global uniqueness and stability for an inverse wave source problem for less regular data abstract by means of a carleman estimate we obtain the global uniqueness and stability of the solution of an inverse problem of determining a wave source within the h 1 space

gaussian interference channel capacity to within one bit the capacity of the two user gaussian interference channel has been open for 30 years the understanding on this problem has been limited the best known achievable region is due to han and kobayashi but its characterization is very complicated it is also not known how tight the existing outer bounds are in this work we show that the existing outer bounds can in fact be arbitrarily loose in some parameter ranges and by deriving new outer bounds we show that a very simple and explicit han kobayashi type scheme can achieve to within a single bit per second per hertz bit s hz of the capacity for all values of the channel parameters we also show that the scheme is asymptotically optimal at certain high signal to noise ratio snr regimes using our results we provide a natural generalization of the point to point classical notion of degrees of freedom to interference limited scenarios

probabilistic analysis of two heuristics for the 3 satisfiability problem an algorithm for the 3 satisfiability problem is presented and a probabilistic analysis is performed the analysis is based on an instance distribution which is parameterized to simulate a variety of sample characteristics the algorithm assigns values to variables appearing in a given instance of 3 satisfiability one at a time using the unit clause heuristic and a maximum occurring literal selection heuristic at each step a variable is chosen randomly from a subset of variables which is usually large the algorithm runs in polynomial time and it is shown that the algorithm finds a solution to a random instance of 3 satisfiability with probability bounded from below by a constant greater than zero for a range of parameter values the heuristics studied here can be used to select variables in a backtrack algorithm for 3 satisfiability experiments have shown that for about the same range of parameters as above the backtrack algorithm using the heuristics finds a solution in polynomial average time

probabilistic analysis of a generalization of the unit clause literal selection heuristics for the k satisfiability problem abstract two algorithms for the k satisfiability problem are presented and a probabilistic analysis is performed the analysis is based on an instance distribution which is parametrized to simulate a variety of sample characteristics the algorithms assign values to literals appearing in a given instance of k satisfiability one at a time until a solution is found or it is discovered that further assignments cannot lead to finding a solution one algorithm chooses the next literal from a unit clause if one exists and randomly from the set of remaining literals otherwise the other algorithm uses a generalization of the unit clause rule as a heuristic for selecting the next literal at each step a literal is chosen randomly from a clause containing the least number of literals the algorithms run in polynomial time and it is shown that they find a solution to a random instance of k satisfiability with probability bounded below by a constant greater than zero for two different ranges of parameter values it is also shown that the second algorithm mentioned finds a solution with probability approaching one for a wide range of parameter values

a general model and thresholds for random constraint satisfaction problems in this paper we study the relation among the parameters in their most general setting that define a large class of random csp models d k csp where d is the domain size and k is the length of the constraint scopes the model d k csp unifies several related models such as the model rb and the model k csp we prove that the model d k csp exhibits exact phase transitions if klnd increases no slower than the logarithm of the number of variables a series of experimental studies with interesting observations are carried out to illustrate the solubility phase transition and the hardness of instances around phase transitions

threshold for monotone symmetric properties through a logarithmic sobolev inequality threshold phenomena are investigated using a general approach following talagrand ann probab 22 1994 1576 1587 and friedgut and kalai proc amer math soc 12 1999 1017 10541 the general upper bound for the threshold width of symmetric monotone properties is improved this follows from a new lower bound on the maximal influence of a variable on a boolean function the method of proof is based on a well known logarithmic sobolev inequality on 0 1 n this new bound is shown to be asymptotically optimal

revisiting the cavity method threshold for random 3 sat a detailed monte carlo study of the satisfiability threshold for random 3 sat has been undertaken in combination with a monotonicity assumption we find that the threshold for random 3 sat satisfie

strong noise sensitivity and random graphs the noise sensitivity of a boolean function describes its likelihood to flip under small perturbations of its input introduced in the seminal work of benjamini kalai and schramm 1999 it was there shown to be governed by the first level of fourier coefficients in the central case of monotone functions at a constant critical probability here we study noise sensitivity and a natural stronger version of it addressing the effect of noise given a specific witness in the original input our main context is the erdos renyi random graph where already the property of containing a given graph is sufficiently rich to separate these notions in particular our analysis implies strong noise sensitivity in settings where the bks criterion involving the first fourier level does not apply e g when the critical value goes to 0 polynomially fast in the number of variables

combinatorial sharpness criterion and phase transition classification for random csps we investigate the nature of the phase transition sharp or coarse for random constraint satisfaction problems we first give a sharp threshold criterion specified for csps which is derived from friedgut bourgain s one thus we get a complete and precise classification of the nature of the threshold for symmetric boolean csps in particular we show that it is governed by two local properties strongly related to the problems 1 sat and 2 xor sat

on the symmetric feedback capacity of the k user cyclic z interference channel the k user cyclic z interference channel models a situation in which the kth transmitter causes interference only to the k 1 th receiver in a cyclic manner e g the first transmitter causes interference only to the kth receiver the impact of noiseless feedback on the capacity of this channel is studied by focusing on the gaussian cyclic z interference channel to this end the symmetric feedback capacity of the linear shift deterministic cyclic z interference channel is completely characterized for all interference regimes using insights from the linear deterministic channel model the symmetric feedback capacity of the gaussian cyclic z interference channel is characterized up to within a constant number of bits as a byproduct of the constant gap result the symmetric generalized degrees of freedom with feedback for the gaussian cyclic z interference channel are also characterized these results highlight that the symmetric feedback capacities for both linear and gaussian channel models are in general functions of k the number of users furthermore the capacity gain obtained due to feedback decreases as k increases

an algorithm for random signed 3 sat with intervals interval k sat k isat is a generalization of classical k sat where the variables can take values in 0 1 instead of 0 1 and the literals are of the form x i for intervals i 0 1 it falls within the class of signed satisfiability problems we propose an algorithm for 3 isat and analyze it on uniformly random formulas the algorithm follows the unit clause paradigm enhanced by a very limited backtracking option using wormald s ode method we prove that if m n 2 3 with high probability our algorithm succeeds in finding an assignment of values to the variables satisfying the formula

a better algorithm for random k sat let boldsymbol phi be a uniformly distributed random k sat formula with n variables and m clauses we present a polynomial time algorithm that finds a satisfying assignment of boldsymbol phi with high probability for constraint densities m n 1 varepsilon k 2 k ln k k where varepsilon k rightarrow0 previously no efficient algorithm was known to find satisfying assignments with a nonvanishing probability beyond m n 1 817 cdot2 k k a frieze and s suen j algorithms 20 1996 pp 312 355

on the critical percolation probabilities we prove that the critical probabilities of site percolation on the square lattice satisfy the relation p c p 1 furthermore we prove the continuity of the function percolation probability

the polynomial trefftz method for solving backward and inverse source wave problems the trefftz method is a truly meshless boundary type method because the trial solutions automatically satisfy the governing equation in order to stably solve the high dimensional backward wave problems and the one dimensional inverse source problems we develop a multiple scale polynomial trefftz method msptm of which the scales are determined a priori by the collocation points the msptm can retrieve the missing initial data and unknown time varying wave source the present method can also be extended to solve the higher dimensional wave equations long term through the introduction of a director in the two dimensional polynomial trefftz bases several numerical examples reveal that the msptm is efficient and stable for solving severely ill posed inverse problems of wave equations under large noises

satisfiability allows no nontrivial sparsification unless the polynomial time hierarchy collapses consider the following two player communication process to decide a language l the first player holds the entire input x but is polynomially bounded the second player is computationally unbounded but does not know any part of x their goal is to decide cooperatively whether x belongs to l at small cost where the cost measure is the number of bits of communication from the first player to the second player for any integer d 3 and positive real e we show that if satisfiability for n variable d cnf formulas has a protocol of cost o nd e then conp is in np poly which implies that the polynomial time hierarchy collapses to its third level the result even holds when the first player is conondeterministic and is tight as there exists a trivial protocol for e 0 under the hypothesis that conp is not in np poly our result implies tight lower bounds for parameters of interest in several areas namely sparsification kernelization in parameterized complexity lossy compression and probabilistically checkable proofs by reduction similar results hold for other np complete problems for the vertex cover problem on n vertex d uniform hypergraphs this statement holds for any integer d 2 the case d 2 implies that no np hard vertex deletion problem based on a graph property that is inherited by subgraphs can have kernels consisting of o k2 e edges unless conp is in np poly where k denotes the size of the deletion set kernels consisting of o k2 edges are known for several problems in the class including vertex cover feedback vertex set and bounded degree deletion

on the capacity of symmetric m user gaussian interference channels with feedback a general time varying feedback coding scheme is proposed for m user fully connected symmetric gaussian interference channels based on the analysis of the general coding scheme we prove a theorem which gives a criterion for designing good time varying feedback codes for gaussian interference channels the proposed scheme improves the suh tse and kramer inner bounds of the channel capacity for the cases of weak and not very strong interference when m 2 this capacity improvement is more significant when the signal to noise ratio snr is not very high in addition our coding scheme can be proved mathematically and numerically to outperform the kramer code for m geq 2 when the snr is equal to the interference to noise ratio inr besides the generalized degrees of freedom gdof of our proposed coding scheme can be proved to be optimal in the all network situations very weak weak strong very strong for any m the numerical results show that our coding scheme can attain better performance than the suh tse coding scheme for m 2 or the mohajer tandon poor lattice coding scheme for m 2 furthermore the simplicity of the encoding decoding algorithms is another strong point of our proposed coding scheme compared with the suh tse coding scheme when m 2 and the mohajer tandon poor lattice coding scheme when m 2 more importantly our results show that an optimal coding scheme for the symmetric gaussian interference channels with feedback can be achieved by only using marginal posterior distributions under a better cooperation strategy between transmitters

phase transition in a random nk landscape model an analysis for the phase transition in a random nk landscape model nk n k z is given this model is motivated from population genetics and the solubility problem for the model is equivalent to a random k 1 sat problem gao and culberson y gao j culberson an analysis of phase transition in nk landscapes journal of artificial intelligence research 17 2002 309 332 showed that a random instance generated by nk n 2 z with z z 0 27 754 is asymptotically insoluble based on empirical results they conjectured that the phase transition occurs around the value z z 0 we prove that an instance generated by nk n 2 z with zz 0 is asymptotically insoluble the results show the phase transition around z z 0 for nk n 2 z in the course of the analysis we introduce a generalized random 2 sat formula which is of self interest and show its phase transition phenomenon

optimal feedback communication via posterior matching in this paper we introduce a fundamental principle for optimal communication over general memoryless channels in the presence of noiseless feedback termed posterior matching using this principle we devise a simple sequential generic feedback transmission scheme suitable for a large class of memoryless channels and input distributions achieving any rate below the corresponding mutual information this provides a unified framework for optimal feedback communication in which the horstein scheme bsc and the schalkwijk kailath scheme awgn channel are special cases thus as a corollary we prove that the horstein scheme indeed attains the bsc capacity settling a longstanding conjecture we further provide closed form expressions for the error probability of the scheme over a range of rates and derive the achievable rates in a mismatch setting where the scheme is designed according to the wrong channel model several illustrative examples of the posterior matching scheme for specific channels are given and the corresponding error probability expressions are evaluated the proof techniques employed utilize novel relations between information rates and contraction properties of iterated function systems

feedback capacity of the gaussian interference channel to within 2 bits we characterize the capacity region to within 2 bits s hz and the symmetric capacity to within 1 bit s hz for the two user gaussian interference channel ic with feedback we develop achievable schemes and derive a new outer bound to arrive at this conclusion one consequence of the result is that feedback provides multiplicative gain at high signal to noise ratio the gain becomes arbitrarily large for certain channel parameters this finding is in contrast to point to point and multiple access channels where feedback provides no gain and only bounded additive gain respectively the result makes use of a linear deterministic model to provide insights into the gaussian channel this deterministic model is a special case of the el gamal costa deterministic model and as a side generalization we establish the exact feedback capacity region of this general class of deterministic ics

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

a new achievable rate region for the interference channel a new achievable rate region for the general interference channel which extends previous results is presented and evaluated the technique used is a generalization of superposition coding to the multivariable case a detailed computation for the gaussian channel case clarifies to what extent the new region improves previous ones the capacity of a class of gaussian interference channels is also established

a multiple direction trefftz method for solving the multi dimensional wave equation in an arbitrary spatial domain in this paper we first express the wave equation in terms of the minkowskian polar coordinates and generate a set of complete hyperbolic type trefftz bases r k cosh k and r k sinh k which are further transformed to wave polynomials as the trial solution bases for the one dimensional wave equation in order to stably solve the wave propagation problems long term we develop a multiple scale trefftz method mstm of which the scales are determined a priori by the collocation points then we derive a very simple method of multi dimensional wave polynomials equipped with different spatial directions which being the normalized wavenumber vectors as the polynomial trefftz bases for solving the multi dimensional wave equations which is named a multiple direction trefftz method mdtm several numerical examples of two and three dimensional wave equations demonstrate that the present method is efficient and stable

models for random constraint satisfaction problems we introduce a class of models for random constraint satisfaction problems this class includes and generalizes many previously studied models we characterize those models from our class which are asymptotically interesting in the sense that the limiting probability of satisfiability changes significantly as the number of constraints increases we also discuss models which exhibit a sharp threshold for satisfiability in the sense that the limiting probability jumps from 0 to 1 suddenly as the number of constraints increases

upper bounds on the satisfiability threshold we present a survey of upper bounds which has been established up to now on the satisfiability threshold of random k sat formulae the ideas which led to these bounds and the techniques used to obtain them are explained a companion paper in this volume present a survey of the lower bounds

storage capacity in symmetric binary perceptrons we study the problem of determining the capacity of the binary perceptron for two variants of the problem where the corresponding constraint is symmetric we call these variants the rectangle binary perceptron rpb and the u function binary perceptron ubp we show that unlike for the usual step function binary perceptron the critical capacity in these symmetric cases is given by the annealed computation in a large region of parameter space for all rectangular constraints and for narrow enough u function constraints k k we conclude that full step replica symmetry breaking would have to be evaluated in order to obtain the exact capacity in this case

gibbs states and the set of solutions of random constraint satisfaction problems abstract an instance of a random constraint satisfaction problem defines a random subset the set of solutions of a large product space x n the set of assignments we consider two prototypical problem ensembles random k satisfiability and q coloring of random regular graphs and study the uniform measure with support on s as the number of constraints per variable increases this measure first decomposes into an exponential number of pure states clusters and subsequently condensates over the largest such states above the condensation point the mass carried by the n largest states follows a poisson dirichlet process for typical large instances the two transitions are sharp we determine their precise location further we provide a formal definition of each phase transition in terms of different notions of correlation between distinct variables in the problem the degree of correlation naturally affects the performances of many search sampling algorithms empirical evidence suggests that local monte carlo markov chain strategies are effective up to the clustering phase transition and belief propagation up to the condensation point finally refined message passing techniques such as survey propagation may also beat this threshold message passing algorithms phase transitions random graphs

optimal flow through the disordered lattice author s aldous d abstract consider routing traffic on the n n torus simultaneously between all source destination pairs to minimize the cost ec e f 2 e where f e is the volume of flow across edge e and the c e form an i i d random environment we prove existence of a rescaled n limit constant for minimum cost by comparison with an appropriate analogous problem about minimum cost flows across a m m subsquare of the lattice institute of mathematical statistics 2007

bicolouring random hypergraphs we study the problem of bicolouring random hypergraphs both numerically and analytically we apply the zero temperature cavity method to find analytical results for the phase transitions dynamic and static in the one step replica symmetry breaking 1rsb approximation these points appear to be in agreement with the results of the numerical algorithm in the second part we implement and test the survey propagation algorithm for specific bicolouring instances in the so called hard sat phase

on the feedback capacity of the fully connected k user interference channel the symmetric k user interference channel with fully connected topology is considered in which 1 each receiver suffers interference from all other k 1 transmitters and 2 each transmitter has causal and noiseless feedback from its respective receiver the number of generalized degrees of freedom ssr gdof is characterized in terms of where the interference to noise ratio ssr inr is given by ssr inr ssr snr it is shown that the per user ssr gdof of this network is the same as that of the two user interference channel with feedback except for 1 for which existence of feedback does not help in terms of ssr gdof the coding scheme proposed for this network termed cooperative interference alignment is based on two key ingredients namely interference alignment and interference decoding moreover an approximate characterization is provided for the symmetric feedback capacity of the network when the ssr snr and ssr inr are far apart from each other

shooting methods for numerical solutions of exact controllability problems constrained by linear and semilinear wave equations with local distributed controls numerical solutions of exact controllability problems for linear and semilinear wave equations with distributed controls are studied exact controllability problems can be solved by the corresponding optimal control problems the optimal control problem is reformulated as a system of equations an optimality system that consists of an initial value problem for the underlying linear or semilinear wave equation and a terminal value problem for the adjoint wave equation the discretized optimality system is solved by a shooting method the convergence properties of the numerical shooting method in the context of exact controllability are illustrated through computational experiments

a quantum lovasz local lemma the lovasz local lemma lll is a powerful tool in probability theory to show the existence of combinatorial objects meeting a prescribed collection of weakly dependent criteria we show that the lll extends to a much more general geometric setting where events are replaced with subspaces and probability is replaced with relative dimension which allows to lower bound the dimension of the intersection of vector spaces under certain independence conditions our result immediately applies to the k qsat problem quantum analog of k sat for instance we show that any collection of rank 1 projectors with the property that each qubit appears in at most 2k e k of them has a joint satisfiable state we then apply our results to the recently studied model of random k qsat recent works have shown that the satisfiable region extends up to a density of 1 in the large k limit where the density is the ratio of projectors to qubits using a hybrid approach building on work by laumann et al 2009 2010 we greatly extend the known satisfiable region for random k qsat to a density of 2k k2 since our tool allows us to show the existence of joint satisfying states without the need to construct them we are able to penetrate into regions where the satisfying states are conjectured to be entangled avoiding the need to construct them which has limited previous approaches to product states

the decomposition method for forward and backward time dependent problems the convergence of the decomposition method as applied to time dependent problems governed by the heat wave and beam equations is investigated for both forward direct and backward inverse problems it is shown that for forward problems the convergence is faster than for backward problems

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

lower bounds for random 3 sat via differential equations it is widely believed that the probability of satisfiability for random k sat formulae exhibits a sharp threshold as a function of their clauses to variables ratio for the most studied case k 3 there have been a number of results during the last decade providing upper and lower bounds for the threshold s potential location all lower bounds in this vein have been algorithmic i e in each case a particular algorithm was shown to satisfy random instances of 3 sat with probability 1 o 1 if the clauses to variables ratio is below a certain value we show how differential equations can serve as a generic tool for analyzing such algorithms by rederiving most of the known lower bounds for random 3 sat in a simple uniform manner

optimally scaled vector regularization method to solve ill posed linear problems instead of the tikhonov regularization method which with a scalar being the regularization parameter liu et al 1 have proposed a novel regularization method with a vector as being the regularization parameter as a continuation we further propose an optimally scaled vector regularization method osvrm to solve the ill posed linear problems which is better than the tikhonov regularization method the presently proposed vector regularization method is shown to be well conditioning to overcome the ill posedness of the linear equations system the osvrm causes a significant improvement of stability and accuracy in the numerical solution of ill posed linear problem and its convergence speed is as fast as by solving the well posed linear problem some tests of the linear inverse problems confirm the efficiency and accuracy of the osvrm

fast simulation of multi dimensional wave problems by the sparse scheme of the method of fundamental solutions in this work a meshless scheme is presented for the fast simulation of multi dimensional wave problems the present method is rather simple and straightforward the houbolt method is used to eliminate the time dependence of spatial variables then the original wave problem is converted into equivalent systems of modified helmholtz equations the sparse scheme of the method of fundamental solutions in combination with the localized method of approximate particular solutions is employed for efficient implementation of spatial variables to demonstrate the effectiveness and simplicity of this new approach three numerical examples have been assessed with excellent performance

on russo s approximate zero one law consider the product measure mu p on 0 1 n when 0 operatorname resp 1 is given weight 1 p operatorname resp p consider a monotone subset a of 0 1 n we give a precise quantitative form to the following statement if a does not depend much on any given coordinate d mu p a dp is large thus in that case there is a threshold effect and mu p a jumps from near 0 to near 1 in a small interval

on the freezing of variables in random constraint satisfaction problems the set of solutions of random constraint satisfaction problems zero energy groundstates of mean field diluted spin glasses undergoes several structural phase transitions as the amount of constraints is increased this set first breaks down into a large number of well separated clusters at the freezing transition which is in general distinct from the clustering one some variables spins take the same value in all solutions of a given cluster in this paper we introduce and study a message passing procedure that allows to compute for generic constraint satisfaction problems the sizes of the rearrangements induced in response to the modification of a variable these sizes diverge at the freezing transition with a critical behavior which is also investigated in details we apply the generic formalism in particular to the random satisfiability of boolean formulas and to the coloring of random graphs the computation is first performed in random tree ensembles for which we underline a connection with percolation models and with the reconstruction problem of information theory the validity of these results for the original random ensembles is then discussed in the framework of the cavity method

bootstrap percolation on the hypercube in the bootstrap percolation on the n dimensional hypercube in the initial position each of the 2n sites is occupied with probability p and empty with probability 1 p independently of the state of the other sites every occupied site remains occupied for ever while an empty site becomes occupied if at least two of its neighbours are occupied if at the end of the process every site is occupied we say that the initial position spans the hypercube we shall show that there are constants c1 c2 0 such that for open image in new window the probability of spanning tends to 1 as n while for open image in new window the probability tends to 0 furthermore we shall show that for each n the transition has a sharp threshold function

on the satisfiability threshold of formulas with three literals per clause in this paper we present a new upper bound for randomly chosen 3 cnf formulas in particular we show that any random formula over n variables with a clauses to variables ratio of at least 4 4898 is as n grows large asymptotically almost surely unsatisfiable the previous best such bound due to dubois in 1999 was 4 506 the first such bound independently discovered by many groups of researchers since 1983 was 5 19 several decreasing values between 5 19 and 4 506 were published in the years between we believe that the probabilistic techniques we use for the proof are of independent interest

techniques from combinatorial approximation algorithms yield efficient algorithms for random 2 k sat we apply techniques from the theory of approximation algorithms to the problem of deciding whether a random k sat formula is satisfiable let form n k m denote a random k sat instance with n variables and m clauses using known approximation algorithms for max cut or min bisection we show how to certify that formn 4 m is unsatisfiable efficiently provided that m cn2 for a sufficiently large constant c 0 in addition we present an algorithm based on the lovasz function that decides within polynomial expected time whether formn k m is satisfiable provided that k is even and m c 4knk 2 finally we present an algorithm that approximates random max 2 sat on input formn 2 m within a factor of 1 o n m 1 2 in expected polynomial time for m cn

the asymptotic k sat threshold abstract since the early 2000s physicists have developed an ingenious but non rigorous formalism called the cavity method to put forward precise conjectures on phase transitions in random problems mezard et al 2002 37 the cavity method predicts that the satisfiability threshold in the random k sat problem is r k sat 2 k ln 2 1 2 1 ln 2 e k with lim k e k 0 mertens et al 2006 35 this paper contains a proof of the conjecture