global optima results for the kauffman nk model the kauffman nk model has been used in theoretical biology physics and business organizations to model complex systems with interacting components recent nk model results have focused on local optima this paper analyzes global optima of the nk model the resulting global optimization problem is transformed into a stochastic network model that is closely related to two well studied problems in operations research this leads to applicable strategies for explicit computation of bounds on the global optima particularly with k either small or close to n a general lower bound which is sharp for k 0 is obtained for the expected value of the global optimum of the nk model a detailed analysis is provided for the expectation and variance of the global optimum when k n 1 the lower and upper bounds on the expectation obtained for this case show that there is a wide gap between the values of the local and the global optima they also indicate that the complexity catastrophe that occurs with the local optima does not arise for the global optima

edge decompositions of graphs with high minimum degree abstract a fundamental theorem of wilson states that for every graph f every sufficiently large f divisible clique has an f decomposition here a graph g is f divisible if e f divides e g and the greatest common divisor of the degrees of f divides the greatest common divisor of the degrees of g and g has an f decomposition if the edges of g can be covered by edge disjoint copies of f we extend this result to graphs g which are allowed to be far from complete in particular together with a result of dross our results imply that every sufficiently large k 3 divisible graph of minimum degree at least 9 n 10 o n has a k 3 decomposition this significantly improves previous results towards the long standing conjecture of nash williams that every sufficiently large k 3 divisible graph with minimum degree at least 3 n 4 has a k 3 decomposition we also obtain the asymptotically correct minimum degree thresholds of 2 n 3 o n for the existence of a c 4 decomposition and of n 2 o n for the existence of a c 2 l decomposition where l 3 our main contribution is a general iterative absorption method which turns an approximate or fractional decomposition into an exact one in particular our results imply that in order to prove an asymptotic version of nash williams conjecture it suffices to show that every k 3 divisible graph with minimum degree at least 3 n 4 o n has an approximate k 3 decomposition

proof of the bandwidth conjecture of bollobas and komlos in this paper we prove the following conjecture by bollobas and komlos for every 0 and integers r 1 and there exists 0 with the following property if g is a sufficiently large graph with n vertices and minimum degree at least r 1 r n and h is an r chromatic graph with n vertices bandwidth at most n and maximum degree at most then g contains a copy of h

approximating the independence number via the j function we describe an approximation algorithm for the independence number of a graph if a graph onn vertices has an independence numbern k m for some fixed integerk 3 and somem 0 the algorithm finds in random polynomial time an independent set of size tilde omega m 3 mathord left vphantom 3 k 1 right kern nulldelimiterspace k 1 improving the best known previous algorithm of boppana and halldorsson that finds an independent set of size m1 k 1 in such a graph the algorithm is based on semi definite programming some properties of the lovasz function of a graph and the recent algorithm of karger motwani and sudan for approximating the chromatic number of a graph if the function of ann vertex graph is at leastmn1 2 k for some absolute constantm we describe another related efficient algorithm that finds an independent set of sizek several examples show the limitations of the approach and the analysis together with some related arguments supply new results on the problem of estimating the largest possible ratio between the function and the independence number of a graph onn vertices 1998 the mathematical programming society inc published by elsevier science b v

on path correlation and pert bias most studies of project time estimation assume that a activity times are mutually independent random variables many also assume that b path completion times are mutually independent in this paper we subject the impact of both these assumptions to close scrutiny using tools from multivariate analysis we make a theoretical study of the direction of the error in the classical pert method of estimating mean project completion time when correlation is ignored we also investigate the effect of activity dependence on the normality of path length via simulation

property testing and its connection to learning and approximation in this paper we consider the question of determining whether a function f has property p or is e far from any function with property p a property testing algorithm is given a sample of the value of f on instances drawn according to some distribution in some cases it is also allowed to query f on instances of its choice we study this question for different properties and establish some connections to problems in learning theory and approximation in particular we focus our attention on testing graph properties given access to a graph g in the form of being able to query whether an edge exists or not between a pair of vertices we devise algorithms to test whether the underlying graph has properties such as being bipartite k colorable or having a p clique clique of density p with respect to the vertex set our graph property testing algorithms are probabilistic and make assertions that are correct with high probability while making a number of queries that is independent of the size of the graph moreover the property testing algorithms can be used to efficiently i e in time linear in the number of vertices construct partitions of the graph that correspond to the property being tested if it holds for the input graph

approximating probability density functions and their convolutions using orthogonal polynomials abstract this paper describes and tests methods for piecewise polynomial approximation of probability density functions using orthogonal polynomials empirical tests indicate that the procedure described in this paper can provide very accurate estimates of probabilities and means when the probability density function cannot be integrated in closed form furthermore the procedure lends itself to approximating convolutions of probability densities such approximations are useful in project management inventory modeling and reliability calculations to name a few applications in these applications decision makers desire an approximation method that is robust rather than customized also for these applications the most appropriate criterion for accuracy is the average percent error over the support of the density function as opposed to the conventional average absolute error or average squared error in this paper we develop methods for using five well known orthogonal polynomials for approximating density functions and recommend one of them as giving the best performance overall

on the complexity of nonoverlapping multivariate marginal bounds for probabilistic combinatorial optimization problems given a combinatorial optimization problem with an arbitrary partition of the set of random objective coefficients we evaluate the tightest possible bound on the expected optimal value for joint distributions consistent with the given multivariate marginals of the subsets in the partition for univariate marginals this bound was first proposed by meilijson and nadas meilijson i a nadas 1979 convex majorization with an application to the length of critical path j appl probab 16 3 671 677 we generalize the bound to nonoverlapping multivariate marginals using multiple choice integer programming new instances of polynomial time computable bounds are identified for discrete distributions for the problem of selecting up to m items out of a set of n items of maximum total weight the multivariate marginal bound is shown to be computable in polynomial time when the size of each subset in the partition is o log n for an activity on arc pert network the partition is naturally defined by subsets of incoming arcs into nodes the multivariate marginal bound on expected project duration is shown to be computable in time polynomial in the maximum number of scenarios for any subset and the size of the network as an application a polynomial time solvable two stage stochastic program for project crashing is identified an important feature of the bound developed in this paper is that it is exactly achievable by a joint distribution unlike many of the existing bounds

computing latest starting times of activities in interval valued networks with minimal time lags abstract this paper deals with problems of determining possible values of earliest and latest starting times of an activity in networks with minimal time lags and imprecise durations that are represented by means of interval or fuzzy numbers although minimal time lags are practical in different projects former researchers have not considered these problems after proposing propositions which reduce the search space a novel polynomial algorithm is presented to compute intervals of possible values of latest starting times in interval valued networks with minimal time lags then the results are extended to networks with fuzzy durations

reduction of multivariate mixtures and its applications abstract we consider fast deterministic algorithms to identify the best linearly independent terms in multivariate mixtures and use them to compute up to a user selected accuracy an equivalent representation with fewer terms importantly the multivariate mixtures do not have to be a separated representation of a function one algorithm employs a pivoted cholesky decomposition of the gram matrix constructed from the terms of the mixture to select what we call skeleton terms and the other uses orthogonalization for the same purpose both algorithms require o r 2 n p d r n operations where n is the initial number of terms in the multivariate mixture r is the number of selected linearly independent terms and p d is the cost of computing the inner product between two terms of a mixture in d variables for general gaussian mixtures p d d 3 since we need to diagonalize a d d matrix whereas for separated representations p d d there is no need for diagonalization due to conditioning issues the resulting accuracy is limited to about one half of the available significant digits for both algorithms we also describe an alternative algorithm that is capable of achieving higher accuracy but is only applicable in low dimensions or to multivariate mixtures in separated form we describe a number of initial applications of these algorithms to solve partial differential and integral equations and to address several problems in data science for data science applications in high dimensions we consider the kernel density estimation kde approach for constructing a probability density function pdf of a cloud of points a far field kernel summation method and the construction of equivalent sources for non oscillatory kernels used in both computational physics and data science and finally show how to use the new algorithm to produce seeds for subdividing a cloud of points into groups

sparse partition universal graphs for graphs of bounded degree abstract in 1983 chvatal trotter and the two senior authors proved that for any there exists a constant b such that for any n any 2 colouring of the edges of the complete graph k n with n b n vertices yields a monochromatic copy of any graph h that has n vertices and maximum degree we prove that the complete graph may be replaced by a sparser graph g that has n vertices and o n 2 1 log 1 n edges with n b n for some constant b that depends only on consequently the so called size ramsey number of any h with n vertices and maximum degree is o n 2 1 log 1 n our approach is based on random graphs in fact we show that the classical erd s renyi random graph with the numerical parameters above satisfies a stronger partition property with high probability namely that any 2 colouring of its edges contains a monochromatic universal graph for the class of graphs on n vertices and maximum degree the main tool in our proof is the regularity method adapted to a suitable sparse setting the novel ingredient developed here is an embedding strategy that allows one to embed bounded degree graphs of linear order in certain pseudorandom graphs crucial to our proof is the fact that regularity is typically inherited at a scale that is much finer than the scale at which it is assumed

extremal results in sparse pseudorandom graphs abstract szemeredi s regularity lemma is a fundamental tool in extremal combinatorics however the original version is only helpful in studying dense graphs in the 1990s kohayakawa and rodl proved an analogue of szemeredi s regularity lemma for sparse graphs as part of a general program toward extending extremal results to sparse graphs many of the key applications of szemeredi s regularity lemma use an associated counting lemma in order to prove extensions of these results which also apply to sparse graphs it remained a well known open problem to prove a counting lemma in sparse graphs the main advance of this paper lies in a new counting lemma proved following the functional approach of gowers which complements the sparse regularity lemma of kohayakawa and rodl allowing us to count small graphs in regular subgraphs of a sufficiently pseudorandom graph we use this to prove sparse extensions of several well known combinatorial theorems including the removal lemmas for graphs and groups the erd s stone simonovits theorem and ramsey s theorem these results extend and improve upon a substantial body of previous work

cycle decompositions v complete graphs into cycles of arbitrary lengths we show that the complete graph on n vertices can be decomposed into t cycles of specified lengths m1 mt if and only if n is odd 3 mi n for i 1 t and we also show that the complete graph on n vertices can be decomposed into a perfect matching and t cycles of specified lengths m1 mt if and only if n is even 3 mi n for i 1 t and m1 mt n2 n 2

an algorithmic regularity lemma for hypergraphs in this paper we will consider the problem of designing an efficient algorithm that finds an epsilon regular partition of an l uniform hypergraph

convergent sequences of dense graphs i subgraph frequencies metric properties and testing we consider sequences of graphs gn and define various notions of convergence related to these sequences left convergence defined in terms of the densities of homomorphisms from small graphs into gn right convergence defined in terms of the densities of homomorphisms from gn into small graphs and convergence in a suitably defined metric in part i of this series we show that left convergence is equivalent to convergence in metric both for simple graphs gn and for graphs gn with nodeweights and edgeweights one of the main steps here is the introduction of a cut distance comparing graphs not necessarily of the same size we also show how these notions of convergence provide natural

quasi randomness and algorithmic regularity for graphs with general degree distributions we deal with two intimately related subjects quasi randomness and regular partitions the purpose of the concept of quasi randomness is to express how much a given graph resembles a random one moreover a regular partition approximates a given graph by a bounded number of quasi random graphs regarding quasi randomness we present a new spectral characterization of low discrepancy which extends to sparse graphs concerning regular partitions we introduce a concept of regularity that takes into account vertex weights and show that if g v e satisfies a certain boundedness condition then g admits a regular partition in addition building on the work of alon and naor proceedings of the 36th acm symposium on theory of computing stoc chicago il acm new york 2004 pp 72 80 we provide an algorithm that computes a regular partition of a given possibly sparse graph g in polynomial time as an application we present a polynomial time approximation scheme for max cut on sparse graphs without dense spots

hamilton decompositions of regular expanders a proof of kelly s conjecture for large tournaments abstract a long standing conjecture of kelly states that every regular tournament on n vertices can be decomposed into n 1 2 edge disjoint hamilton cycles we prove this conjecture for large n in fact we prove a far more general result based on our recent concept of robust expansion and a new method for decomposing graphs we show that every sufficiently large regular digraph g on n vertices whose degree is linear in n and which is a robust outexpander has a decomposition into edge disjoint hamilton cycles this enables us to obtain numerous further results e g as a special case we confirm a conjecture of erd s on packing hamilton cycles in random tournaments as corollaries to the main result we also obtain several results on packing hamilton cycles in undirected graphs giving e g the best known result on a conjecture of nash williams we also apply our result to solve a problem on the domination ratio of the asymmetric travelling salesman problem which was raised e g by glover and punnen as well as alon gutin and krivelevich

parallel tensor methods for high dimensional linear pdes abstract high dimensional partial differential equations pdes arise in a number of fields of science and engineering where they are used to describe the evolution of joint probability functions their examples include the boltzmann and fokker planck equations we develop new parallel algorithms to solve such high dimensional pdes the algorithms are based on canonical and hierarchical numerical tensor methods combined with alternating least squares and hierarchical singular value decomposition both implicit and explicit integration schemes are presented and discussed we demonstrate the accuracy and efficiency of the proposed new algorithms in computing the numerical solution to both an advection equation in six variables plus time and a linearized version of the boltzmann equation

a fast approximation algorithm for computing thefrequencies of subgraphs in a given graph in this paper we give an algorithm which given a labeled graph on n vertices and a list of all labeled graphs on k vertices provides for each graph h of this list an approximation to the number of induced copies of h in g with total error small this algorithm has running time o n 1 over log log n cdot m n where m n is the time needed to square a n by n matrix with 0 1 entries over the integers the main tool in designing this algorithm is a variant of the regularity lemma of szemeredi

tensor methods for the boltzmann bgk equation abstract we present a tensor decomposition method to solve the boltzmann transport equation bte in the bhatnagar gross krook approximation the method represents the six dimensional bte as a set of six one dimensional problems which are solved with the alternating least squares algorithm and the discrete fourier transform at n collocation points we use this method to predict the equilibrium distribution steady state simulation and a non equilibrium distribution returning to the equilibrium transient simulation our numerical experiments demonstrate n log n scaling unlike many bte specific numerical techniques the numerical tensor decomposition method we propose is a general technique that can be applied to other high dimensional systems

a new structural mechanism for reducibility of stochastic pert networks abstract in recent years many techniques were developed to compute completion time distribution function of stochastic pert networks none of the existing techniques clearly consider the subject from structural viewpoint in this paper a structural mechanism based on graph theory is presented this mechanism changes structure of network to series parallel network with contraction of some arcs in contraction operation one arc is removed and its two nodes are changed to individual node in this operation probably dependency between activities and number of existing paths in the network would be different from the original network an algorithm is presented in correspondence with the developed structural mechanism through this algorithm the mechanism acts as an effective approach to estimate f t the paper contains a comparative analysis this comparison is made between the computational results of developed algorithm and those of dodin s and kleindorfer s approaches the comparative analysis demonstrates that the mechanism has capability to be used for the networks with actual sizes also to show that the method works for practical problems in the real world a real case from a car factory of iran is considered again for this case of the developed method dodin s and kleindorfer s approaches are applied whose results approve the acceptability of proposed method completely

arc reduction and path preference in stochastic acyclic networks the paper presents a heuristic for determining the path that maximizes the expected utility of a stochastic acyclic network the focus is on shortest route problems where a general nonlinear utility function is used to measure outcomes for such problems enumerating all feasible paths is the only way to guarantee that the global optimum has been found alternatively we develop a reduction algorithm based on stochastic dominance to speed up the computations monte carlo simulation is used to evaluate the approach in all 70 test problems comprising 20 to 60 nodes are randomly generated and analyzed the results indicate that the heuristic produces significant computational saving as the size of the network grows and that the quality of the reduced network solutions are better than those obtained from the original formulation

the most critical path in a pert network a heuristic approach abstract the classical pert approach uses a deterministic critical path to estimate the probability of completing a project by a given deadline in general such a path is not the most critical path and does not provide the lowest probability of completion time this paper studies the most critical path problem and formulates it as an optimal path problem through a deterministic network with a two attribute fractional objective function a heuristic approach is developed for the optimal path problem which gives a solution to the most critical path problem the illustrative examples and computational results demonstrate that the proposed procedure provides critical activities and estimates for the probabilities of completion time that are more accurate than those resulting from the classical pert approach further our computational experiments show that the proposed heuristic performs well in identifying the most critical paths

additive approximation for edge deletion problems a graph property is monotone if it is closed under removal of vertices and edges in this paper we consider the following algorithmic problem called the edge deletion problem given a monotone property p and a graph g compute the smallest number of edge deletions that are needed in order to turn g into a graph satisfying p we denote this quantity by e p g the first result of this paper states that the edge deletion problem can be efficiently approximated for any monotone property for any fixed e 0 and any monotone property p there is a deterministic algorithm which given a graph g v e of size n approximates e p g in linear time o v e to within an additive error of en 2 given the above a natural question is for which monotone properties one can obtain better additive approximations of e p our second main result essentially resolves this problem by giving a precise characterization of the monotone graph properties for which such approximations exist 1 if there is a bipartite graph that does not satisfy p then there is a 0 for which it is possible to approximate e p to within an additive error of n 2 in polynomial time 2 on the other hand if all bipartite graphs satisfy p then for any 0 it is np hard to approximate e p to within an additive error of n 2 while the proof of 1 is relatively simple the proof of 2 requires several new ideas and involves tools from extremal graph theory together with spectral techniques interestingly prior to this work it was not even known that computing e p precisely for the properties in 2 is np hard we thus answer in a strong form a question of yannakakis who asked in 1981 if it is possible to find a large and natural family of graph properties for which computing e p is np hard

approximating the cut norm via grothendieck s inequality the cut norm a c of a real matrix a a ij i in r j in s is the maximum over all i subset r j subset s of the quantity sum i in i j in j a ij this concept plays a major role in the design of efficient approximation algorithms for dense graph and matrix problems here we show that the problem of approximating the cut norm of a given real matrix is max snp hard and we provide an efficient approximation algorithm this algorithm finds for a given matrix a a ij i in r j in s two subsets i subset r and j subset s such that sum i in i j in j a ij geq rho a c where rho 0 is an absolute constant satisfying rho 0 56 the algorithm combines semidefinite programming with a rounding technique based on grothendieck s inequality we present three known proofs of grothendieck s inequality with the necessary modifications which emphasize their algorithmic aspects these proofs contain rounding techniques which go beyond the random hyperplane rounding of goemans and williamson j acm 42 1995 pp 1115 1145 allowing us to transfer various algorithms for dense graph and matrix problems to the sparse case

approximate hypergraph partitioning and applications szemeredi s regularity lemma is a cornerstone result in extremal combinatorics it roughly asserts that any dense graph is composed of a finite number of pseudorandom graphs the regularity lemma has found many applications in theoretical computer science and thus a lot of attention was given to designing algorithmic versions of this lemma our main results in this paper are the following i we introduce a new approach to the problem of constructing regular partitions of graphs which results in a surprisingly simple o n time algorithmic version of the regularity lemma thus improving over the previous o n 2 time algorithms furthermore unlike all the previous approaches for this problem see n alon and a naor siam j comput 35 2006 pp 787 803 r a duke h lefmann and v rodl siam j comput 24 1995 pp 598 620 a frieze and r kannan electron j combin 6 1999 article 17 a frieze and r kannan the regularity lemma and approximation schemes for dense problems in proceedings of the 37th annual symposium on foundations of computer science burlington vt 1996 ieee computer society press los alamitos ca 1996 pp 12 20 and y kohayakawa v rodl and l thoma siam j comput 32 2003 pp 1210 1235 which only guaranteed to find tower size partitions our algorithm will find a small regular partition if one exists in the graph ii for any constant r geq3 we give an o n time randomized algorithm for constructing regular partitions of r uniform hypergraphs thus improving the previous o n 2r 1 time deterministic algorithms a czygrinow and v rodl siam j comput 30 2000 pp 1041 1066 a frieze and r kannan the regularity lemma and approximation schemes for dense problems in proceedings of the 37th annual symposium on foundations of computer science burlington vt 1996 ieee computer society press los alamitos ca 1996 pp 12 20 these two results are obtained as an application of an efficient algorithm for approximating partition problems of hypergraphs which we obtain here given a directed hypergraph with bounded edge arities a set of constraints on the set sizes and densities of a possible partition of its vertex set and an approximation parameter we provide in o n time a partition approximating the constraints if a partition satisfying them exists we can also test in o 1 time for the existence of such a partition given the approximation parameter this algorithm extends the result of goldreich goldwasser and ron for graph partition problems o goldreich s goldwasser and d ron j acm 45 1998 pp 653 750 and encompasses more recent hypergraph related results such as the maximal constraint satisfaction approximation of g andersson and l engebretsen random structures algorithms 21 2002 pp 14 32

the primes contain arbitrarily long arithmetic progressions we prove that there are arbitrarily long arithmetic progressions of primes there are three major ingredients the first is szemeredi s theorem which asserts that any subset of the integers of positive density contains progressions of arbitrary length the second which is the main new ingredient of this paper is a certain transference principle this allows us to deduce from szemeredi s theorem that any subset of a suficiently pseudorandom set or measure of positive relative density contains progressions of arbitrary length the third ingredient is a recent result of goldston and yildirim which we reproduce here using this one may place a large fraction of the primes inside a pseudorandom set of almost primes or more precisely a pseudorandom measure concentrated on almost primes with positive relative density

risk taking in stochastic pert networks abstract this paper studies a pert problem which incorporates project managers risk taking behavior in stochastic environments in this stochastic pert network which we refer to as riskpert activity durations are nonnegative random variables and a project manager uses a disutility function to evaluate each path and chooses that path which maximizes his expected disutility as an optimal or critical path this path as opposed to the classical critical path is used as a basis to analyze the pert network and to estimate the probability of completion time the general riskpert is difficult to solve however special cases when activity durations are statistically independent and disutility functions are linear exponential quadratic or linear exponential are solvable exactly the illustrative examples demonstrate that riskpert captures managers risk taking behavior and provides results that are more realistic than those of the classical pert models furthermore this paper extends some of the riskpert models to include multi dimensional pert networks

a new upper bound for diagonal ramsey numbers we prove a new upper bound for diagonal two colour ramsey numbers showing that there exists a constant c such that r k 1 k 1 k clog k log log k 2k k

bounding the project completion time distribution in pert networks we consider the pert model of a project composed of activities whose durations are random variables with known distributions for the situations in which the activity durations are completely independent we present a new method for obtaining a probability distribution function that bounds the exact probability distribution of the project completion time from below the bounding distribution can be used to obtain an upper bound on the mean completion time of the project we also prove and illustrate that this bounding distribution is better tighter than any of the existing lower bounds implying that the corresponding upper bound on the mean completion time is tighter than any of the existing upper bounds

a random activity network generator exact and heuristic procedures are often developed to obtain optimal and near optimal solutions to decision problems modeled as activity networks testing the accuracy and efficiency of these procedures requires the use of activity networks with various sizes structures and parameters the size of the network is determined by its number of nodes and arcs where the structure is chosen from the set of all structures for the specified network size the network parameters depend on the nature of the decision problem often it is desirable for test problems to be generated at random from the space of all feasible networks this paper deals with the problem of generating the size and structure of the network at random from the space of all feasible networks it develops a theory which guarantees the randomness of the network structure the theory is the basis for two methods one can be used to generate dense networks where the other is used to generate nondense networks the methods which are practical a

on latest starting times and floats in activity networks with ill known durations abstract this paper deals with fuzzy activity networks where fuzzy intervals model uncertain durations of tasks while it is easy to compute fuzzy earliest starting times of activities using the critical path method the problem of determining latest starting dates and slack times is much more tricky and has never been solved in a fully satisfactory manner in the past here we propose a rigorous treatment of this problem in the framework of possibility theory the main difficulty lies in the fact that the behavior of latest starting dates and slack times as a function of task durations is not straightforward to predict for general network topologies however it is easier in the case of series parallel graphs the case of interval valued durations is first addressed and then extended to fuzzy intervals

least squares approximation to the distribution of project completion times with gaussian uncertainty this paper is motivated by the following question how to construct good approximation for the distribution of the solution value to linear optimization problem when the random objective coefficients follow a multivariate normal distribution using stein s identity we show that the least squares normal approximation of the random optimal value can be computed by estimating the persistency values of the corresponding optimization problem we further extend our method to construct a least squares quadratic estimator to improve the accuracy of the approximation in particular to capture the skewness of the objective computational studies show that the new approach provides more accurate estimates of the distributions of project completion times compared to existing methods

randomized alternating least squares for canonical tensor decompositions application to a pde with random data this paper introduces a randomized variation of the alternating least squares als algorithm for rank reduction of canonical tensor formats the aim is to address the potential numerical ill conditioning of least squares matrices at each als iteration the proposed algorithm dubbed randomized als mitigates large condition numbers via projections onto random tensors a technique inspired by well established randomized projection methods for solving overdetermined least squares problems in a matrix setting a probabilistic bound on the condition numbers of the randomized als matrices is provided demonstrating reductions relative to their standard counterparts additionally results are provided that guarantee comparable accuracy of the randomized als solution at each iteration the performance of the randomized algorithm is studied with three examples including manufactured tensors and an elliptic pde with random inputs in particular for the latter tests illustrate not only improvements in condition

packing degenerate graphs abstract given d and 0 whenever c 0 is sufficiently small and n sufficiently large if g is a family of d degenerate graphs of individual orders at most n maximum degrees at most c n log n and total number of edges at most 1 n 2 then g packs into the complete graph k n our proof proceeds by analysing a natural random greedy packing algorithm

optimization via separated representations and the canonical tensor decomposition abstract we introduce a new quadratically convergent algorithm for finding maximum absolute value entries of tensors represented in the canonical format the computational complexity of the algorithm is linear in the dimension of the tensor we show how to use this algorithm to find global maxima of non convex multivariate functions in separated form we demonstrate the performance of the new algorithms on several examples

quasi random set systems there are many properties of mathematical objects that satisfy what is sometimes called a 0 1 law in the following sense under some natural probability measure on the set of objects the measure of the subset of objects having the given property is either 0 or 1 in the latter case we can say that almost all the objects have the property familiar examples of this phenomenon are the following almost all real numbers are transcendental or normal to every base almost all integers are composite almost all continuous real functions are nondifferentiable etc it is often the case that the objects under consideration can be partitioned into a countable number of finite classes cn with the probability assigned to an object in cn being just 1 icn i in this case we say that a property pn satisfies a 0 1 law if the fraction of the number of objects in cn that satisfy pn either tends to 0 or tends to 1 as n x 0o for example almost all graphs on n vertices have maximum cliques and maximum independent sets of size at most 2 log n almost all boolean functions with n variables have circuit complexity 1 o 1 2n and almost all binary codes of length n with at most 2nr codewords with r less than the binary symmetric channel capacity c have arbitrarily small error probability a special case of shannon s coding theorem see s48 one of the first general results of this type was the theorem of fagin f76 and glebskii et al gklt69 which asserts that every property of graphs that can be expressed in first order logic satisfies a 0 1 law see ss88 for recent striking developments in this topic one obvious method for finding explicit objects having some property pn shared by almost all objects in cn is simply to select one at random with overwhelming probability tending to 1 as n x oc the selected object will have property pn unfortunately it may be and often is extremely difficult to prove that any particular object does indeed satisfy pn it is our purpose in this paper to describe a method that can to a certain extent circumvent this difficulty we will show that for a variety of families it is possible to identify a natural hierarchy of equivalence classes of properties all of which are shared by almost all objects in the family any object satisfying

systematic identification of statistically significant network measures we present a graph embedding space i e a set of measures on graphs for performing statistical analyses of networks key improvements over existing approaches include discovery of motif hubs multiple overlapping significant subgraphs computational efficiency relative to subgraph census and flexibility the method is easily generalizable to weighted and signed graphs the embedding space is based on scalars functionals of the adjacency matrix representing the network scalars are global involving all nodes although they can be related to subgraph enumeration there is not a one to one mapping between scalars and subgraphs improvements in network randomization and significance testing we learn the distribution rather than assuming gaussianity are also presented the resulting algorithm establishes a systematic approach to the identification of the most significant scalars and suggests machine learning techniques for network classification

the exact overall time distribution of a project with uncertain task durations abstract this paper presents a new technique for computing the exact overall duration of a project when task durations have independent distributions a project is represented as an activity on arc aoa graph where a task begins as soon as all its predecessor tasks have finished task durations use a probability density function p d f which combines piecewise polynomial segments and dirac delta functions defined over a finite interval a semi analytical procedure is proposed to compute the cumulative distribution function c d f directly by integrating a linear transformation of the p d f of the task durations graph reduction techniques by hopcroft and tarjan and by valdes allow the problem to be broken into a series of smaller subproblems improving computational efficiency examples are presented to illustrate the proposed method

the hypergraph regularity method and its applications szemeredi s regularity lemma asserts that every graph can be decomposed into relatively few random like subgraphs this random like behavior enables one to find and enumerate subgraphs of a given isomorphism type yielding the so called counting lemma for graphs the combined application of these two lemmas is known as the regularity method for graphs and has proved useful in graph theory combinatorial geometry combinatorial number theory and theoretical computer science here we report on recent advances in the regularity method for k uniform hypergraphs for arbitrary k 2 this method purely combinatorial in nature gives alternative proofs of density theorems originally due to e szemeredi h furstenberg and y katznelson further results in extremal combinatorics also have been obtained with this approach the two main components of the regularity method for k uniform hypergraphs the regularity lemma and the counting lemma have been obtained recently rodl and skokan based on earlier work of frankl and rodl generalized szemeredi s regularity lemma to k uniform hypergraphs and nagle rodl and schacht succeeded in proving a counting lemma accompanying the rodl skokan hypergraph regularity lemma the counting lemma is proved by reducing the counting problem to a simpler one previously investigated by kohayakawa rodl and skokan similar results were obtained independently by w t gowers following a different approach

an optimal algorithm for checking regularity we present a deterministic algorithm cal a that in o m2 time verifies whether a given m by m bipartite graph g is regular in the sense of szemeredi regular partitions of graphs in problemes combinatoires et theorie des graphes orsay 1976 colloques internationaux cnrs 260 cnrs paris 1978 pp 399 401 in the case in which g is not regular enough our algorithm outputs a witness to this irregularity algorithm cal a may be used as a subroutine in an algorithm that finds an varepsilon regular partition of a given n vertex graph gamma in time o n2 this time complexity is optimal up to a constant factor and improves upon the bound o m n proved by alon et al the algorithmic aspects of the regularity lemma j algorithms 16 1994 pp 80 109 where m n o n2 376 is the time required to square a 0 1 matrix over the integers our approach is elementary except that it makes use of linear sized expanders to accomplish a suitable form of deterministic sampling