratio of clauses per variable for SAT, average vertex degree for COL), and the branching heuristics.
cs.CC↗
Circuits in random graphs: from local trees to global loops
We compute the number of circuits and of loops with multiple crossings in random regular graphs. We discuss the importance of this issue for the validity of the cavity approach. On the one side we obtain analytic results for the infinite volume limit in agreement with existing exact results. On the other side we implement a counting algorithm, enumerate circuits at finite N and draw some general conclusions about the finite N behavior of the circuits.
cond-mat.stat-mech↗
Heuristic average-case analysis of the backtrack resolution of random 3-Satisfiability instances
An analysis of the average-case complexity of solving random 3-Satisfiability (SAT) instances with backtrack algorithms is presented. We first interpret previous rigorous works in a unifying framework based on the statistical physics notions of dynamical trajectories, phase diagram and growth process. It is argued that, under the action of the Davis--Putnam--Loveland--Logemann (DPLL) algorithm, 3-SAT instances are turned into 2+p-SAT instances whose characteristic parameters (ratio alpha of clauses per variable, fraction p of 3-clauses) can be followed during the operation, and define resolution trajectories. Depending on the location of trajectories in the phase diagram of the 2+p-SAT model, easy (polynomial) or hard (exponential) resolutions are generated. Three regimes are identified, depending on the ratio alpha of the 3-SAT instance to be solved. Lower sat phase: for small ratios, DPLL almost surely finds a solution in a time growing linearly with the number N of variables. Upper sat phase: for intermediate ratios, instances are almost surely satisfiable but finding a solution requires exponential time (2 ^ (N omega) with omega>0) with high probability. Unsat phase: for large ratios, there is almost always no solution and proofs of refutation are exponential. An analysis of the growth of the search tree in both upper sat and unsat regimes is presented, and allows us to estimate omega as a function of alpha. This analysis is based on an exact relationship between the average size of the search tree and the powers of the evolution operator encoding the elementary steps of the search heuristic.
cs.DS↗
The dynamics of proving uncolourability of large random graphs I. Symmetric Colouring Heuristic
We study the dynamics of a backtracking procedure capable of proving uncolourability of graphs, and calculate its average running time T for sparse random graphs, as a function of the average degree c and the number of vertices N. The analysis is carried out by mapping the history of the search process onto an out-of-equilibrium (multi-dimensional) surface growth problem. The growth exponent of the average running time is quantitatively predicted, in agreement with simulations.
cond-mat.stat-mech↗
Approximate analysis of search algorithms with "physical" methods
An overview of some methods of statistical physics applied to the analysis of algorithms for optimization problems (satisfiability of Boolean constraints, vertex cover of graphs, decoding, ...) with distributions of random inputs is proposed. Two types of algorithms are analyzed: complete procedures with backtracking (Davis-Putnam-Loveland-Logeman algorithm) and incomplete, local search procedures (gradient descent, random walksat, ...). The study of complete algorithms makes use of physical concepts such as phase transitions, dynamical renormalization flow, growth processes, ... As for local search procedures, the connection between computational complexity and the structure of the cost function landscape is questioned, with emphasis on the notion of metastability.
cs.CC↗
Relaxation and Metastability in the RandomWalkSAT search procedure
An analysis of the average properties of a local search resolution procedure for the satisfaction of random Boolean constraints is presented. Depending on the ratio alpha of constraints per variable, resolution takes a time T_res growing linearly (T_res \sim tau(alpha) N, alpha < alpha_d) or exponentially (T_res \sim exp(N zeta(alpha)), alpha > alpha_d) with the size N of the instance. The relaxation time tau(alpha) in the linear phase is calculated through a systematic expansion scheme based on a quantum formulation of the evolution operator. For alpha > alpha_d, the system is trapped in some metastable state, and resolution occurs from escape from this state through crossing of a large barrier. An annealed calculation of the height zeta(alpha) of this barrier is proposed. The polynomial/exponentiel cross-over alpha_d is not related to the onset of clustering among solutions.
cond-mat↗
Analysis of the computational complexity of solving random satisfiability problems using branch and bound search algorithms
The computational complexity of solving random 3-Satisfiability (3-SAT) problems is investigated. 3-SAT is a representative example of hard computational tasks; it consists in knowing whether a set of alpha N randomly drawn logical constraints involving N Boolean variables can be satisfied altogether or not. Widely used solving procedures, as the Davis-Putnam-Loveland-Logeman (DPLL) algorithm, perform a systematic search for a solution, through a sequence of trials and errors represented by a search tree. In the present study, we identify, using theory and numerical experiments, easy (size of the search tree scaling polynomially with N) and hard (exponential scaling) regimes as a function of the ratio alpha of constraints per variable. The typical complexity is explicitly calculated in the different regimes, in very good agreement with numerical simulations. Our theoretical approach is based on the analysis of the growth of the branches in the search tree under the operation of DPLL. On each branch, the initial 3-SAT problem is dynamically turned into a more generic 2+p-SAT problem, where p and 1-p are the fractions of constraints involving three and two variables respectively. The growth of each branch is monitored by the dynamical evolution of alpha and p and is represented by a trajectory in the static phase diagram of the random 2+p-SAT problem. Depending on whether or not the trajectories cross the boundary between phases, single branches or full trees are generated by DPLL, resulting in easy or hard resolutions.
cond-mat.stat-mech↗
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.
cond-mat.dis-nn↗
From inherent structures to pure states: some simple remarks and examples
The notions of pure states and inherent structures, i.e. stable configurations against 1-spin flip are discussed. We explain why these different concepts accidentally coincide in mean-field models with infinite connectivity and present an exactly solvable unidimensional model where they do not. At zero temperature pure states are to some extent related to k-spin flip stable configurations with k going to infinity after the thermodynamical limit has been taken. This relationship is supported by an explicit analysis of the TAP equations and calculation of the number of pure states and k-spin flips stable configurations in a mean-field model with finite couplings. Finally we discuss the relevance of the concepts of pure states and inherent structures in finite dimensional glassy systems.
cond-mat.dis-nn↗
A variational description of the ground state structure in random satisfiability problems
A variational approach to finite connectivity spin-glass-like models is developed and applied to describe the structure of optimal solutions in random satisfiability problems. Our variational scheme accurately reproduces the known replica symmetric results and also allows for the inclusion of replica symmetry breaking effects. For the 3-SAT problem, we find two transitions as the ratio $α$ of logical clauses per Boolean variables increases. At the first one $α_s \simeq 3.96$, a non-trivial organization of the solution space in geometrically separated clusters emerges. The multiplicity of these clusters as well as the typical distances between different solutions are calculated. At the second threshold $α_c \simeq 4.48$, satisfying assignments disappear and a finite fraction $B_0 \simeq 0.13$ of variables are overconstrained and take the same values in all optimal (though unsatisfying) assignments. These values have to be compared to $α_c \simeq 4.27, B_0 \simeq 0.4$ obtained from numerical experiments on small instances. Within the present variational approach, the SAT-UNSAT transition naturally appears as a mixture of a first and a second order transition. For the mixed $2+p$-SAT with $p<2/5$, the behavior is as expected much simpler: a unique smooth transition from SAT to UNSAT takes place at $α_c=1/(1-p)$.
cond-mat.dis-nn↗
Theoretical study of collective modes in DNA at ambient temperature
The instantaneous normal modes corresponding to base pair vibrations (radial modes) and twist angle fluctuations (angular modes) of a DNA molecule model at ambient temperature are theoretically investigated. Due to thermal disorder, normal modes are not plane waves with a single wave number q but have a finite and frequency dependent damping width. The density of modes rho(nu), the average dispersion relation nu(q) as well as the coherence length xi(nu) are analytically calculated. The Gibbs averaged resolvent is computed using a replicated transfer matrix formalism and variational wave functions for the ground and first excited state. Our results for the density of modes are compared to Raman spectroscopy measurements of the collective modes for DNA in solution and show a good agreement with experimental data in the low frequency regime nu < 150 cm^{-1}. Radial modes extend over frequencies ranging from 50 cm^{-1} to 110 cm^{-1}. Angular modes, related to helical axis vibrations are limited to nu < 25 cm^{-1}. Normal modes are highly disordered and coherent over a few base pairs only (xi < 2 nm) in good agreement with neutron scattering experiments.
cond-mat.stat-mech↗
Statistical Mechanics of Torque Induced Denaturation of DNA
A unifying theory of the denaturation transition of DNA, driven by temperature T or induced by an external mechanical torque Gamma is presented. Our model couples the hydrogen-bond opening and the untwisting of the helicoidal molecular structure. We show that denaturation corresponds to a first-order phase transition from B-DNA to d-DNA phases and that the coexistence region is naturally parametrized by the degree of supercoiling sigma. The denaturation free energy, the temperature dependence of the twist angle, the phase diagram in the T,Gamma plane and isotherms in the sigma, Gamma plane are calculated and show a good agreement with experimental data.
cond-mat.stat-mech↗
Diffusion, localization and dispersion relations on ``small-world'' lattices
The spectral properties of the Laplacian operator on ``small-world'' lattices, that is mixtures of unidimensional chains and random graphs structures are investigated numerically and analytically. A transfer matrix formalism including a self-consistent potential a la Edwards is introduced. In the extended region of the spectrum, an effective medium calculation provides the density of states and pseudo relations of dispersion for the eigenmodes in close agreement with the simulations. Localization effects, which are due to connectivity fluctuations of the sites are shown to be quantitatively described by the single defect approximation recently introduced for random graphs.
cond-mat.dis-nn↗