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

maps between non commutative spaces let j be a graded ideal in a not necessarily commutative graded k algebra a a 0 a 1 in which dim k a i for all i we show that the map a a j induces a closed immersion i proj nc a j proj nc a between the non commutative projective spaces with homogeneous coordinate rings a and a j we also examine two other kinds of maps between non commutative spaces first a homomorphism a b between not necessarily commutative n graded rings induces an affine map proj nc b u proj nc a from a non empty open subspace u c proj nc b second if a is a right noetherian connected graded algebra not necessarily generated in degree one and a n is a veronese subalgebra of a there is a map proj nc a proj nc a n we identify open subspaces on which this map is an isomorphism applying these general results when a is a quotient of a weighted polynomial ring produces a non commutative resolution of a closed subscheme of a weighted projective space

computational analysis and improvement of sirt iterative x ray computed tomography ct algorithms have the potential for producing high quality images but are computationally very demanding especially when applied to high resolution problems focusing on simultaneous iterative reconstruction technique sirt we provide an eigenvalue based scheme for automatically determining a near optimal value of the relaxation parameter this accelerates the convergence rate of sirt to the point where only half the number of iterations normally required is needed we also modify the way sirt uses preconditioning to solve a weighted least squares problem the resulting algorithm which we call psirt is associated with a smaller memory footprint and calls for less data to be communicated in a distributed memory implementation experimental residual norm and timing results are provided based on cone beam micro ct mouse data including for an ordered subsets study

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

a splitting based iterative algorithm for accelerated statistical x ray ct reconstruction statistical image reconstruction using penalized weighted least squares pwls criteria can improve image quality in x ray computed tomography ct however the huge dynamic range of the statistical weights leads to a highly shift variant inverse problem making it difficult to precondition and accelerate existing iterative algorithms that attack the statistical model directly we propose to alleviate the problem by using a variable splitting scheme that separates the shift variant and nearly invariant components of the statistical data model and also decouples the regularization term this leads to an equivalent constrained problem that we tackle using the classical method of multipliers framework with alternating minimization the specific form of our splitting yields an alternating direction method of multipliers admm algorithm with an inner step involving a nearly shift invariant linear system that is suitable for fft based preconditioning using cone type filters the proposed method can efficiently handle a variety of convex regularization criteria including smooth edge preserving regularizers and non smooth sparsity promoting ones based on the l1 norm and total variation numerical experiments with synthetic and real in vivo human data illustrate that cone filter preconditioners accelerate the proposed admm resulting in fast convergence of admm compared to conventional nonlinear conjugate gradient ordered subsets and state of the art mfista split bregman algorithms that are applicable for ct

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

approximate formula and bounds for the time varying susceptible infected susceptible prevalence in networks based on a recent exact differential equation the time dependence of the sis prevalence the average fraction of infected nodes in any graph is first studied and then upper and lower bounded by an explicit analytic function of time that new approximate tanh formula obeys a riccati differential equation and bears resemblance to the classical expression in epidemiology of kermack and mckendrick proc r soc london a 115 700 1927 1364 502110 1098 rspa 1927 0118 but enhanced with graph specific properties such as the algebraic connectivity the second smallest eigenvalue of the laplacian of the graph we further revisit the challenge of finding tight upper bounds for the sis and sir epidemic threshold for all graphs we propose two new upper bounds and show the importance of the variance of the number of infected nodes finally a formula for the epidemic threshold in the cycle or ring graph is presented

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

expander flows geometric embeddings and graph partitioning we give a o slog n approximation algorithm for the sparsest cut edge expansion balanced separator and graph conductance problems this improves the o log n approximation of leighton and rao 1988 we use a well known semidefinite relaxation with triangle inequality constraints central to our analysis is a geometric theorem about projections of point sets in rd whose proof makes essential use of a phenomenon called measure concentration we also describe an interesting and natural approximate certificate for a graph s expansion which involves embedding an n node expander in it with appropriate dilation and congestion we call this an expander flow

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

a translation principle for the four dimensional sklyanin algebras abstract in this paper we prove a translation principle for the central quotients of four dimensional sklyanin algebras which is analogous to the translation principle for semi simple lie algebras in the course of the proof we construct an elliptic analog of the sheaf of differential operators on2 1 the translation principle may be used to construct the fat points of a four dimensional non pi sklyanin algebra

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

noetherian connected graded algebras of global dimension 3 abstract let a be a noetherian connected graded algebra of global dimension 3 we show that a is regular in the sense of artin and schelter if one of the following conditions holds 1 a is generated by two elements 2 the graded simple module has a standard resolution 3 the degrees of minimal relations are the same this includes the quadratic case some general properties of a are studied without assuming regularity

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

algebras associated to elliptic curves this paper completes the classification of artin schelter regular algebras of global dimension three for algebras generated by elements of degree one this has been achieved by artin schelter tate and van den bergh we are therefore concerned with algebras which are not generated in degree one we show that there exist some exceptional algebras each of which has geometric data consisting of an elliptic curve together with an automorphism just as in the case where the algebras are assumed to be generated in degree one in particular we study the elliptic algebras a a and a a where a p2 which were first defined in an earlier paper we omit a set s p2 consisting of 11 specified points where the algebras a a become too degenerate to be regular theorem let a represent a a or a a where a p2 s then a is an artin schelter regular algebra of global dimension three moreover a is a noetherian domain with the same hilbert series as the appropriately graded commutative polynomial ring in three variables this combined with our earlier results completes the classification

accelerated image reconstruction using ordered subsets of projection data the authors define ordered subset processing for standard algorithms such as expectation maximization em for image restoration from projections ordered subsets methods group projection data into an ordered sequence of subsets or blocks an iteration of ordered subsets em is defined as a single pass through all the subsets in each subset using the current estimate to initialize application of em with that data subset this approach is similar in concept to block kaczmarz methods introduced by eggermont et al 1981 for iterative reconstruction simultaneous iterative reconstruction sirt and multiplicative algebraic reconstruction mart techniques are well known special cases ordered subsets em os em provides a restoration imposing a natural positivity condition and with close links to the em algorithm os em is applicable in both single photon spect and positron emission tomography pet in simulation studies in spect the os em algorithm provides an order of magnitude acceleration over em with restoration quality maintained

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

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

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

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

basic properties of generalized down up algebras abstract we introduce a large class of infinite dimensional associative algebras which generalize down up algebras let k be a field and fix f k x and r s k define l l f r s to be the algebra generated by d u and h with defining relations d h r d 0 h u r u 0 d u s f h 0 included in this family are smith s class of algebras similar to u sl 2 le bruyn s conformal sl 2 enveloping algebras and the algebras studied by rueda the algebras l have gelfand kirillov dimension 3 and are noetherian domains if and only if rs 0 we calculate the global dimension of l and for rs 0 classify the simple weight modules for l including all finite dimensional simple modules simple weight modules need not be classical highest weight modules

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

noncommutative projective schemes an analogue of the concept of projective scheme is defined for noncommutative n graded algebras using the quotient category c of graded right a modules modulo its full subcategory of torsion modules we define proj a c a s where a is the object corresponding to the module aa and s is the autoequivalence defined by the shift of degrees the triples equivalent to proj a for a right noetherian graded algebra a are characterized in terms of a condition on extensions

blowing up quantum weighted projective planes abstract quantum weighted projective planes of degree 1 1 n were classified by stephenson we show that the blowups of some of these spaces are quantum ruled surfaces as defined by patrick phd thesis mit 1997 this is a quantum version of the fact that projective cones over rational normal curves blowup to rational ruled surfaces the class of algebras which we can exhibit as blowdowns of quantum ruled surfaces are those which can be twisted to ore extensions of k x y

quantum planes of weight 1 1 n 1 in this article we study the categories tails a arising from artin schelter regular algebras of global dimension 3 of weight 1 1 n where n is an integer greater than 1 we determine generators and relations for all such algebras and this information is used to classify the points and lines in tails a

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

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

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

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

accelerated iterative transmission ct reconstruction using an ordered subsets convex algorithm iterative maximum likelihood ml transmission computed tomography algorithms have distinct advantages over fourier based reconstruction but unfortunately require increased computation time the convex algorithm is a relatively fast iterative ml algorithm but it is nevertheless too slow for many applications therefore an acceleration of this algorithm by using ordered subsets of projections is proposed ordered subsets convex algorithm osc osc applies the convex algorithm sequentially to subsets of projections osc was compared with the convex algorithm using simulated and physical thorax phantom data reconstructions were performed for osc using eight and 16 subsets eight and four projections subset respectively global errors image noise contrast recovery and likelihood increase were calculated results show that osc is faster than the convex algorithm the amount of acceleration being approximately proportional to the number of subsets in osc and it causes only a slight increase of noise and global errors in the reconstructions images and image profiles of the reconstructions were in good agreement in conclusion osc and the convex algorithm result in similar image quality but osc is more than an order of magnitude faster

tighter spectral bounds for the cut size based on laplacian eigenvectors abstract the cut set v in a graph is defined as the set of all links between a set of nodes v and all other nodes in that graph finding bounds for the size of a cut set v is an important problem and is related to mixing times connectedness and spreading processes on networks a standard way to bound the number of links in a cut set v relies on laplacian eigenvalues which approximate the largest and smallest possible cut sets for a given size of the set v in this article we extend the standard spectral approximations by including information about the laplacian eigenvectors this additional information leads to provably tighter bounds compared to the standard spectral bounds we apply our new method to find improved spectral bounds for the well known cheeger constant the max cut problem and the expander mixing lemma we also apply our bounds to study cut sizes in the hypercube graph and describe an application related to the spreading of epidemics on networks we further illustrate the performance of our new bounds using simulations revealing that a significant improvement over the standard bounds is possible

hybrid poisson polynomial objective functions for tomographic image reconstruction from transmission scans this paper describes rapidly converging algorithms for computing attenuation maps from poisson transmission measurements using penalized likelihood objective functions we demonstrate that an under relaxed cyclic coordinate ascent algorithm converges faster than the convex algorithm of lange see ibid vol 4 no 10 p 1430 1438 1995 which in turn converges faster than the expectation maximization em algorithm for transmission tomography to further reduce computation one could replace the log likelihood objective with a quadratic approximation however we show with simulations and analysis that the quadratic objective function leads to biased estimates for low count measurements therefore we introduce hybrid poisson polynomial objective functions that use the exact poisson log likelihood for detector measurements with low counts but use computationally efficient quadratic or cubic approximations for the high count detector measurements we demonstrate that the hybrid objective functions reduce computation time without increasing estimation bias

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

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

globally convergent algorithms for maximum a posteriori transmission tomography this paper reviews and compares three maximum likelihood algorithms for transmission tomography one of these algorithms is the em algorithm one is based on a convexity argument devised by de pierro see ieee trans med imaging vol 12 p 328 333 1993 in the context of emission tomography and one is an ad hoc gradient algorithm the algorithms enjoy desirable local and global convergence properties and combine gracefully with bayesian smoothing priors preliminary numerical testing of the algorithms on simulated data suggest that the convex algorithm and the ad hoc gradient algorithm are computationally superior to the em algorithm this superiority stems from the larger number of exponentiations required by the em algorithm the convex and gradient algorithms are well adapted to parallel computing

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

convergent incremental optimization transfer algorithms application to tomography no convergent ordered subsets os type image reconstruction algorithms for transmission tomography have been proposed to date in contrast in emission tomography there are two known families of convergent os algorithms methods that use relaxation parameters and methods based on the incremental expectation maximization em approach this paper generalizes the incremental em approach by introducing a general framework incremental optimization transfer the proposed algorithms accelerate convergence speeds and ensure global convergence without requiring relaxation parameters the general optimization transfer framework allows the use of a very broad family of surrogate functions enabling the development of new algorithms this paper provides the first convergent os type algorithm for nonconcave penalized likelihood pl transmission image reconstruction by using separable paraboloidal surrogates sps which yield closed form maximization steps we found it is very effective to achieve fast convergence rates by starting with an os algorithm with a large number of subsets and switching to the new transmission incremental optimization transfer triot algorithm results show that triot is faster in increasing the pl objective than nonincremental ordinary sps and even os sps yet is convergent

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

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

laplacian eigenvalues and the maximum cut problem we introduce and study an eigenvalue upper bound g on the maximum cut mc g of a weighted graph the function g has several interesting properties that resemble the behaviour of mc g the following results are presented we show that is subadditive with respect to amalgam and additive with respect to disjoint sum and 1 sum we prove that g is never worse that 1 131 mc g for a planar or more generally a weakly bipartite graph with nonnegative edge weights we give a dual characterization of g and show that g is computable in polynomial time with an arbitrary precision

unified mean field framework for susceptible infected susceptible epidemics on networks based on graph partitioning and the isoperimetric inequality we propose an approximation framework that unifies and generalizes a number of existing mean field approximation methods for the susceptible infected susceptible sis epidemic model on complex networks we derive the framework which we call the unified mean field framework umff as a set of approximations of the exact markovian sis equations our main novelty is that we describe the mean field approximations from the perspective of the isoperimetric problem which results in bounds on the umff approximation error these new bounds provide insight in the accuracy of existing mean field methods such as the n intertwined mean field approximation and heterogeneous mean field method which are contained by umff additionally the isoperimetric inequality relates the umff approximation accuracy to the regularity notions of szemeredi s regularity lemma

the ordered subsets mirror descent optimization method with applications to tomography we describe an optimization problem arising in reconstructing three dimensional medical images from positron emission tomography pet a mathematical model of the problem based on the maximum likelihood principle is posed as a problem of minimizing a convex function of several million variables over the standard simplex to solve a problem of these characteristics we develop and implement a new algorithm ordered subsets mirror descent and demonstrate theoretically and computationally that it is well suited for solving the pet reconstruction problem

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

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

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

rectification for cone beam projection and backprojection the purpose of this paper is to derive a technique for accelerating the computation of cone beam forward and backward projections that are the basic steps of tomographic reconstruction the cone beam geometry of c arm systems is commonly described with projection matrices such matrices provide a continuous framework for analyzing the flow of operations needed to compute backprojection for analytical reconstruction as well as the combination of forward and backward projections for iterative reconstruction the proposed rectification technique resamples the original data to planes that are aligned with two of the reconstructed volume main axes so that the original cone beam geometry can be replaced by a simpler geometry where succession of plane magnifications are involved only rectification generalizes previous independent results to the cone beam backprojection of preprocessed data as well as to cone beam iterative reconstruction the memory access pattern of simple magnifications provides superior predictability and is therefore easier to optimize independently of the choice of the interpolation technique rectification is also shown to provide control over interpolation errors through oversampling allowing tradeoffs between computation speed and precision to be set experimental results are provided for linear and nearest neighbor interpolations based on simulations as well as phantom and patient data acquired on a digital c arm system

statistical interior tomography this paper presents a statistical interior tomography sit approach making use of compressed sensing cs theory with the projection data modeled by the poisson distribution an objective function with a total variation tv regularization term is formulated in the maximization of a posteriori map framework to solve the interior problem an alternating minimization method is used to optimize the objective function with an initial image from the direct inversion of the truncated hilbert transform the proposed sit approach is extensively evaluated with both numerical and real datasets the results demonstrate that sit is robust with respect to data noise and down sampling and has better resolution and less bias than its deterministic counterpart in the case of low count data

alternating minimization algorithms for transmission tomography a family of alternating minimization algorithms for finding maximum likelihood estimates of attenuation functions in transmission x ray tomography is described the model from which the algorithms are derived includes polyenergetic photon spectra background events and nonideal point spread functions the maximum likelihood image reconstruction problem is reformulated as a double minimization of the i divergence a novel application of the convex decomposition lemma results in an alternating minimization algorithm that monotonically decreases the objective function each step of the minimization is in closed form the family of algorithms includes variations that use ordered subset techniques for increasing the speed of convergence simulations demonstrate the ability to correct the cupping artifact due to beam hardening and the ability to reduce streaking artifacts that arise from beam hardening and background events

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

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