a polynomial time approximation algorithm for the permanent of a matrix with nonnegative entries we present a polynomial time randomized algorithm for estimating the permanent of an arbitrary n n matrix with nonnegative entries this algorithm technically a fully polynomial randomized approximation scheme computes an approximation that is with high probability within arbitrarily small specified relative error of the true value of the permanent

kazhdan projections random walks and ergodic theorems in this paper we investigate generalizations of kazhdan s property t to the setting of uniformly convex banach spaces we explain the interplay between the existence of spectral gaps and that of kazhdan projections our methods employ markov operators associated to a random walk on the group for which we provide new norm estimates and convergence results this construction exhibits useful properties and flexibility and allows to reclassify the existence of kazhdan projections from a c algebraic phenomenon to a natural object associated to random walks on groups we give a number of applications of these results in particular we address several open questions we give a direct comparison of properties te and fe with lafforgue s reinforced banach property t we obtain shrinking target theorems for orbits of kazhdan groups and apply them to a question of kleinbock and margulis finally we construct non compact ghost projections for warped cones answering a question of willett and yu in this last case we conjecture that such warped cones provide new counterexamples to the coarse baum connes conjecture

design of capacity approaching irregular low density parity check codes we design low density parity check ldpc codes that perform at rates extremely close to the shannon capacity the codes are built from highly irregular bipartite graphs with carefully chosen degree patterns on both sides our theoretical analysis of the codes is based on the work of richardson and urbanke see ibid vol 47 no 2 p 599 618 2000 assuming that the underlying communication channel is symmetric we prove that the probability densities at the message nodes of the graph possess a certain symmetry using this symmetry property we then show that under the assumption of no cycles the message densities always converge as the number of iterations tends to infinity furthermore we prove a stability condition which implies an upper bound on the fraction of errors that a belief propagation decoder can correct when applied to a code induced from a bipartite graph with a given degree distribution our codes are found by optimizing the degree structure of the underlying graphs we develop several strategies to perform this optimization we also present some simulation results for the codes found which show that the performance of the codes is very close to the asymptotic theoretical bounds

hopf algebras having a dense big cell we discuss some axioms that ensure that a hopf algebra has its simple comodules classified using an analogue of the borel weil construction more precisely we show that a hopf algebra having a dense big cell satisfies the above requirement this method has its roots in the work of parshall and wang in the case of q deformed quantum groups gl and sl here we examine the example of universal cosovereign hopf algebras for which the weight group is the free group on two generators

moments of the inverse participation ratio for the laplacian on finite regular graphs we investigate the rst and second moments of the inverse participation ratio ipr for all eigenvectors of the laplacian on nite random regular graphs with n vertices and degree z by exactly diagonalizing a large set of z regular graphs we nd that as n becomes large the mean of the inverse participation ratio on each graph when averaged over a large ensemble of graphs approaches the numerical value 3 this universal number is understood as the large n limit of the average of the quartic polynomial corresponding to the ipr over an appropriate n 2 dimensional hypersphere of r n the graph ensemble mean variance of the inverse participation ratio for all graph laplacian eigenvectors deviates from its continuous hypersphere average due to large graph to graph uctuations that arise from the existence of localized modes

bounds on the bisection width for random d regular graphs in this paper we provide an explicit way to compute asymptotically almost sure upper bounds on the bisection width of random d regular graphs for any value of d the upper bounds are obtained from the analysis of the performance of a randomized greedy algorithm to find bisections of d regular graphs we provide bounds for 5 d 12 we also give empirical values of the size of the bisection found by the algorithm for some small values of d and compare them with numerical approximations of our theoretical bounds our analysis also gives asymptotic lower bounds for the size of the maximum bisection

proof verification and the hardness of approximation problems we show that every language in np has a probablistic verifier that checks membership proofs for it using logarithmic number of random bits and by examining a constant number of bits in the proof if a string is in the language then there exists a proof such that the verifier accepts with probability 1 i e for every choice of its random string for strings not in the language the verifier rejects every provided proof with probability at least 1 2 our result builds upon and improves a recent result of arora and safra 1998 whose verifiers examine a nonconstant number of bits in the proof though this number is a very slowly growing function of the input length as a consequence we prove that no max snp hard problem has a polynomial time approximation scheme unless np p the class max snp was defined by papadimitriou and yannakakis 1991 and hard problems for this class include vertex cover maximum satisfiability maximum cut metric tsp steiner trees and shortest superstring we also improve upon the clique hardness results of feige et al 1996 and arora and safra 1998 and show that there exists a positive e such that approximating the maximum clique size in an n vertex graph to within a factor of n e is np hard

self similar groups and the zig zag and replacement products of graphs abstract every finitely generated self similar group naturally produces an infinite sequence of finite d regular graphs n we construct self similar groups whose graphs n can be represented as an iterated zig zag product and graph powering n 1 n k z k 1 we also construct self similar groups whose graphs n can be represented as an iterated replacement product and graph powering n 1 n k r k 1 this gives simple explicit examples of self similar groups whose graphs n form an expanding family and examples of automaton groups whose graphs n have linear diameters diam n o n and bounded girth

multicommodity max flow min cut theorems and their use in designing approximation algorithms in this paper we establish max flow min cut theorems for several important classes of multicommodity flow problems in particular we show that for any n node multicommodity flow problem with uniform demands the max flow for the problem is within an o log n factor of the upper bound implied by the min cut the result which is existentially optimal establishes an important analogue of the famous 1 commodity max flow min cut theorem for problems with multiple commodities the result also has substantial applications to the field of approximation algorithms for example we use the flow result to design the first polynomial time polylog n times optimal approximation algorithms for well known np hard optimization problems such as graph partitioning min cut linear arrangement crossing number vlsi layout and minimum feedback arc set applications of the flow results to path routing problems network reconfiguration communication in distributed networks scientific computing and rapidly mixing markov chains are also described in the paper categories and subject descriptors f 2 2 analysis of algorithms and problem complexity

randomized oblivious integral routing for minimizing power cost given an undirected network g v e and a set of traffic requests r the minimum power cost routing problem requires that each r k r be routed along a single path to minimize e e l e where l e is the traffic load on edge e and is a constant greater than 1 typically 1 3 this problem is important in optimizing the energy consumption of networks to address this problem we propose a randomized oblivious routing algorithm an oblivious routing algorithm makes decisions independently of the current traffic in the network this feature enables the efficient implementation of our algorithm in a distributed manner which is desirable for large scale high capacity networks an important feature of our work is that our algorithm can satisfy the integral constraint which requires that each traffic request r k should follow a single path we prove that given this constraint no randomized oblivious routing algorithm can guarantee a competitive ratio bounded by o e 1 1 by contrast our approach provides a competitive ratio of o e 1 1 log 2 1 v log 1 d where d is the maximum demand of traffic requests furthermore our results also hold for a more general case where the objective is to minimize e l e p where p 1 is an arbitrary unknown parameter with a given upper bound 1 the theoretical results established in proving these bounds can be further generalized to a framework of designing and analyzing oblivious integral routing algorithms which is significant for research on minimizing e l e in specific scenarios with simplified problem settings for instance we prove that this framework can generate an oblivious integral routing algorithm whose competitive ratio can be bounded by o log v log 1 d and o log 3 v log 1 d on expanders and hypercubes respectively original study on energy saving in oblivious integral routing an e 1 1 lower bound on competitive ratio a random oblivious integral routing algorithm with polylog tight competitive ratio a general framework to design and analyze oblivious integral routing algorithms polylog bound on competitive ratio for expanders and hypercubes

expander graphs and their applications a major consideration we had in writing this survey was to make it accessible to mathematicians as well as to computer scientists since expander graphs the protagonists of our story come up in numerous and often surprising contexts in both fields but perhaps we should start with a few words about graphs in general they are of course one of the prime objects of study in discrete mathematics however graphs are among the most ubiquitous models of both natural and human made structures in the natural and social sciences they model relations among species societies companies etc in computer science they represent networks of communication data organization computational devices as well as the flow of computation and more in mathematics cayley graphs are useful in group theory graphs carry a natural metric and are therefore useful in geometry and though they are just one dimensional complexes they are useful in certain parts of topology e g knot theory in statistical physics graphs can represent local connections between interacting parts of a system as well as the dynamics of a physical process on such systems the study of these models calls then for the comprehension of the significant structural properties of the relevant graphs but are there nontrivial structural properties which are universally important expansion of a graph requires that it is simultaneously sparse and highly connected expander graphs were first defined by bassalygo and pinsker and their existence first proved by pinsker in the early 70s the property of being an expander seems significant in many of these mathematical computational and physical contexts it is not surprising that expanders are useful in the design and analysis of communication networks what is less obvious is that expanders have surprising utility in other computational settings such as in the theory of error correcting codes and the theory of pseudorandomness in mathematics we will encounter e g their role in the study of metric embeddings and in particular in work around the baum connes conjecture expansion is closely related to the convergence rates of markov chains and so they play a key role in the study of monte carlo algorithms in statistical mechanics and in a host of practical computational applications the list of such interesting and fruitful connections goes on and on with so many applications we will not even

tannaka krein duality for compact matrix pseudogroups twisted su n groups the notion of concrete monoidalw category is introduced and investigated a generalization of the tannaka krein duality theorem is proved it leads to new examples of compact matrix pseudogroups among them we have twistedsu n groups denoted bys u n it is shown that the representation theory fors u n is similar to that ofsu n irreducible representations are labeled by young diagrams and formulae for dimensions and multiplicity are the same as in the classical case

eigenvalues and expansion of regular graphs the spectral method is the best currently known technique to prove lower bounds on expansion ramanujan graphs which have asymptotically optimal second eigenvalue are the best known explicit expanders the spectral method yielded a lower bound of k 4 on the expansion of linear sized subsets of k regular ramanujan graphs we improve the lower bound on the expansion of ramanujan graphs to approximately k 2 moreover we construct a family of k regular graphs with asymptotically optimal second eigenvalue and linear expansion equal to k 2 this shows that k 2 is the best bound one can obtain using the second eigenvalue method we also show an upper bound of roughly 1 k 1 on the average degree of linear sized induced subgraphs of ramanujan graphs this compares positively with the classical bound 2 k 1 as a byproduct we obtain improved results on random walks on expanders and construct selection networks respectively extrovert graphs of smaller size respectively degree than was previously known

are bitvectors optimal we study the it static membership problem given a set s of at most n keys drawn from a universe u of size m store it so that queries of the form is u in s can be answered by making few accesses to the memory we study schemes for this problem that use space close to the information theoretic lower bound of omega n log frac m n bits and yet answer queries by reading a small number of bits of the memory we show that for epsilon 0 there is a scheme that stores o frac n epsilon 2 log m bits and answers membership queries using a randomized algorithm that reads just one bit of memory and errs with probability at most epsilon we consider schemes that make no error for queries in s but are allowed to err with probability at most epsilon for queries not in s we show that there exist such schemes that store o frac n epsilon 2 log m bits and answer queries using just one bitprobe if multiple probes are allowed then the number of bits stored can be reduced to o n 1 delta log m for any delta 0 the schemes mentioned above are based on probabilistic constructions of set systems with small intersections we show lower bounds that come close to our upper bounds for a large range of n and epsilon schemes that answer queries with just one bitprobe and error probability epsilon must use omega frac n epsilon log 1 epsilon log m bits of storage if the error is restricted to queries not in s then the scheme must use omega frac n 2 epsilon 2 log n epsilon log m bits of storage we also consider deterministic schemes for the static membership problem and show tradeoffs between space and the number of probes

grothendieck rings of universal quantum groups abstract we determine the grothendieck ring of finite dimensional comodules for the free hopf algebra on a matrix coalgebra and similarly for the free hopf algebra with bijective antipode and other related universal quantum groups the results turn out to parallel those for wang and van daele s universal compact quantum groups and bichon s generalization of those results to universal cosovereign hopf algebras in all cases the rings are isomorphic to those of non commutative polynomials over certain sets these sets varying from case to case in most cases we are able to give more precise information about the multiplication table of the grothendieck ring

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 20 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

a recursive approach to low complexity codes a method is described for constructing long error correcting codes from one or more shorter error correcting codes referred to as subcodes and a bipartite graph a graph is shown which specifies carefully chosen subsets of the digits of the new codes that must be codewords in one of the shorter subcodes lower bounds to the rate and the minimum distance of the new code are derived in terms of the parameters of the graph and the subeodes both the encoders and decoders proposed are shown to take advantage of the code s explicit decomposition into subcodes to decompose and simplify the associated computational processes bounds on the performance of two specific decoding algorithms are established and the asymptotic growth of the complexity of decoding for two types of codes and decoders is analyzed the proposed decoders are able to make effective use of probabilistic information supplied by the channel receiver e g reliability information without greatly increasing the number of computations required it is shown that choosing a transmission order for the digits that is appropriate for the graph and the subcodes can give the code excellent burst error correction abilities the construction principles

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

bounds on the max and min bisection of random cubic and random 4 regular graphs in this paper we present a randomized algorithm to compute the bisection width of cubic and 4 regular graphs the analysis of the proposed algorithms on random graphs provides asymptotic upper bounds for the bisection width of random cubic and random 4 regular graphs with n vertices giving upper bounds of 0 174039n for random cubic and of 0 333333n for random 4 regular we also obtain asymptotic lower bounds for the size of the maximum bisection for random cubic and random 4 regular graphs with n vertices of 1 32697n and 1 66667n respectively the randomized algorithms are derived from initial greedy algorithm and their analysis is based on the differential equation method

boolean compressed sensing and noisy group testing the fundamental task of group testing is to recover a small distinguished subset of items from a large population while efficiently reducing the total number of tests measurements the key contribution of this paper is in adopting a new information theoretic perspective on group testing problems we formulate the group testing problem as a channel coding decoding problem and derive a single letter characterization for the total number of tests used to identify the defective set although the focus of this paper is primarily on group testing our main result is generally applicable to other compressive sensing models

galois reconstruction of finite quantum groups let c be a small category and let f c malgf be a functor where malgf is the category of finite dimensional measured algebras over a field k or frobenius algebras we construct a universal hopf algebra aaut f such that f factorizes through a functor f c mcoalgf aaut f where mcoalgf aaut f is the category of finite dimensional measured aaut f comodule algebras this general reconstruction result allows us to recapture a finite dimensional hopf algebra a from the category mcoalgf a and the forgetful functor mcoalgf a malgf we have a aaut our universal construction is also done in a c algebra framework and we get compact quantum groups in the sense of woronowicz

expander codes using expander graphs we construct a new family of asymptotically good linear error correcting codes these codes have linear time sequential decoding algorithms and logarithmic time parallel decoding algorithms that use a linear number of processors we present both randomized and explicit constructions of these codes experimental results demonstrate the good performance of the randomly chosen codes

three dimensional gravity inversion using graph theory to delineate the skeleton of homogeneous sources abstractwe developed a new strategy based on graph theory concepts to invert gravity data using an ensemble of simple point masses our method consisted of a genetic algorithm with elitism to generate a set of possible solutions each estimate was associated to a graph to solve the minimum spanning tree mst problem to produce unique and stable estimates we restricted the position of the point masses by minimizing the statistical variance of the distances of an mst jointly with the data misfit function during the iterations of the genetic algorithm hence the 3d spatial distribution of the point masses identified the skeleton of homogeneous gravity sources in addition our method also gave an estimation of the anomalous mass of the source so together with the anomalous mass the skeleton could aid other 3d methods with promising geometric a priori parameters several tests with different values of regularizing parameter were made to bespeak this new regularizing strategy the inversion results ap

approximate counting uniform generation and rapidly mixing markov chains the paper studies effective approximate solutions to combinatorial counting and uniform generation problems using a technique based on the simulation of ergodic markov chains it is shown that for self reducible structures almost uniform generation is possible in polynomial time provided only that randomised approximate counting to within some arbitrary polynomial factor is possible in polynomial time it follows that for self reducible structures polynomial time randomised algorithms for counting to within factors of the form 1 n are available either for all fl e r or for no fi e r a substantial part of the paper is devoted to investigating the rate of convergence of finite ergodic markov chains and a simple but powerful characterisation of rapid convergence for a broad class of chains based on a structural property of the underlying graph is established finally the general techniques of the paper are used to derive an almost uniform generation procedure for labelled graphs with a given degree sequence which is valid over a much wider range of degrees than previous methods this in turn leads to randomised approximate counting algorithms for these graphs with very good

concentration of the kirchhoff index for erdos renyi graphs given an undirected graph the resistance distance between two nodes is the resistance one would measure between these two nodes in an electrical network if edges were resistors summing these distances over all pairs of nodes yields the so called kirchhoff index of the graph which measures its overall connectivity in this work we consider erdos renyi random graphs since the graphs are random their kirchhoff indices are random variables we give formulas for the expected value of the kirchhoff index and show it concentrates around its expectation we achieve this by studying the trace of the pseudoinverse of the laplacian of erdos renyi graphs for synchronization a class of estimation problems on graphs our results imply that acquiring pairwise measurements uniformly at random is a good strategy even if only a vanishing proportion of the measurements can be acquired

entropy waves the zig zag graph product and new constant degree expanders the main contribution of this work is a new type of graph product which we call the zig zag product taking a product of a large graph with a small graph the resulting graph inherits roughly its size from the large one its degree from the small one and its expansion properties from both iteration yields simple explicit constructions of constant degree expanders of arbitrary size starting from one constant size expander crucial to our intuition and simple analysis of the properties of this graph product is the view of expanders as functions which act as entropy wave propagators they transform probability distributions in which entropy is concentrated in one area to distributions where that concentration is dissipated in these terms the graph product affords the constructive interference of two such waves

finite simple groups as expanders abstract we prove that there exist k and 0 e such that every non abelian finite simple group g which is not a suzuki group has a set of k generators for which the cayley graph cay g s is an e expander expander graphs ramanujan complexes

compact matrix pseudogroups the compact matrix pseudogroup is a non commutative compact space endowed with a group structure the precise definition is given and a number of examples is presented among them we have compact group of matrices duals of discrete groups and twisted deformed su n groups the representation theory is developed it turns out that the tensor product of representations depends essentially on their order the existence and the uniqueness of the haar measure is proved and the orthonormality relations for matrix elements of irreducible representations are derived the form of these relations differs from that in the group case this is due to the fact that the haar measure on pseudogroups is not central in general the corresponding modular properties are discussed the haar measures on the twistedsu 2 group and on the finite matrix pseudogroup are found

expansion in sl d z q z q arbitrary let s be a fixed finite symmetric subset of sl d z and assume that it generates a zariski dense subgroup g we show that the cayley graphs of q g with respect to the generating set q s form a family of expanders where q is the projection map z z q z

upper bound on the characters of the symmetric groups let c be a conjugacy class in the symmetric group s n and be a partition of n let f be the degree of the irreducible representation s c the character of s at c and r c the normalized character c f we prove that there exist constants b 0 and 1 q 0 such that for n 4 for every conjugacy class c in s n and every irreducible representation s of s n r c max q 1 n 1 n b supp c where supp c is the number of non fixed digits under the action of a permutation in c 1 is the size of the largest part in and 1 is the number of parts in the proof is obtained by enumeration of rim hook tableaux the hook formula and probabilistic arguments combinatorial algebraic and statistical applications follow this result in particular we estimate the rate of mixing of random walks on the alternating groups with respect to conjugacy classes

graph constrained group testing nonadaptive group testing involves grouping arbitrary subsets of n items into different pools each pool is then tested and defective items are identified a fundamental question involves minimizing the number of pools required to identify at most d defective items motivated by applications in network tomography sensor networks and infection propagation a variation of group testing problems on graphs is formulated unlike conventional group testing problems each group here must conform to the constraints imposed by a graph for instance items can be associated with vertices and each pool is any set of nodes that must be path connected in this paper a test is associated with a random walk in this context conventional group testing corresponds to the special case of a complete graph on n vertices for interesting classes of graphs a rather surprising result is obtained namely that the number of tests required to identify d defective items is substantially similar to what is required in conventional group testing problems where no such constraints on pooling is imposed specifically if t n corresponds to the mixing time of the graph g it is shown that with m o d2t2 n log n d nonadaptive tests one can identify the defective items consequently for the erdos renyi random graph g n p as well as expander graphs with constant spectral gap it follows that m o d2 log3 n non adaptive tests are sufficient to identify d defective items next a specific scenario is considered that arises in network tomography for which it is shown that m o d3 log3 n nonadaptive tests are sufficient to identify d defective items noisy counterparts of the graph constrained group testing problem are considered for which parallel results are developed we also briefly discuss extensions to compressive sensing on graphs

the non backtracking spectrum of the universal cover of a graph a non backtracking walk on a graph h is a directed path of directed edges of h such that no edge is the inverse of its preceding edge non backtracking walks of a given length can be counted using the non backtracking adjacency matrix b indexed by h s directed edges and related to ihara s zeta function we show how to determine b s spectrum in the case where h is a tree covering a finite graph we show that when h is not regular this spectrum can have positive measure in the complex plane unlike the regular case we show that outside of b s spectrum the corresponding green function has periodic decay ratios the existence of such a ratio system can be effectively checked and is equivalent to being outside the spectrum we also prove that the spectral radius of the non backtracking walk operator on the tree covering a finite graph is exactly gr where gr is the growth rate of the tree this further motivates the definition of the graph theoretical riemann hypothesis proposed by stark and terras st finally we give experimental evidence that for a fixed finite graph h a ran dom lift of large degree has non backtracking new spectrum near that of h s universal cover this suggests a new generalization of alon s second eigenvalue conjecture

on the complexity of approximating the independent set problem we show that for some positive constant c it is not feasible to approximate independent set for graphs of n nodes within a factor of n c provided maximum 2 satisfiability does not have a randomized polynomial time approximation scheme we also study reductions preserving the quality of approximations and exhibit complete problems

the spectra of infinite hypertrees a model of regular infinite hypertrees is developed to mimic for hypergraphs what infinite trees do for graphs two notions of spectra or first eigenvalue are then examined for the infinite tree obtaining a precise value for the first notion and obtaining some estimates for the second the results indicate agreement of the first eigenvalue of the infinite hypertree with the second eigenvalue of a random hypergraph of the same degree to within logarithmic factors at least for the first notion of first eigenvalue author s abstract

hopf galois systems abstract we introduce the concept of hopf galois system a reformulation of the notion of galois extension of the base field for a hopf algebra the main feature of our definition is a generalization of the antipode of an ordinary hopf algebra we present several examples which indicate that although our axiomatic is slightly more complicated than the classical one it is also more natural and easier to handle with the main application of hopf galois systems is the construction of monoidal equivalences between comodule categories

the manin hopf algebra of a koszul artin schelter regular algebra is quasi hereditary abstract for any koszul artin schelter regular algebra a we consider the universal hopf algebra aut a coacting on a introduced by manin to study the representations i e finite dimensional comodules of this hopf algebra we use the tannaka krein formalism specifically we construct an explicit combinatorial rigid monoidal category u equipped with a functor m to finite dimensional vector spaces such that aut a coend u m using this pair u m we show that aut a is quasi hereditary as a coalgebra and in addition is derived equivalent to the representation category of u

invertible extractors and wiretap protocols a wiretap protocol is a pair of randomized encoding and decoding functions such that knowledge of a bounded fraction of the encoding of a message reveals essentially no information about the message while knowledge of the entire encoding reveals the message using the decoder in this paper the notion of efficiently invertible extractors is studied and it is shown that a wiretap protocol can be constructed from such an extractor then invertible extractors for symbol fixing affine and general sources are constructed and used to create wiretap protocols with asymptotically optimal trade offs between their rate ratio of the length of the message versus its encoding and resilience ratio of the observed positions of the encoding and the length of the encoding the results are further applied to create wiretap protocols for challenging communication problems such as active intruders who change portions of the encoding network coding and intruders observing arbitrary boolean functions of the encoding

on the diameter of permutation groups given a finite group g and a set a of generators the diameter diam g g a of the cayley graph g g a is the smallest l such that every element of g can be expressed as a word of length at most l in a a 1 we are concerned with bounding diam g max a diam g g a it has long been conjectured that the diameter of the symmetric group of degree n is polynomially bounded in n but the best previously known upper bound was exponential in nlogn v we give a quasipolynomial upper bound namely diam g exp o logn 4 loglogn exp loglog g o 1 for g sym n or g alt n where the implied constants are absolute this addresses a key open case of babai s conjecture on diameters of simple groups by a result of babai and seress 1992 our bound also implies a quasipolynomial upper bound on the diameter of all transitive permutation groups of degree n

symmetric groups and expander graphs we construct explicit generating sets sn and tilde s n of the alternating and the symmetric groups which turn the cayley graphs mathcal c text alt n s n and mathcal c text sym n tilde s n into a family of bounded degree expanders for all n

difference equations isoperimetric inequality and transience of certain random walks the difference laplacian on a square lattice in rn has been stud ied by many authors in this paper an analogous difference operator is studied for an arbitrary graph it is shown that many properties of the laplacian in the continuous setting e g the maximum principle the harnack inequality and cheeger s bound for the lowest eigenvalue hold for this difference oper ator the difference laplacian governs the random walk on a graph just as the laplace operator governs the brownian motion as an application of the theory of the difference laplacian it is shown that the random walk on a class of graphs is transient the random walks we consider are defined as follows let k be a connected graph i e a one dimensional simplicial complex for a vertex x e k let m x denote the number of edges emanating from x the probability that a particle moves from x to another vertex y e k is l m x if x and y are connected by an edge and it is zero otherwise as observed by courant friedrichs and lewy cfl for the case of a square lattice in the plane this random walk is intimately related to the difference analog of the laplacian

a polylogarithmic approximation of the minimum bisection a bisection of a graph with n vertices is a partition of its vertices into two sets each of size n 2 the bisection cost is the number of edges connecting the two sets it is known that finding a bisection of minimum cost is np hard we present an algorithm that finds a bisection whose cost is within ratio of o log2 n from the minimum for graphs excluding any fixed graph as a minor e g planar graphs we obtain an improved approximation ratio of o log n the previously known approximation ratio for bisection was roughly sqrt n

lower bounds for matrix product in bounded depth circuits with arbitrary gates we prove superlinear lower bounds for the number of edges in constant depth circuits with n inputs and up to n outputs our lower bounds are proved for all types of constant depth circuits e g constant depth arithmetic circuits and constant depth boolean circuits with arbitrary gates the bounds apply for several explicit functions and most importantly for matrix product in particular we obtain the following results we show that the number of edges in any constant depth arithmetic circuit for matrix product over any field is superlinear in m2 where m m is the size of each matrix that is the lower bound is superlinear in the number of input variables moreover if the circuit is bilinear the result applies also for the case in which the circuit gets any product of two linear functions for free we show that the number of edges in any constant depth arithmetic circuit for the trace of the product of three matrices over fields with characteristic 0 is superlinear in m2 note that the trace is a single output function we give explicit examples for n boolean functions f1 fn such that any constant depth boolean circuit with arbitrary gates for f1 fn has a superlinear number of edges the lower bound is also proved for circuits with arbitrary gates over any finite field the bound applies for matrix product over finite fields as well as for several other explicit functions

linear time encodable and decodable error correcting codes we present a new class of asymptotically good linear error correcting codes these codes can be both encoded and decoded in linear time they can also be encoded by logarithmic depth circuits of linear size and decoded by logarithmic depth circuits of size o nlogn we present both randomized and explicit constructions of these codes

flexible queueing architectures we study a multiserver model with n flexible servers and n queues connected through a bipartite graph where the level of flexibility is captured by an upper bound on the graph s average degree dn applications in content replication in data centers skill based routing in call centers and flexible supply chains are among our main motivations we focus on the scaling regime where the system size n tends to infinity while the overall traffic intensity stays fixed we show that a large capacity region and an asymptotically vanishing queueing delay are simultaneously achievable even under limited flexibility dn n our main results demonstrate that when dn ln n a family of expander graph based flexibility architectures has a capacity region that is within a constant factor of the maximum possible while simultaneously ensuring a diminishing queueing delay for all arrival rate vectors in the capacity region our analysis is centered around a new class of virtual queue based scheduling policies that

discrete fundamental groups of warped cones and expanders in this paper we compute the discrete fundamental groups of warped cones as an immediate consequence this allows us to show that there exist coarsely simply connected expanders and superexpanders this also provides a strong coarse invariant of warped cones and implies that many warped cones cannot be coarsely equivalent to any box space

some properties of graphs constructed from 2 designs abstract let d p b be any nontrivial nonsymmetric 2 v k design in this paper we construct a connected regular and bipartite graph say ig v b d from d some graphic properties of ig v b d are investigated in particular the results in the paper bridges the equivalency between edge transitivity of ig v b d and flag transitivity of d with the equivalency it is obtained that ig v b d is a semisymmetric graph as long as d is a nonsymmetric flag transitive 2 v k design

a chernoff bound for random walks on expander graphs we consider a finite random walk on a weighted graph g we show that the fraction of time spent in a set of vertices a converges to the stationary probability pi a with error probability exponentially small in the length of the random walk and the square of the size of the deviation from pi a the exponential bound is in terms of the expansion of g and improves previous results of d aldous probab engrg inform sci 1 1987 pp 33 46 l lovasz and m simonovits it random structures algorithms 4 1993 pp 359 412 m ajtai j komlos and e szemeredi deterministic simulation of logspace in proc 19th acm symp on theory of computing 1987 we show that taking the sample average from one trajectory gives a more efficient estimate of pi a than the standard method of generating independent sample points from several trajectories using this more efficient sampling method we improve the algorithms of jerrum and sinclair for approximating the number of perfect matchings in a dense graph and for approximating the partition function of a ferromagnetic ising system and we give an efficient algorithm to estimate the entropy of a random walk on an unweighted graph

polynomial time approximation schemes for max bisection on planar and geometric graphs the max bisection and min bisection problems are to find a partition of the vertices of a graph into two equal size subsets that respectively maximizes or minimizes the number of edges with endpoints in both subsets we design the first polynomial time approximation scheme for the max bisection problem on arbitrary planar graphs solving a long standing open problem the method of solution involves designing exact polynomial time algorithms for computing optimal partitions of bounded treewidth graphs in particular max and min bisection which could be of independent interest using a similar method we design also the first polynomial time approximation scheme for max bisection on unit disk graphs which could also be easily extended to other geometrically defined graphs

how to compute the volume in high dimension in some areas of theoretical computer science we feel that randomized algorithms are better and in some others we can prove that they are more efficient than the deterministic ones

zeta functions of finite graphs and coverings iii abstract a graph theoretical analog of brauer siegel theory for zeta functions of number fields is developed using the theory of artin l functions for galois coverings of graphs from parts i and ii in the process we discuss possible versions of the riemann hypothesis for the ihara zeta function of an irregular graph

topological structural classes of complex networks we use theoretical principles to study how complex networks are topologically organized at large scale using spectral graph theory we predict the existence of four different topological structural classes of networks these classes correspond respectively to highly homogenous networks lacking structural bottlenecks networks organized into highly interconnected modules with low inter community connectivity networks with a highly connected central core surrounded by a sparser periphery and networks displaying a combination of highly connected groups quasicliques and groups of nodes partitioned into disjoint subsets quasibipartites here we show by means of the spectral scaling method that these classes really exist in real world ecological biological informational technological and social networks we show that neither of three network growth mechanisms random with uniform distribution preferential attachment and random with the same degree sequence as real network is able to reproduce the four structural classes of complex networks these models reproduce two of the network classes as a function of the average degree but completely fail in reproducing the other two classes of networks

born again group testing multiaccess communications a brief summary of the basic notions of group testing is presented together with a brief historical account one of the early papers on group testing is shown to include a description of the tree search polling algorithm of hayes the classical group testing problem is formulated including a criterion for optimality of test plans a restricted class of tests called nested testing is described and a complete description for an optimal nested strategy is given for both a finite number and an infinite number of bernoulli distributed random variables a generalization of group testing applicable to the random access communications problem is presented

a polynomial time algorithm for rayleigh ratio on discrete variables replacing spectral techniques for expander ratio normalized cut and cheeger constant a general form of minimizing the rayleigh ratio on discrete variables is shown here for the first time to be polynomial time solvable this is significant because major problems in clustering partitioning and imaging can be presented as the rayleigh ratio minimization on discrete variables and an orthogonality constraint these challenging problems are modeled as the normalized cut problem the graph expander ratio problem the cheeger constant problem or the conductance problem all of which are np hard these problems have traditionally been solved heuristically using the spectral technique a unified framework is provided here whereby all these problems are formulated as a constrained minimization form of a quadratic ratio referred to here as the rayleigh ratio the quadratic ratio is to be minimized on discrete variables and a single sum constraint that we call the balance or orthogonality constraint when the discreteness constraints on the variables are disregarded the resulting continuous relaxation is solved by the spectral method it is shown here that the rayleigh ratio minimization subject to the discreteness constraints requiring each variable to assume one of two values in b 1 is solvable in strongly polynomial time equivalent to a single minimum s t cut algorithm on a graph of same size as the input graph for any nonnegative value of b this discrete form for the rayleigh ratio problem was often assumed to be np hard not only is it shown here that the discrete rayleigh ratio problem is polynomial time solvable but also the algorithm is more efficient than the spectral algorithm furthermore an experimental study demonstrates that the new algorithm provides in practice an improvement often dramatic on the quality of the results of the spectral method both in terms of approximating the true optimum of the rayleigh ratio problem on both the discrete variables and the balance constraint and in terms of the subjective partition quality a further contribution here is the introduction of a problem the quantity normalized cut generalizing all the rayleigh ratio problems the discrete version of that problem is also solved with the efficient algorithm presented this problem is shown in a companion paper to enable the modeling of features essential to clustering that are valuable in practical applications

the spectrum of the random environment and localization of noise we consider random walk on a mildly random environment on finite transitive d regular graphs of increasing girth after scaling and centering the analytic spectrum of the transition matrix converges in distribution to a gaussian noise an interesting phenomenon occurs at d 2 as the limit graph changes from a regular tree to the integers the noise becomes localized open image in new window the graphs of the noise covariance structure for d 4 3 2 1 from above

fast matrix rank algorithms and applications we consider the problem of computing the rank of an m n matrix a over a field we present a randomized algorithm to find a set of r rank a linearly independent columns in o vav p r field operations where vav denotes the number of nonzero entries in a and

sieve methods in group theory i powers in linear groups a general sieve method for groups is formulated it enables one to measure subsets of a finitely generated group as an application we show that if is a finitely generated non virtually solvable linear group of characteristic zero then the set of proper powers in is exponentially small this is a far reaching strengthening of the main result of hkls

short proofs are narrow resolution made simple the width of a resolution proof is defined to be the maximal number of literals in any clause of the proof in this paper we relate proof width to proof length size in both general resolution and its tree like variant the following consequences of these relations reveal width as a crucial resource of resolution proofs in one direction the relations allow us to give simple unified proofs for almost all known exponential lower bounds on size of resolution proofs as well as several interesting new ones they all follow from width lower bounds and we show how these follow from natural expansion property of clauses of the input tautology in the other direction the width size relations naturally suggest a simple dynamic programming procedure for automated theorem proving one which simply searches for small width proofs this relation guarantees that the runnuing time and thus the size of the produced proof is at most quasi polynomial in the smallest tree like proof this algorithm is never much worse than any of the recursive automated provers such as dll used in practice in contrast we present a family of tautologies on which it is exponentially faster

compression functions of uniform embeddings of groups into hilbert and banach spaces we construct finitely generated groups with arbitrary prescribed hilbert space compression alpha from the interval 0 1 for a large class of banach spaces e including all uniformly convex banach spaces the e compression of these groups coincides with their hilbert space compression moreover the groups that we construct have asymptotic dimension at most 3 hence they are exact in particular the first examples of groups that are uniformly embeddable into a hilbert space respectively exact of finite asymptotic dimension with hilbert space compression 0 are given these groups are also the first examples of groups with uniformly convex banach space compression 0

pseudorandom generators in propositional proof complexity we call a pseudorandom generator g n 0 1 n to 0 1 m hard for a propositional proof system p if p cannot efficiently prove the properly encoded statement g n x 1 ldots x n neq b for any string b in 0 1 m we consider a variety of combinatorial pseudorandom generators inspired by the nisan wigderson generator on the one hand and by the construction of tseitin tautologies on the other we prove that under certain circumstances these generators are hard for such proof systems as resolution polynomial calculus and polynomial calculus with resolution pcr

improved low density parity check codes using irregular graphs we construct new families of error correcting codes based on gallager s 1973 low density parity check codes we improve on gallager s results by introducing irregular parity check matrices and a new rigorous analysis of hard decision decoding of these codes we also provide efficient methods for finding good irregular structures for such decoding algorithms our rigorous analysis based on martingales our methodology for constructing good irregular codes and the demonstration that irregular structure improves performance constitute key points of our contribution we also consider irregular codes under belief propagation we report the results of experiments testing the efficacy of irregular codes on both binary symmetric and gaussian channels for example using belief propagation for rate 1 4 codes on 16000 bits over a binary symmetric channel previous low density parity check codes can correct up to approximately 16 errors while our codes correct over 17 in some cases our results come very close to reported results for turbo codes suggesting that variations of irregular low density parity check codes may be able to match or beat turbo code performance

random latin squares and 2 dimensional expanders let x be a 2 dimensional simplicial complex the degree of an edge e is the number of 2 faces of x containing e the complex x is an expander if the coboundary d1 of every f2 valued 1 cochain c 1 x f2 satisfies supp d1 min supp d0 c 0 x f2