random constraint satisfaction easy generation of hard satisfiable instances in this paper we show that the models of random csp instances proposed by xu and li k xu w li exact phase transitions in random constraint satisfaction problems journal of artificial intelligence research 12 2000 93 103 k xu w li many hard examples in exact phase transitions with application to generating hard satisfiable instances technical report corr report cs cc 0302001 revised version in theoretical computer science 355 2006 291 302 are of theoretical and practical interest indeed these models called rb and rd present several nice features first it is quite easy to generate random instances of any arity since no particular structure has to be integrated or property enforced in such instances then the existence of an asymptotic phase transition can be guaranteed while applying a limited restriction on domain size and on constraint tightness in that case a threshold point can be precisely located and all instances have the guarantee to be hard at the threshold i e to have an exponential tree resolution complexity next a formal analysis shows that it is possible to generate forced satisfiable instances whose hardness is similar to unforced satisfiable ones this analysis is supported by some representative results taken from an intensive experimentation that we have carried out using complete and incomplete search methods

slow motion manifolds far from the attractor in multistable reaction diffusion equations abstract we consider a scalar reaction diffusion equation with multistable nonlinearity with a particular symmetry by reduction to a family of transmission problems in r and by contraction arguments a manifold close to an invariant manifold formed by functions exhibiting a pattern of transition layers is constructed an approximation for the associated vector field is also provided this shows that the motion on those manifolds is exponentially slow as in the well known case of the bistable equation however in opposition to the bistable case some of these manifolds are far from the attractor since these manifolds correspond to metastable patterns this shows the importance of the transient motion toward the attractor and the importance of these manifolds in organizing that motion it is also shown that by a suitable perturbation we can obtain new equilibria on those manifolds

on the phase transitions of random k constraint satisfaction problems constraint satisfaction has received increasing attention over the years intense research has focused on solving all kinds of constraint satisfaction problems csps in this paper first we propose a random csp model named k csp that guarantees the existence of phase transitions under certain circumstances the exact location of the phase transition is quantified and experimental results are provided to illustrate the performance of the proposed model second we revise the model k csp to a random linear csp by incorporating certain linear structure to constraint relations we also prove the existence of the phase transition and exhibit its exact location for this random linear csp model

metastable patterns for the cahn hilliard equation part ii layer dynamics and slow invariant manifold abstract in this paper we study the dynamics of the 1 dimensional cahn hilliard equation u t 2 u xx w u xx in a finite interval in a neighborhood of an equilibrium with n 1 transition layers where is a small parameter and w is a double well energy density function with equal minima the lower bound of the layer motion speed is given explictly and the layer motion directions are described precisely if a solution of the cahn hilliard equation starts outside a neighborhood of the equilibrium of size o ln 1 it is proved that there is an n dimensional unstable invariant manifold which is a smooth graph over the approximate manifold constructed in j differential equations 111 1994 421 457 with its global lipschitz constant exponentially small and this unstable invariant manifold attracts solutions exponentially fast uniformly in

classifying the complexity of constraints using finite algebras many natural combinatorial problems can be expressed as constraint satisfaction problems this class of problems is known to be np complete in general but certain restrictions on the form of the constraints can ensure tractability here we show that any set of relations used to specify the allowed forms of constraints can be associated with a finite universal algebra and we explore how the computational complexity of the corresponding constraint satisfaction problem is connected to the properties of this algebra hence we completely translate the problem of classifying the complexity of restricted constraint satisfaction problems into the language of universal algebra we introduce a notion of tractable algebra and investigate how the tractability of an algebra relates to the tractability of the smaller algebras which may be derived from it including its subalgebras and homomorphic images this allows us to reduce significantly the types of algebras which need to be classified using our results we also show that if the decision problem associated with a given collection of constraint types can be solved efficiently then so can the corresponding search problem we then classify all finite strictly simple surjective algebras with respect to tractability obtaining a dichotomy theorem which generalizes schaefer s dichotomy for the generalized satisfiability problem finally we suggest a possible general algebraic criterion for distinguishing the tractable and intractable cases of the constraint satisfaction problem

slow dynamics for the cahn hilliard equation in higher space dimensions the motion of bubbles it is known that the van der waals cahn hilliard w c h dynamics can be approximated by a quasi static stefan problem with surface tension it turns out that the stefan problem has a manifold of equilibria equal in dimension to that of the domain any sphere of fixed radius with interface contained in the domain is an equilibrium indistinguishable from the point of view of the perimeter functional we resolve this degeneracy by showing that at the w c h level this manifold is replaced by a quasi invariant stable manifold on which the typical solution preserves its bubble like shape until it reaches the boundary moreover we show that the bubble moves superslowly we also obtain an equation that determines those special spheres that correspond to equilibria at the w c h level our work establishes the phenomenon of superslow motion in higher space dimensions in the class of single interface solutions

on the slowness of phase boundary motion in one space dimension we study the limiting behavior of the solution of u t e 2 u xx u 3 u 0 a x b with a neumann boundary condition or an appropriate dirichlet condition the analysis is based on energy methods we assume that the initial data has a transition layer structure i e ue 1 except near finitely many transition points we show that in the limit as e 0 the solution maintains its transition layer structure and the transition points move slower than any power of e

locating the phase transition in binary constraint satisfaction problems abstract the phase transition in binary constraint satisfaction problems i e the transition from a region in which almost all problems have many solutions to a region in which almost all problems have no solutions as the constraints become tighter is investigated by examining the behaviour of samples of randomly generated problems in contrast to theoretical work which is concerned with the asymptotic behaviour of problems as the number of variables becomes larger this paper is concerned with the location of the phase transition in finite problems the accuracy of a prediction based on the expected number of solutions is discussed it is shown that the variance of the number of solutions can be used to set bounds on the phase transition and to indicate the accuracy of the prediction a class of sparse problems for which the prediction is known to be inaccurate is considered in detail it is shown that for these problems the phase transition depends on the topology of the constraint graph as well as on the tightness of the constraints

metastable patterns for the cahn hilliard equation part i abstract in this paper we study the dynamics of the one dimensional cahn hilliard equation in a neighborhood of an equilibrium having n 1 transition layers an approximation for an n dimensional invariant manifold and slow channel are defined and it is proved that ff a solution of the cahn hilliard equation starts outside but close to the slow channel around the approximate manifold then it will approach the channel at an exponentially large speed after it enters the slow channel it will follow the approximate manifold with speed o e c and stay in the channel for exponentially large time

layer oscillations in reaction diffusion systems this paper considers a two component system of reaction diffusion equations involving two parameters varepsilon and tau varepsilon tau u 1 varepsilon 2 u xx f u v qquad v

generalized satisfiability problems minimal elements and phase transitions we develop a probabilistic model on the generalized satisfiability problems defined by schaefer in proceedings of the 10th stoc san diego ca usa association for computing machinery new york 1978 pp 216 226 for which the arity of the constraints is fixed in order to study the associated phase transition we establish new results on minimal elements associated with such generalized satisfiability problems these results are the keys of the exploration we conduct on the location and on the nature of the phase transition for generalized satisfiability we first prove that the phase transition occurs at the same scale for every reasonable problem and we provide lower and upper bounds for the associated critical ratio our framework allows one to get these bounds in a uniform way in particular we obtain a lower bound proportional to the number of variables for k sat without analyzing any algorithm finally we reveal the seed of coarseness for the phase transition of generalized satisfiability 2 xor sat

fast reaction slow diffusion and curve shortening the reaction diffusion problem u 1 varepsilon delta u varepsilon 1 v n u quad u x 0 varepsilon g x quad partial n u 0 text on partial omega for a vector u x t varepsilon is considered in a domain omega in r m an asymptotic solution is constructed for varepsilon small it shows that at each x u tends quickly to a minimum of v u when v has several minima when u tends to a piecewise constant function boundary layer expansions are constructed around the resulting surfaces of discontinuity or fronts each front is found to move along its normal with a constant velocity determined by the discontinuity v in v across it when v 0 the front s normal velocity is varepsilon kappa where kappa is its mean curvature the motion of fronts in this manner is studied for arcs in the plane which are normal to partial omega at their endpoints and for fronts that are closed curves it is shown a front can shrink to a point in

constructing an asymptotic phase transition in random binary constraint satisfaction problems the standard models used to generate random binary constraint satisfaction problems are described at the problem sizes studied experimentally a phase transition is seen as the constraint tightness is varied however achlioptas et al showed that if the problem size number of variables increases while the remaining parameters are kept constant asymptotically almost all instances are unsatisfiable in this paper an alternative scheme for one of the standard models is proposed in which both the number of values in each variable s domain and the average degree of the constraint graph are increased with problem size it is shown that with this scheme there is asymptotically a range of values of the constraint tightness in which instances are trivially satis able with probability at least 0 5 and a range in which instances are almost all unsatisfiable hence there is a crossover point at some value of the constraint tightness between these two ranges this scheme is compared to a similar scheme due to xu and li

an empirical study of phase transitions in binary constraint satisfaction problems abstract an empirical study of randomly generated binary constraint satisfaction problems reveals that for problems with a given number of variables domain size and connectivity there is a critical level of constraint tightness at which a phase transition occurs at the phase transition problems change from being soluble to insoluble and the difficulty of problems increases dramatically a theory developed by williams and hogg 44 and independently developed by smith 37 predicts where the hardest problems should occur it is shown that the theory is in close agreement with the empirical results except when constraint graphs are sparse

on the exponentially slow motion of a viscous shock using formal asymptotic methods we study the internal layer behavior associated with the following viscous shock problem in the limit e 0 the convex nonlinearity f u satisfies f f for the steady problem we show that the method of matched asymptotic expansions fails to uniquely determine the location of the equilibrium shock layer solution this indeterminacy resulting from neglecting certain exponentially small effects is eliminated by using the projection method which exploits certain properties of the spectrum associated with the linearized operator for the time dependent problem we show that the viscous shock which is formed from initial data drifts towards the equilibrium solution on an exponentially long time interval of the order o ec e for some c 0 this exponentially slow behavior is analyzed by deriving an equation of motion for the location of the viscous shock for burgers equation f u u2 2 the results give an analytical characterization of the slow shock layer motion observed numerically in kreiss and kreiss see 11 we also show that the shock layer behavior is very sensitive to small changes in the boundary operator in addition using a wkb type method the slow viscous shock motion is studied numerically for small e the results comparing favorably with corresponding analytical results finally we relate the slow viscous shock motion to similar slow internal layer motion for the allen cahn equation

metastable patterns in solutions of ut ϵ2uxx f u we consider the equation in question on the interval 0 x 1 having neumann boundary conditions with f u f u where f is a double well energy density with equal minima at u 1 the only stable states of the system are patternless constant solutions but given two phase initial data a pattern of interfacial layers typically forms far out of equilibrium the ensuing nonlinear relaxation process is extremely slow patterns persist for exponentially long times proportional to exp a l where a f 1 and l is the minimum distance between layers physically a tiny potential jump across a layer drives its motion we prove the existence and persistence of these metastable patterns and characterise accurately the equations governing their motion the point of view is reminiscent of center manifold theory a manifold parametrising slowly evolving states is introduced a neighbourhood is shown to be normally attracting and the parallel flow is characterised to high relative accuracy proofs involve a detailed study of the dirichlet problem spectral gap analysis and energy estimates

randomization derandomization and antirandomization three games abstract three games are given between two players paul and carole with a common theme in each round paul does a split and carole chooses random play by carole allows a bound for the game value through derandomization this becomes a deterministic strategy for carole minimizing a weight function paul can use that same weight function to give a bound for the game value in the other direction

sharp thresholds of graph properties and the sat problem consider g n p to be the probability space of random graphs on n vertices with edge probability p we will be considering subsets of this space defined by monotone graph properties a monotone graph property p is a property of graphs such that a p is invariant under graph automorphisims b if graph h has property p then so does any graph g having h as a subgraph a monotone symmetric family of graphs is a family defined by such a property one of the first observations made about random graphs by erdos and renyi in their seminal work on random graph theory 12 was the existence of threshold phenomena the fact that for many interesting properties p the probability of p appearing in g n p exhibits a sharp increase at a certain critical value of the parameter p bollobas and thomason proved the existence of threshold functions for all monotone set properties 6 and in 14 it is shown that this behavior is quite general and that all monotone graph properties exhibit threshold behavior i e the probability of their appearance increases from values very close to 0 to values close to 1 in a very small interval more precise analysis of the size of the threshold interval is done in 7 this threshold behavior which occurs in various settings which arise in combinatorics and computer science is an instance of the phenomenon of phase transitions which is the subject of much interest in statistical physics one of the main questions that arises in studying phase transitions is how sharp is the transition for example one of the motivations for this paper arose from the question of the sharpness of the phase transition for the property of satisfiability of a random kcnf boolean formula nati linial who introduced me to this problem suggested that although much concrete analysis was being performed on this problem the best approach would be to find general conditions for sharpness of the phase transition answering the question posed in 14 as to the relation between the length of the threshold interval and the value of the critical probability in this paper we indeed introduce a simple condition and prove it is sufficient stated roughly in the setting of random graphs the main theorem states that if a property has a coarse threshold then it can be approximated by the property of having certain given graphs as a subgraph this condition can be applied in a more

invariant manifolds for metastable patterns in ut e2uxx f u we consider the above equation on the interval 0 x 1 subject to neumann boundary conditions with f u f u where f is a double well energy density function with equal minima our previous work 3 proved the existence and persistence of very slowly evolving patterns metastable states in solutions with two phase initial data here we characterise these metastable states in terms of the global unstable manifolds of equilibria as conjectured by fusco and hale 6

front interaction and nonhomogeneous equilibria for tristable reaction diffusion equations the authors investigate the interfacial dynamics associated with three phase systems and the possible occurrence of steady multiphase patterns in one and higher space dimensions the model for this study is a ginzburg landau model for phase transitions given by the partial differential equation u 1 varepsilon delta u frac 1 varepsilon v u varepsilon u defined for x in a domain omega subset r and t 0 the potential v varepsilon is assumed to possess three wells having depths within mathcal o varepsilon of each other the evolution of the solution can be described by tracking the motion of fronts connecting adjacent wells of v varepsilon the primary interest is to understand how the self induced motion of a given front is affected by the presence of a second nearby front this interaction is characterized precisely using the formal method of matched asymptotic expansions in some settings this leads to the prediction of steady waves linking the outer two phases

slow motion for gradient systems with equal depth multiple well potentials abstract for scalar reaction diffusion in one space dimension it has been known for a long time that fronts move with an exponentially small speed for potentials with several distinct minimizers the purpose of this paper is to provide a similar result in the case of systems our method relies on a careful study of the evolution of localized energy this approach also has the advantage of relaxing the preparedness assumptions on the initial datum

many hard examples in exact phase transitions this paper analyzes the resolution complexity of two random constraint satisfaction problem csp models i e model rb rd for which we can establish the existence of phase transitions and identify the threshold points exactly by encoding csps into cnf formulas it is proved that almost all instances of model rb rd have no tree like resolution proofs of less than exponential size thus we not only introduce new families of csps and cnf formulas hard to solve which can be useful in the experimental evaluation of csp and sat algorithms but also propose models with both many hard instances and exact phase transitions finally conclusions are presented as well as a detailed comparison of model rb rd with the hamiltonian cycle problem and random 3 sat which respectively exhibit three different kinds of phase transition behavior in np complete problems