optimal network configuration for maximal coherence resonance in excitable systems we analyze the coherence resonance phenomenon in an ensemble of noise driven excitable neurons giving special attention to the role of the interaction topology the neural architecture is modeled using a spatially embedded network in which we can tune the network organization between scale free like with dominating long range connections and a network with mostly adjacent neurons connected we found that besides an optimal noise intensity also an optimal network configuration exists at which the largest average coherence of noise induced spikes is achieved furthermore we show that long as well as short range interactions between neurons should exist in order to achieve the optimal response of the neuronal network

interactive proofs and the hardness of approximating cliques the contribution of this paper is two fold first a connection is established between approximating the size of the largest clique in a graph and multi prover interactive proofs second an efficient multi prover interactive proof for np languages is constructed where the verifier uses very few random bits and communication bits last the connection between cliques and efficient multi prover interaction proofs is shown to yield hardness results on the complexity of approximating the size of the largest clique in a graph of independent interest is our proof of correctness for the multilinearity test of functions

effects of attachment preferences on coevolution of opinions and networks abstract in the coevolution of network structures and opinion formation we investigate the effects of a mixed population with distinctive relinking preferences on both the convergence time and the network structures it has been found that a heterogeneous network structure is easier to be reached with more high degree preferential hdp nodes there exists high correlation between the convergence time and the network heterogeneity the heterogeneous degree distribution caused by preferential attachment accelerates the convergence to a consensus state and the shortened convergence time inhibits the occurrence of the following disquieting situation that occurs in a continuously evolving network with preferential attachment and long time evolvement most of the nodes would become separated and only a few leaders would have immediate neighbors analytical calculations based on mean field theory reveal that both the transition point p t r and the consensus time depend upon the standard deviation of the degree distribution d p t r increases while decreases with the rise of d functions of p t r k k 1 and a n k d 2 k are found theoretical analyses are in accordance with simulation data

the complexity of counting cuts and of computing the probability that a graph is connected several enumeration and reliability problems are shown to be p complete and hence at least as hard as np complete problems included are important problems in network reliability analysis namely computing the probability that a graph is connected and counting the number of minimum cardinality s t cuts or directed network cuts also shown to be p complete are counting vertex covers in a bipartite graph counting antichains in a partial order and approximating the probability that a graph is connected and the probability that a pair of vertices is connected

the shift operator technique for so 7 in an su 2 3 basis ii applications reduced matrix elements of the tensor operator which forms part of so 7 generator basis are calculated for the three respective classes of so 7 representations v 0 0 v n0 v 0 1 v n0 0 0 v v n0 the calculation is based on the relations between quadratic products of shift operators established in a previous paper by the same authors

probabilistic checking of proofs a new characterization of np we give a new characterization of np the class np contains exactly those languages l for which membership proofs a proof that an input x is in l can be verified probabilistically in polynomial time using logarithmic number of random bits and by reading sublogarithmic number of bits from the proof we discuss implications of this characterization specifically we show that approximating clique and independent set even in a very weak sense is np hard

analyzing glauber dynamics by comparison of markov chains a popular technique for studying random properties of a combinatorial set is to design a markov chain monte carlo algorithm for many problems there are natural markov chains connecting the set of allowable configurations which are based on local moves or glauber dynamics typically these single site update algorithms are difficult to analyze so often the markov chain is modified to update several sites simultaneously recently there has been progress in analyzing these more complicated algorithms for several important combinatorial problems in this work we use the comparison technique of diaconis and saloff coste to show that several of the natural single point update algorithms are efficient the strategy is to relate the mixing rate of these algorithms to the corresponding nonlocal algorithms which have already been analyzed this allows us to give polynomial time bounds for single point update algorithms for problems such as generating planar tilings and random triangulations of convex polygons we also survey several other comparison techniques along with specific applications which have been used in the context of estimating mixing rates of markov chains

approximate counting for complex weighted boolean constraint satisfaction problems constraint satisfaction problems or csps have been extensively studied in for instance artificial intelligence database theory graph theory and statistical physics from a practical viewpoint it is beneficial to approximately solve those csps when one tries to approximate the total number of truth assignments that satisfy all boolean valued constraints for unweighted boolean csps there is a known trichotomy theorem by which all such counting problems are neatly classified into exactly three categories under polynomial time randomized approximation preserving reductions in contrast we obtain a dichotomy theorem of approximate counting for complex weighted boolean csps provided that all complex valued unary constraints are freely available to use it is the expressive power of free unary constraints that enables us to prove such a stronger complete classification theorem this discovery makes a step forward in the quest for the approximation complexity classification of all counting csps to deal with complex weights we employ proof techniques of factorization and arity reduction along the line of solving holant problems moreover we introduce a novel notion of t constructibility that naturally induces approximation preserving reducibility our result also gives an approximation analogue of the dichotomy theorem on the complexity of exact counting for complex weighted boolean csps

approximating the permanent of graphs with large factors let g u v e be a bipartite graph with u v n the factor size of g f is the maximum number of edge disjoint perfect matchings in g we characterize the complexity of counting the number of perfect matchings in classes of graphs parameterized by factor size we describe the simple algorithm which is an approximation algorithm for the permanent that is a natural simplification of the algorithm suggested by broder 1986 and analyzed by jerrum and sinclair 1988a b compared to the algorithm by jerrum and sinclair 1988a b the simple algorithm achieves a polynomial speed up in the running time to compute the permanent a combinatorial lemma is used to prove that the simple algorithm runs in time no n f thus 1 for all constants 0 the simple algorithm runs in polynomial time for graphs with factor size at least n 2 for some constant c the simple algorithm is the fastest known approximation for graphs with factor size at least c log n compare with the approximation algorithms described in karmarkar 1988 we prove the following complementary hardness results for functions f such that 3 f n n 3 the exact counting problem for f n regular bipartite graphs is p complete for and e 0 for any function f such that 3 f n n1 e approximate counting for f n regular bipartite graphs is as hard as approximate counting for all bipartite graphs an announcement of these results appears in dagum 1988

weak signal transmission in complex networks and its application in detecting connectivity we present a network model of coupled oscillators to study how a weak signal is transmitted in complex networks through both theoretical analysis and numerical simulations we find that the response of other nodes to the weak signal decays exponentially with their topological distance to the signal source and the coupling strength between two neighboring nodes can be figured out by the responses this finding can be conveniently used to detect the topology of unknown network such as the degree distribution clustering coefficient and community structure etc by repeatedly choosing different nodes as the signal source through four typical networks i e the regular one dimensional small world random and scale free networks we show that the features of network can be approximately given by investigating many fewer nodes than the network size thus our approach to detect the topology of unknown network may be efficient in practical situations with large network size

on the hardness of approximate reasoning abstract many ai problems when formalized reduce to evaluating the probability that a propositional expression is true in this paper we show that this problem is computationally intractable even in surprisingly restricted cases and even if we settle for an approximation to this probability we consider various methods used in approximate reasoning such as computing degree of belief and bayesian belief networks as well as reasoning techniques such as constraint satisfaction and knowledge compilation that use approximation to avoid computational difficulties and reduce them to model counting problems over a propositional domain we prove that counting satisfying assignments of propositional languages is intractable even for horn and monotone formulae and even when the size of clauses and number of occurrences of the variables are extremely limited this should be contrasted with the case of deductive reasoning where horn theories and theories with binary clauses are distinguished by the existence of linear time satisfiability algorithms what is even more surprising is that as we show even approximating the number of satisfying assignments i e approximating approximate reasoning is intractable for most of these restricted theories we also identify some restricted classes of propositional formulae for which efficient algorithms for counting satisfying assignments can be given

complexity of generalized satisfiability counting problems abstract the class of generalized satisfiability problems introduced in 1978 by schaefer presents a uniform way of studying the complexity of satisfiability problems with special conditions the complexity of each decision and counting problem in this class depends on the involved logical relations in 1979 valiant defined the class p the class of functions definable as the number of accepting computations of a polynomial time nondeterministic turing machine clearly all satisfiability counting problems belong to this class p we prove a dichotomy theorem for generalized satisfiability counting problems that is if all logical relations involved in a generalized satisfiability counting problem are affine then the number of satisfying assignments of this problem can be computed in polynomial time otherwise this function is p complete this gives us a comparison between decision and counting generalized satisfiability problems we can determine exactly the polynomial satisfiability decision problems whose number of solutions can be computed in polynomial time and also the polynomial satisfiability decision problems whose counting counterparts are already p complete moreover taking advantage of a similar dichotomy result proved in 1978 by schaefer for generalized satisfiability decision problems we get as a corollary the implication that the counting counterpart of each np complete generalized satisfiability decision problem is p complete

inapproximability of the tutte polynomial the tutte polynomial of a graph g is a two variable polynomial t g x y that encodes many interesting properties of the graph we study the complexity of the following problem for rationals x and y take as input a graph g and output a value which is a good approximation to t g x y jaeger et al have completely mapped the complexity of exactly computing the tutte polynomial they have shown that this is p hard except along the hyperbola x 1 y 1 1 and at four special points we are interested in determining for which points x y there is a fully polynomial randomised approximation scheme fpras for t g x y under the assumption rp np we prove that there is no fpras at x y if x y is in one of the half planes x np there is no fpras at the point x y 0 email protected when l 2 is a positive integer thus there is no fpras for counting nowhere zero l flows for l 2 this is an interesting consequence of our work since the corresponding decision problem is in p for example for l 6 although our main concern is to distinguish regions of the tutte plane that admit an fpras from those that do not we also note that the latter regions exhibit different levels of intractability at certain points x y for example the integer points on the x axis or any point in the positive quadrant there is a randomised approximation scheme for t g x y that runs in polynomial time using an oracle for an np predicate on the other hand we identify a region of points x y at which even approximating t g x y is as hard as p

quantum circuits that can be simulated classically in polynomial time a model of quantum computation based on unitary matrix operations was introduced by feynman and deutsch it has been asked whether the power of this model exceeds that of classical turing machines we show here that a significant class of these quantum computations can be simulated classically in polynomial time in particular we show that two bit operations characterized by 4 4 matrices in which the sixteen entries obey a set of five polynomial relations can be composed according to certain rules to yield a class of circuits that can be simulated classically in polynomial time this contrasts with the known universality of two bit operations and demonstrates that efficient quantum computation of restricted classes is reconcilable with the polynomial time turing hypothesis therefore it is possible that the techniques introduced bring the quantum computational model within the realm of algebraic complexity theory in a manner consistent with one view of quantum physics the wave function is simulated deterministically and randomization arises only in the course of making measurements the results generalize the quantum model in that they do not require the matrices to be unitary in a different direction these techniques also yield deterministic polynomial time algorithms for the decision and parity problems for certain classes of read twice boolean formulae all our results are based on the use of gates that are defined in terms of their graph matching properties

on a class of spinor representations of so 7 the reduction of the irreducible representations v 0 0 1 of so 2n 1 with respect to su 2 su 2 so 2n 3 is considered for the n 3 case all the reduced matrix elements of the so 7 generators in the su 2 2 so 3 basis are calculated with the use of recursion relations

so 2n 1 in an so 2n 3 su 2 su 2 basis ii detailed study of the symmetric representations of the so 7 group for pt i see ibid vol 15 p 2665 76 1982 the so 7 down arrow su 2 3 reduction is analysed in detail explicit forms for the different su 2 generators and the remaining ones are derived using the results of the previous paper basis states describing the symmetric irreducible unitary representations are introduced expressions for the reduced matrix elements of the occurring generators with respect to these basis states are determined

approximating the permanent a randomised approximation scheme for the permanent of a 0 1s presented the task of estimating a permanent is reduced to that of almost uniformly generating perfect matchings in a graph the latter is accomplished by simulating a markov chain whose states are the matchings in the graph for a wide class of 0 1 matrices the approximation scheme is fully polynomial i e runs in time polynomial in the size of the matrix and a parameter that controls the accuracy of the output this class includes all dense matrices those that contain sufficiently many 1 s and almost all sparse matrices in some reasonable probabilistic model for 0 1 matrices of given density for the approach sketched above to be computationally efficient the markov chain must be rapidly mixing informally it must converge in a short time to its stationary distribution a major portion of the paper is devoted to demonstrating that the matchings chain is rapidly mixing apparently the first such result for a markov chain with genuinely c

the complexity of counting in sparse regular and planar graphs we show that a number of graph theoretic counting problems remain cal np hard indeed cal p complete in very restricted classes of graphs in particular we prove that the problems of counting matchings vertex covers independent sets and extremal variants of these all remain hard when restricted to planar bipartite graphs of bounded degree or regular graphs of constant degree we obtain corollaries about counting cliques in restricted classes of graphs and counting satisfying assignments to restricted classes of monotone 2 cnf formulae to achieve these results a new interpolation based reduction technique which preserves properties such as constant degree is introduced

scale free topology induced double resonance in networked two state systems we study numerically the effect of a scale free topology on the signal to noise ratio of networked two state systems and find a double resonance phenomenon i e a resonance on coupling strength and a stochastic resonance on noise strength this finding suggests an alternative approach of self tuning i e tuning from the scale free topology instead of the self tuning of potential a heuristic theory through a starlike network is presented to explain the double resonance

so 2n 1 in an so 2n 3 su 2 su 2 basis i reduction of the symmetric representations the branching rule for the reduction of symmetric irreducible unitary representations iur of the simple lie group so 2n 1 into iur of its maximal subgroup so 2n 3 x su 2 x su 2 is established for all n or 3 after the particular case n 3 is analysed in detail a general proof is presented which is valid for all n or 3 all branching rules n 3 4 can be summed up in one formula also a dimension verification is carried out the generators of so 2n 1 not belonging to the semi simple subgroup can be combined into a mixed tensor spinor representation with respect to the simple groups which occur in the direct product the precise nature of that representation is indicated and discussed

polynomial time approximation algorithms for the ising model the paper presents a randomised algorithm which evaluates the partition function of an arbitrary ferromagnetic ising system to any specified degree of accuracy the running time of the algorithm in

two dimensional monomer dimer systems are computationally intractable the classic problem of counting monomer dimer arrangements on a two dimensional lattice is analyzed using techniques from theoretical computer science under a certain assumption made precise in the text it can be shown that the general problem is computationally intractable this negative result contrasts with the special case of a system with monomer density zero for which efficient solutions have been known for some time a second much easier result obtained under the same assumption is that the partition function of a three dimensional ising system is computationally intractable again the negative result contrasts with known efficient techniques for evaluating the partition function of a two dimensional system

on unapproximable versions of np complete problems we prove that all of karp s 21 original np complete problems have a version that is hard to approximate these versions are obtained from the original problems by adding essentially the same simple constraint we further show that these problems are absurdly hard to approximate in fact no polynomial time algorithm can even approximate log k of the magnitude of these problems to within any constant factor where logk denotes the logarithm iterated k times unless np is recognized by slightly superpolynomial randomized machines we use the same technique to improve the constant epsilon such that max clique is hard to approximate to within a factor of n epsilon finally we show that it is even harder to approximate two counting problems counting the number of satisfying assignments to a monotone 2sat formula and computing the permanent of 1 0 1 matrices

on approximation algorithms for p the theme of this paper is to investigate to what extent approximation possibly together with randomization can reduce the complexity of problems in valiant s class p in general any function in p can be approximated to within any constant factor by a function in the class delta 3 p of the polynomial time hierarchy relative to a particular oracle delta 3 p cannot be replaced by delta 2 p in this result another part of the paper introduces a model of random sampling where the size of a set x is estimated by checking for various sample sets s whether or not s intersects x for various classes of sample sets upper and lower bounds on the number of samples required to estimate the size of x are discussed this type of sampling is motivated by particular problems in p such as computing the size of a backtrack search tree in the case of backtrack search trees a sample amounts to checking whether a certain path exists in the tree one of the lower bounds suggests that such tests

the complexity of planar counting problems we prove the p hardness of the counting problems associated with various satisfiability graph and combinatorial problems when restricted to planar instances these problems include 3sat 1 3sat 1 ex3sat minimum vertex cover minimum dominating set minimum feedback vertex set x3c partition into triangles and clique cover we also prove the np completeness of the ambiguous satisfiability problems j b saxe two papers on graph embedding problems tech report cmu cs 80 102 dept of computer science carnegie mellon univ pittsburgh pa 1980 and the dp completeness with respect to random polynomial reducibility of the unique satisfiability problems l g valiant and v v vazirani np is as easy as detecting unique solutions in proc 17th acm symp on theory of computing 1985 pp 458 463 associated with several of the above problems when restricted to planar instances previously very few p hardness results no sf np hardness results and no dp completeness results were known for counting problems ambiguous satisfiability problems and unique satisfiability problems respectively when restricted to planar instances assuming sf p neq np one corollary of the above results is that there are no epsilon approximation algorithms for the problems of maximizing or minimizing a linear objective function subject to a planar system of linear inequality constraints over the integers

expressiveness of matchgates matchgates have been used to simulate classically in polynomial time assemblies of quantum gates in this paper it is shown that every matchgate for 2 input 2 output functions has to obey a certain set of five polynomial identities it is also shown that no matchgate can realize a nontrivial control gate for any number of inputs on the other hand it is proved that classical boolean formulae can be expressed as matchcircuits of polynomial size

the complexity of weighted boolean csp this paper gives a dichotomy theorem for the complexity of computing the partition function of an instance of a weighted boolean constraint satisfaction problem the problem is parameterized by a finite set mathcal f of nonnegative functions that may be used to assign weights to the configurations feasible solutions of a problem instance classical constraint satisfaction problems correspond to the special case of 0 1 valued functions we show that computing the partition function i e the sum of the weights of all configurations is text sf fp text sf p complete unless either 1 every function in mathcal f is of product type or 2 every function in mathcal f is pure affine in the remaining cases computing the partition function is in p