Application of statistical mechanics to NP-complete problems in combinatorial optimisation

Effect of constraint relaxation on the minimum vertex cover problem in random graphs

A statistical-mechanical study of the effect of constraint relaxation on the minimum vertex cover problem in Erdős-Rényi random graphs is presented. Using a penalty-method formulation for constraint relaxation, typical properties of solutions, including infeasible solutions that violate the constraints, are analyzed by means of the replica method and cavity method. The problem involves a competition between reducing the number of vertices to be covered and satisfying the edge constraints. The analysis under the replica-symmetric (RS) ansatz clarifies that the competition leads to degeneracies in the vertex and edge states, which determine the quantitative properties of the system, such as the cover and penalty ratios. A precise analysis of these effects improves the accuracy of RS approximation for the minimum cover ratio in the replica symmetry-breaking (RSB) region. Furthermore, the analysis based on the RS cavity method indicates that the RS/RSB boundary of the ground states with respect to the mean degree of the graphs is expanded, and the critical temperature is lowered by constraint relaxation.

Phase transition in the computational complexity of the shortest common superstring and genome assembly

Genome assembly, the process of reconstructing a long genetic sequence by aligning and merging short fragments, or reads, is known to be NP-hard, either as a version of the shortest common superstring problem or in a Hamiltonian-cycle formulation. That is, the computing time is believed to grow exponentially with the problem size in the worst case. Despite this fact, high-throughput technologies and modern algorithms currently allow bioinformaticians to handle datasets of billions of reads. Using methods from statistical mechanics, we address this conundrum by demonstrating the existence of a phase transition in the computational complexity of the problem and showing that practical instances always fall in the "easy" phase (solvable by polynomial-time algorithms). In addition, we propose a Markov-chain Monte Carlo method that outperforms common deterministic algorithms in the hard regime.

Characterizing the Asymmetry in Hardness between Synthesis and Destruction of Heteropolymers

We present a simple model describing the assembly and disassembly of heteropolymers consisting of two types of monomers A and B. We prove that no matter how we manipulate the concentrations of A and B, it takes longer than the exponential function of d to synthesize a fixed amount of the desired heteropolymer, where d is the number of A-B connections. We also prove the decomposition time is linear for chain length n. When d is proportional to n, synthesis and destruction have an exponential asymmetry. Our findings may facilitate research on the more general asymmetry of operational hardness.

Benchmark of quantum-inspired heuristic solvers for quadratic unconstrained binary optimization

Recently, inspired by quantum annealing, many solvers specialized for unconstrained binary quadratic programming problems have been developed. For further improvement and application of these solvers, it is important to clarify the differences in their performance for various types of problems. In this study, the performance of four quadratic unconstrained binary optimization problem solvers, namely D-Wave Hybrid Solver Service (HSS), Toshiba Simulated Bifurcation Machine (SBM), Fujitsu Digital Annealer (DA), and simulated annealing on a personal computer, was benchmarked. The problems used for benchmarking were instances of real problems in MQLib, instances of the SAT-UNSAT phase transition point of random not-all-equal 3-SAT (NAE 3-SAT), and the Ising spin glass Sherrington-Kirkpatrick (SK) model. Concerning MQLib instances, the HSS performance ranked first; for NAE 3-SAT, DA performance ranked first; and regarding the SK model, SBM performance ranked first. These results may help understand the strengths and weaknesses of these solvers.

Analytical Expressions for Ising Models on High Dimensional Lattices

We use an m-vicinity method to examine Ising models on hypercube lattices of high dimensions d≥3. This method is applicable for both short-range and long-range interactions. We introduce a small parameter, which determines whether the method can be used when calculating the free energy. When we account for interaction with the nearest neighbors only, the value of this parameter depends on the dimension of the lattice d. We obtain an expression for the critical temperature in terms of the interaction constants that is in a good agreement with the results of computer simulations. For d=5,6,7, our theoretical estimates match the numerical results both qualitatively and quantitatively. For d=3,4, our method is sufficiently accurate for the calculation of the critical temperatures; however, it predicts a finite jump of the heat capacity at the critical point. In the case of the three-dimensional lattice (d=3), this contradicts the commonly accepted ideas of the type of the singularity at the critical point. For the four-dimensional lattice (d=4), the character of the singularity is under current discussion. For the dimensions d=1, 2 the m-vicinity method is not applicable.

The overlap gap property: A topological barrier to optimizing over random structures

The problem of optimizing over random structures emerges in many areas of science and engineering, ranging from statistical physics to machine learning and artificial intelligence. For many such structures, finding optimal solutions by means of fast algorithms is not known and often is believed not to be possible. At the same time, the formal hardness of these problems in the form of the complexity-theoretic NP-hardness is lacking. A new approach for algorithmic intractability in random structures is described in this article, which is based on the topological disconnectivity property of the set of pairwise distances of near-optimal solutions, called the Overlap Gap Property. The article demonstrates how this property 1) emerges in most models known to exhibit an apparent algorithmic hardness; 2) is consistent with the hardness/tractability phase transition for many models analyzed to the day; and, importantly, 3) allows to mathematically rigorously rule out a large class of algorithms as potential contenders, specifically the algorithms that exhibit the input stability (insensitivity).

Cluster Structure of Optimal Solutions in Bipartitioning of Small Worlds

Using simulated annealing, we examine a bipartitioning of small worlds obtained by adding a fraction of randomly chosen links to a one-dimensional chain or a square lattice. Models defined on small worlds typically exhibit a mean-field behavior, regardless of the underlying lattice. Our work demonstrates that the bipartitioning of small worlds does depend on the underlying lattice. Simulations show that for one-dimensional small worlds, optimal partitions are finite size clusters for any fraction of additional links. In the two-dimensional case, we observe two regimes: when the fraction of additional links is sufficiently small, the optimal partitions have a stripe-like shape, which is lost for a larger number of additional links as optimal partitions become disordered. Some arguments, which interpret additional links as thermal excitations and refer to the thermodynamics of Ising models, suggest a qualitative explanation of such a behavior. The histogram of overlaps suggests that a replica symmetry is broken in a one-dimensional small world. In the two-dimensional case, the replica symmetry seems to hold, but with some additional degeneracy of stripe-like partitions.

Ground State Properties of the Diluted Sherrington-Kirkpatrick Spin Glass

We present a numerical study of ground states of the dilute versions of the Sherrington-Kirkpatrick (SK) mean-field spin glass. In contrast to so-called "sparse" mean-field spin glasses that have been studied widely on random networks of finite (average or regular) degree, the networks studied here are randomly bond diluted to an overall density p, such that the average degree diverges as ∼pN with the system size N. Ground state energies are obtained with high accuracy for random instances over a wide range of fixed p. Since this is an NP-hard combinatorial problem, we employ the extremal optimization heuristic to that end. We find that the exponent describing the finite-size corrections ω varies continuously with p, a somewhat surprising result, as one would not expect that gradual bond dilution would change the T=0 universality class of a statistical model. For p→1, the familiar result of ω(p=1)≈2/3 for the SK model is obtained.

Fluctuations in the random-link matching problem

Using the replica approach and the cavity method, we study the fluctuations of the optimal cost in the random-link matching problem. By means of replica arguments, we derive the exact expression of its variance. Moreover, we study the large deviation function, deriving its expression in two different ways, namely using both the replica method and the cavity method.

Experimental investigation of performance differences between coherent Ising machines and a quantum annealer

Physical annealing systems provide heuristic approaches to solving combinatorial optimization problems. Here, we benchmark two types of annealing machines-a quantum annealer built by D-Wave Systems and measurement-feedback coherent Ising machines (CIMs) based on optical parametric oscillators-on two problem classes, the Sherrington-Kirkpatrick (SK) model and MAX-CUT. The D-Wave quantum annealer outperforms the CIMs on MAX-CUT on cubic graphs. On denser problems, however, we observe an exponential penalty for the quantum annealer [exp(-α_DW N ²)] relative to CIMs [exp(-α_CIM N)] for fixed anneal times, both on the SK model and on 50% edge density MAX-CUT. This leads to a several orders of magnitude time-to-solution difference for instances with over 50 vertices. An optimal-annealing time analysis is also consistent with a substantial projected performance difference. The difference in performance between the sparsely connected D-Wave machine and the fully-connected CIMs provides strong experimental support for efforts to increase the connectivity of quantum annealers.

Statistical mechanical analysis of linear programming relaxation for combinatorial optimization problems

Typical behavior of the linear programming (LP) problem is studied as a relaxation of the minimum vertex cover (min-VC), a type of integer programming (IP) problem. A lattice-gas model on the Erdös-Rényi random graphs of α-uniform hyperedges is proposed to express both the LP and IP problems of the min-VC in the common statistical mechanical model with a one-parameter family. Statistical mechanical analyses reveal for α=2 that the LP optimal solution is typically equal to that given by the IP below the critical average degree c=e in the thermodynamic limit. The critical threshold for good accuracy of the relaxation extends the mathematical result c=1 and coincides with the replica symmetry-breaking threshold of the IP. The LP relaxation for the minimum hitting sets with α≥3, minimum vertex covers on α-uniform random graphs, is also studied. Analytic and numerical results strongly suggest that the LP relaxation fails to estimate optimal values above the critical average degree c=e/(α-1) where the replica symmetry is broken.

Ground-state energy of the q-state Potts model: The minimum modularity

A wide range of interacting systems can be described by complex networks. A common feature of such networks is that they consist of several communities or modules, the degree of which may quantified as the modularity. However, even a random uncorrelated network, which has no obvious modular structure, has a finite modularity due to the quenched disorder. For this reason, the modularity of a given network is meaningful only when it is compared with that of a randomized network with the same degree distribution. In this context, it is important to calculate the modularity of a random uncorrelated network with an arbitrary degree distribution. The modularity of a random network has been calculated [Reichardt and Bornholdt, Phys. Rev. E 76, 015102 (2007)PLEEE81539-375510.1103/PhysRevE.76.015102]; however, this was limited to the case whereby the network was assumed to have only two communities, and it is evident that the modularity should be calculated in general with q(≥2) communities. Here we calculate the modularity for q communities by evaluating the ground-state energy of the q-state Potts Hamiltonian, based on replica symmetric solutions assuming that the mean degree is large. We found that the modularity is proportional to 〈sqrt[k]〉/〈k〉 regardless of q and that only the coefficient depends on q. In particular, when the degree distribution follows a power law, the modularity is proportional to 〈k〉^{-1/2}. Our analytical results are confirmed by comparison with numerical simulations. Therefore, our results can be used as reference values for real-world networks.

Quantum annealing with antiferromagnetic fluctuations

We introduce antiferromagnetic quantum fluctuations into quantum annealing in addition to the conventional transverse-field term. We apply this method to the infinite-range ferromagnetic p-spin model, for which the conventional quantum annealing has been shown to have difficulties in finding the ground state efficiently due to a first-order transition. We study the phase diagram of this system both analytically and numerically. Using the static approximation, we find that there exists a quantum path to reach the final ground state from the trivial initial state that avoids first-order transitions for intermediate values of p. We also study numerically the energy gap between the ground state and the first excited state and find evidence for intermediate values of p for which the time complexity scales polynomially with the system size at a second-order transition point along the quantum path that avoids first-order transitions. These results suggest that quantum annealing would be able to solve this problem with intermediate values of p efficiently, in contrast to the case with only simple transverse-field fluctuations.

Physics and complexity

This paper is concerned with complex macroscopic behaviour arising in many-body systems through the combinations of competitive interactions and disorder, even with simple ingredients at the microscopic level. It attempts to indicate and illustrate the richness that has arisen, in conceptual understanding, in methodology and in application, across a large range of scientific disciplines, together with a hint of some of the further opportunities that remain to be tapped. In doing so, it takes the perspective of physics and tries to show, albeit rather briefly, how physics has contributed and been stimulated.

Theory of minimum spanning trees. I. Mean-field theory and strongly disordered spin-glass model

The minimum spanning tree (MST) is a combinatorial optimization problem: given a connected graph with a real weight ("cost") on each edge, find the spanning tree that minimizes the sum of the total cost of the occupied edges. We consider the random MST, in which the edge costs are (quenched) independent random variables. There is a strongly disordered spin-glass model due to Newman and Stein [Phys. Rev. Lett. 72, 2286 (1994)], which maps precisely onto the random MST. We study scaling properties of random MSTs using a relation between Kruskal's greedy algorithm for finding the MST, and bond percolation. We solve the random MST problem on the Bethe lattice (BL) with appropriate wired boundary conditions and calculate the fractal dimension D=6 of the connected components. Viewed as a mean-field theory, the result implies that on a lattice in Euclidean space of dimension d , there are of order W(d-D) large connected components of the random MST inside a window of size W , and that d=d(c)=D=6 is a critical dimension. This differs from the value 8 suggested by Newman and Stein. We also critique the original argument for 8, and provide an improved scaling argument that again yields d(c)=6 . The result implies that the strongly disordered spin-glass model has many ground states for d>6 , and only of order one below six. The results for MSTs also apply on the Poisson-weighted infinite tree, which is a mean-field approach to the continuum model of MSTs in Euclidean space, and is a limit of the BL. In a companion paper we develop an epsilon=6-d expansion for the random MST on critical percolation clusters.

Detecting network communities by propagating labels under constraints

We investigate the recently proposed label-propagation algorithm (LPA) for identifying network communities. We reformulate the LPA as an equivalent optimization problem, giving an objective function whose maxima correspond to community solutions. By considering properties of the objective function, we identify conceptual and practical drawbacks of the label-propagation approach, most importantly the disparity between increasing the value of the objective function and improving the quality of communities found. To address the drawbacks, we modify the objective function in the optimization problem, producing a variety of algorithms that propagate labels subject to constraints; of particular interest is a variant that maximizes the modularity measure of community quality. Performance properties and implementation details of the proposed algorithms are discussed. Bipartite as well as unipartite networks are considered.

Broad lifetime distributions for ordering dynamics in complex networks

We search for conditions under which a characteristic time scale for ordering dynamics toward either of two absorbing states in a finite complex network of interactions does not exist. With this aim, we study random networks and networks with mesoscale community structure built up from randomly connected cliques. We find that large heterogeneity at the mesoscale level of the network appears to be a sufficient mechanism for the absence of a characteristic time for the dynamics. Such heterogeneity results in dynamical metastable states that survive at any time scale.

Statistical mechanics of the quantum K -satisfiability problem

We study the quantum version of the random K -satisfiability problem in the presence of an external magnetic field Gamma applied in the transverse direction. We derive the replica-symmetric free-energy functional within the static approximation and the saddle-point equation for the order parameter: the distribution P[h(m)] of functions of magnetizations. The order parameter is interpreted as the histogram of probability distributions of individual magnetizations. In the limit of zero temperature and small transverse fields, to leading order in Gamma magnetizations m approximately 0 become relevant in addition to purely classical values of m approximately +/-1 . Self-consistency equations for the order parameter are solved numerically using a quasi-Monte Carlo method for K=3 . It is shown that for an arbitrarily small Gamma quantum fluctuations destroy the phase transition present in the classical limit Gamma=0 , replacing it with a smooth crossover transition. The implications of this result with respect to the expected performance of quantum optimization algorithms via adiabatic evolution are discussed. The replica-symmetric solution of the classical random K -satisfiability problem is briefly reexamined. It is shown that the phase transition at T=0 predicted by the replica-symmetric theory is of continuous type with atypical critical exponents.

(Un)detectable cluster structure in sparse networks

Can a cluster structure in a sparse relational data set, i.e., a network, be detected at all by unsupervised clustering techniques? We answer this question by means of statistical mechanics making our analysis independent of any particular algorithm used for clustering. We find a sharp transition from a phase in which the cluster structure is not detectable at all to a phase in which it can be detected with high accuracy. We calculate the transition point and the shape of the transition, i.e., the theoretically achievable accuracy, analytically. This illuminates theoretical limitations of data mining in networks and allows for an understanding and evaluation of the performance of a variety of algorithms.

Cooperative updating in the Hopfield model

We propose a new method for updating units in the Hopfield model. With this method two or more units change at the same time, so as to become the lowest energy state among all possible states. Since this updating algorithm is based on the detailed balance equation, convergence to the Boltzmann distribution is guaranteed. If our algorithm is applied to finding the minimum energy in constraint satisfaction and combinatorial optimization problems, then there is a faster convergence than those with the usual algorithm in the neural network. This is shown by experiments with the travelling salesman problem, the four-color problem, the N-queen problem, and the graph bi-partitioning problem. In constraint satisfaction problems, for which earlier neural networks are effective in some cases, our updating scheme works fine. Even though we still encounter the problem of ending up in local minima, our updating scheme has a great advantage compared with the usual updating scheme used in combinatorial optimization problems. Also, we discuss parallel computing using our updating algorithm.

Landscape analysis of constraint satisfaction problems

We discuss an analysis of constraint satisfaction problems, such as sphere packing, K-SAT, and graph coloring, in terms of an effective energy landscape. Several intriguing geometrical properties of the solution space become in this light familiar in terms of the well-studied ones of rugged (glassy) energy landscapes. A benchmark algorithm naturally suggested by this construction finds solutions in polynomial time up to a point beyond the clustering and in some cases even the thermodynamic transitions. This point has a simple geometric meaning and can be in principle determined with standard statistical mechanical methods, thus pushing the analytic bound up to which problems are guaranteed to be easy. We illustrate this for the graph 3- and 4-coloring problem. For packing problems the present discussion allows to better characterize the J-point, proposed as a systematic definition of random close packing, and to place it in the context of other theories of glasses.

Partitioning and modularity of graphs with arbitrary degree distribution

We solve the graph bipartitioning problem in dense graphs with arbitrary degree distribution using the replica method. We find the cut size to scale universally with <square root k> . In contrast, earlier results studying the problem in graphs with a Poissonian degree distribution had found a scaling with square root <k> [Fu and Anderson, J. Phys. A 19, 1605 (1986)]. Our results also generalize to the problem of q partitioning. They can be used to find the expected modularity Q [Newman and Girvan, Phys. Rev. E 69, 026113 (2004)] of random graphs and allow for the assessment of the statistical significance of the output of community detection algorithms.

Complexity of the predecessor problem in Kauffman networks

Kauffman nets, also known as N-K models, have been studied extensively because their dynamics can be used to model a variety of interesting dynamical processes. This paper investigates the properties of the problem of determining whether or not a given configuration of a Kauffman net has a predecessor. Here it is shown that when the parameter K that governs the number of connections grows as ln(N) , where N is the number of elements, the problem of finding a solution is extremely sensitive to small changes in the problem statement. This result has implications for studies of the physics of random systems and also may have applications for questions in computational complexity theory.

Statistical mechanics of community detection

Starting from a general ansatz, we show how community detection can be interpreted as finding the ground state of an infinite range spin glass. Our approach applies to weighted and directed networks alike. It contains the ad hoc introduced quality function from [J. Reichardt and S. Bornholdt, Phys. Rev. Lett. 93, 218701 (2004)] and the modularity Q as defined by Newman and Girvan [Phys. Rev. E 69, 026113 (2004)] as special cases. The community structure of the network is interpreted as the spin configuration that minimizes the energy of the spin glass with the spin states being the community indices. We elucidate the properties of the ground state configuration to give a concise definition of communities as cohesive subgroups in networks that is adaptive to the specific class of network under study. Further, we show how hierarchies and overlap in the community structure can be detected. Computationally efficient local update rules for optimization procedures to find the ground state are given. We show how the ansatz may be used to discover the community around a given node without detecting all communities in the full network and we give benchmarks for the performance of this extension. Finally, we give expectation values for the modularity of random graphs, which can be used in the assessment of statistical significance of community structure.

Statistical mechanics of combinatorial optimization problems with site disorder

We study the statistical mechanics of a class of problems whose phase space is the set of permutations of an ensemble of quenched random positions. Specific examples analyzed are the finite-temperature traveling salesman problem on several different domains and various problems in one dimension such as the so-called descent problem. We first motivate our method by analyzing these problems using the annealed approximation. Then in the limit of a large number of points we develop a formalism to carry out the quenched calculation. This formalism does not require the replica method, and its predictions are found to agree with Monte Carlo simulations. In addition our method reproduces an exact mathematical result for the maximum traveling salesman problem in two dimensions and suggests its generalization to higher dimensions. The general approach may provide an alternative method to study certain systems with quenched disorder.

Biased random satisfiability problems: from easy to hard instances

In this paper we study biased random K -satisfiability ( K -SAT) problems in which each logical variable is negated with probability p . This generalization provides us a crossover from easy to hard problems and would help us in a better understanding of the typical complexity of random K -SAT problems. The exact solution of 1-SAT case is given. The critical point of K -SAT problems and results of replica method are derived in the replica symmetry framework. It is found that in this approximation alpha(c) proportional p(-(K-1)) for p --> 0. Solving numerically the survey propagation equations for K = 3 we find that for p < p* approximately 0.17 there is no replica symmetry breaking and still the SAT-UNSAT transition is discontinuous.

Rigorous location of phase transitions in hard optimization problems

It is widely believed that for many optimization problems, no algorithm is substantially more efficient than exhaustive search. This means that finding optimal solutions for many practical problems is completely beyond any current or projected computational capacity. To understand the origin of this extreme 'hardness', computer scientists, mathematicians and physicists have been investigating for two decades a connection between computational complexity and phase transitions in random instances of constraint satisfaction problems. Here we present a mathematically rigorous method for locating such phase transitions. Our method works by analysing the distribution of distances between pairs of solutions as constraints are added. By identifying critical behaviour in the evolution of this distribution, we can pinpoint the threshold location for a number of problems, including the two most-studied ones: random k-SAT and random graph colouring. Our results prove that the heuristic predictions of statistical physics in this context are essentially correct. Moreover, we establish that random instances of constraint satisfaction problems have solutions well beyond the reach of any analysed algorithm.

Detecting fuzzy community structures in complex networks with a Potts model

A fast community detection algorithm based on a q-state Potts model is presented. Communities (groups of densely interconnected nodes that are only loosely connected to the rest of the network) are found to coincide with the domains of equal spin value in the minima of a modified Potts spin glass Hamiltonian. Comparing global and local minima of the Hamiltonian allows for the detection of overlapping ("fuzzy") communities and quantifying the association of nodes with multiple communities as well as the robustness of a community. No prior knowledge of the number of communities has to be assumed.

Quantum adiabatic optimization and combinatorial landscapes

In this paper we analyze the performance of the Quantum Adiabatic Evolution algorithm on a variant of the satisfiability problem for an ensemble of random graphs parametrized by the ratio of clauses to variables, gamma=M/N . We introduce a set of macroscopic parameters (landscapes) and put forward an ansatz of universality for random bit flips. We then formulate the problem of finding the smallest eigenvalue and the excitation gap as a statistical mechanics problem. We use the so-called annealing approximation with a refinement that a finite set of macroscopic variables (instead of only energy) is used, and are able to show the existence of a dynamic threshold gamma= gamma(d) starting with some value of K -the number of variables in each clause. Beyond the dynamic threshold, the algorithm should take an exponentially long time to find a solution. We compare the results for extended and simplified sets of landscapes and provide numerical evidence in support of our universality ansatz. We have been able to map the ensemble of random graphs onto another ensemble with fluctuations significantly reduced. This enabled us to obtain tight upper bounds on the satisfiability transition and to recompute the dynamical transition using the extended set of landscapes.

Network bipartivity

Systems with two types of agents with a preference for heterophilous interaction produce networks that are more or less close to bipartite. We propose two measures quantifying the notion of bipartivity. The two measures-one well known and natural, but computationally intractable, and the other computationally less complex, but also less intuitive-are examined on model networks that continuously interpolate between bipartite graphs and graphs with many odd circuits. We find that the bipartivity measures increase as we tune the control parameters of the test networks to intuitively increase the bipartivity, and thus conclude that the measures are quite relevant. We also measure and discuss the values of our bipartivity measures for empirical social networks (constructed from professional collaborations, Internet communities, and field surveys). Here we find, as expected, that networks arising from romantic online interaction have high, and professional collaboration networks have low, bipartivity values. In some other cases, probably due to low average degree of the network, the bipartivity measures cannot distinguish between romantic and friendship oriented interaction.

Disappearance of spurious states in analog associative memories

We show that symmetric n-mixture states, when they exist, are almost never stable in autoassociative networks with threshold-linear units. Only with a binary coding scheme, we could find a limited region of the parameter space in which either 2-mixture or 3-mixture states are stable attractors of the dynamics.

Random K-satisfiability problem: from an analytic solution to an efficient algorithm

We study the problem of satisfiability of randomly chosen clauses, each with K Boolean variables. Using the cavity method at zero temperature, we find the phase diagram for the K=3 case. We show the existence of an intermediate phase in the satisfiable region, where the proliferation of metastable states is at the origin of the slowdown of search algorithms. The fundamental order parameter introduced in the cavity method, which consists of surveys of local magnetic fields in the various possible states of the system, can be computed for one given sample. These surveys can be used to invent new types of algorithms for solving hard combinatorial optimizations problems. One such algorithm is shown here for the K=3 satisfiability problem, with very good performances.

Analytic and algorithmic solution of random satisfiability problems

We study the satisfiability of random Boolean expressions built from many clauses with K variables per clause (K-satisfiability). Expressions with a ratio alpha of clauses to variables less than a threshold alphac are almost always satisfiable, whereas those with a ratio above this threshold are almost always unsatisfiable. We show the existence of an intermediate phase below alphac, where the proliferation of metastable states is responsible for the onset of complexity in search algorithms. We introduce a class of optimization algorithms that can deal with these metastable states; one such algorithm has been tested successfully on the largest existing benchmark of K-satisfiability.

Extreme value statistics and traveling fronts: application to computer science

We study the statistics of height and balanced height in the binary search tree problem in computer science. The search tree problem is first mapped to a fragmentation problem that is then further mapped to a modified directed polymer problem on a Cayley tree. We employ the techniques of traveling fronts to solve the polymer problem and translate back to derive exact asymptotic properties in the original search tree problem. The second mapping allows us not only to again derive the already known results for random binary trees but to obtain exact results for search trees where the entries arrive according to an arbitrary distribution, not necessarily randomly. Besides it allows us to derive the asymptotic shape of the full probability distribution of height and not just its moments. Our results are then generalized to m-ary search trees with arbitrary distribution.

Extremal optimization for graph partitioning

Extremal optimization is a new general-purpose method for approximating solutions to hard optimization problems. We study the method in detail by way of the computationally hard (NP-hard) graph partitioning problem. We discuss the scaling behavior of extremal optimization, focusing on the convergence of the average run as a function of run time and system size. The method has a single free parameter, which we determine numerically and justify using a simple argument. On random graphs, our numerical results demonstrate that extremal optimization maintains consistent accuracy for increasing system sizes, with an approximation error decreasing over run time roughly as a power law t(-0.4). On geometrically structured graphs, the scaling of results from the average run suggests that these are far from optimal with large fluctuations between individual trials. But when only the best runs are considered, results consistent with theoretical arguments are recovered.

Competition between glassiness and order in a multispin glass

A mean-field multispin interaction spin glass model is analyzed in the presence of a ferromagnetic coupling. The static and dynamical phase diagrams contain four phases (paramagnet, spin glass, ordinary ferromagnet, and glassy ferromagnet) and exhibit reentrant behavior. The glassy ferromagnet phase has anomalous dynamical properties. The results are consistent with a nonequilibrium thermodynamics that has been proposed for glasses.