characterizations of locally testable linear and affine invariant families the linear or affine invariance is the property of a function family that is closed under linear or affine transformations on the domain and closed under linear combinations of functions respectively both the linear and affine invariant families of functions are generalizations of many symmetric families for instance the low degree polynomials kaufman and sudan 2007 21 introduced the notions of constraint and characterization to characterize the locally testable affine and linear invariant families of functions over finite fields of constant size in this article it is shown that for any finite field f of size q and characteristic p and its arbitrary extension field k of size q if an affine invariant family k n f has a k local constraint then it is k locally testable for k k 2 q pq 2 q p 4 and that if a linear invariant family k n f has a k local characterization then it is k locally testable for k 2k 2 q pq 4 q p 1 consequently for any prime field f of size q any positive integer k we have that for any affine invariant family over field f the four notions of the constraint the characterization the formal characterization and the local testability are equivalent modulo a poly k q of the corresponding localities and that for any linear invariant family the notions of the characterization the formal characterization and the local testability are equivalent modulo a poly k q of the corresponding localities the results significantly improve and are in contrast to the characterizations in 21 which have locality exponential in q even if the field k is prime in the research above a missing result is a characterization of linear invariant function families by the more natural notion of constraint for this we show that a single strong local constraint is sufficient to characterize the local testability of a linear invariant boolean function family and that for any finite field f of size q greater than 2 there exists a linear invariant function family over f such that it has a strong 2 local constraint but is not q d q 1 1 locally testable the proof for this result provides an appealing approach toward more negative results in the theme of characterization of locally testable algebraic properties which is rare and of course significant

using dual approximation algorithms for scheduling problems theoretical and practical results the problem of scheduling a set of n jobs on m identical machines so as to minimize the makespan time is perhaps the most well studied problem in the theory of approximation algorithms for np hard optimization problems in this paper the strongest possible type of result for this problem a polynomial approximation scheme is presented more precisely for each e an algorithm that runs in time o n e 1 e 2 and has relative error at most e is given in addition more practical algorithms for e 1 5 2 k and e 1 6 2 k which have running times o n k log n and o n km 4 log n are presented the techniques of analysis used in proving these results are extremely simple especially in comparison with the baroque weighting techniques used previously the scheme is based on a new approach to constructing approximation algorithms which is called dual approximation algorithms where the aim is to find superoptimal but infeasible solutions and the performance is measured by the degree of infeasibility allowed this notion should find wide applicability in its own right and should be considered for any optimization problem where traditional approximation algorithms have been particularly elusive

a new rounding procedure for the assignment problem with applications to dense graph arrangement problems we present a randomized procedure for rounding fractional perfect matchings to integral matchings if the original fractional matching satisfies any linear inequality then with high probability the new matching satisfies that linear inequality in an approximate sense this extends the well known lp rounding procedure of raghavan and thompson which is usually used to round fractional solutions of linear programs we use our rounding procedure to design an additive approximation algorithm to the quadratic assignment problem the approximation error of the algorithm is en 2 and it runs in n o log n e2 time we also describe polynomial time approximation schemes ptass for dense subcases of many well known np hard arrangement problems including minimum linear arrangement minimum cut linear arrangement maximum acyclic subgraph and betweenness

robust characterizations of polynomials withapplications to program testing the study of self testing and self correcting programs leads to the search for robust characterizations of functions here the authors make this notion precise and show such a characterization for polynomials from this characterization the authors get the following applications simple and efficient self testers for polynomial functions are constructed the characterizations provide results in the area of coding theory by giving extremely fast and efficient error detecting schemes for some well known codes this error detection scheme plays a crucial role in subsequent results on the hardness of approximating some np optimization problems

applying property testing to an image partitioning problem property testing is a rapidly growing field of research typically a property testing algorithm proceeds by quickly determining whether an input can satisfy some condition under the assumption that most inputs do not satisfy it if the input is far from satisfying the condition the algorithm is guaranteed to reject it with high probability applying this paradigm to image detection is desirable since images are large objects and a lot of time can be saved by quickly rejecting images which are far from satisfying a certain condition the user is interested in further typically most inputs are indeed far from the sought images we demonstrate this by analyzing the problem of deciding whether a binary image can be partitioned according to a template represented by a rectangular grid and introduce a quick rejector which tests an image extracted from the input image but whose size as well as the time required to construct it are constants which are independent of the input image size with high probability the rejector dismisses the inputs which are far from the template

analysis and implementation issues for the numerical approximation of parabolic equations with random coefficients we consider the problem of numerically approximating statistical moments of the solution of a time dependent linear parabolic partial differential equation pde whose coefficients and or forcing terms are spatially correlated random fields the stochastic coefficients of the pde are approximated by truncated karhunen loeve expansions driven by a finite number of uncorrelated random variables after approximating the stochastic coefficients the original stochastic pde turns into a new deterministic parametric pde of the same type the dimension of the parameter set being equal to the number of random variables introduced after proving that the solution of the parametric pde problem is analytic with respect to the parameters we consider global polynomial approximations based on tensor product total degree or sparse polynomial spaces and constructed by either a stochastic galerkin or a stochastic collocation approach we derive convergence rates for the different cases and present numerical results that show how these approaches are a valid alternative to the more traditional monte carlo method for this class of problems copyright 2009 john wiley sons ltd

error analysis of a stochastic collocation method for parabolic partial differential equations with random input data a stochastic collocation method for solving linear parabolic partial differential equations with random coefficients forcing terms and initial conditions is analyzed the input data are assumed to depend on a finite number of random variables unlike previous analyses a wider range of situations are considered including input data that depend nonlinearly on the random variables and random variables that are correlated or even unbounded we provide a rigorous convergence analysis and demonstrate the exponential decay of the interpolation error in the probability space for both finite element semidiscrete spatial discretizations and for finite element crank nicolson fully discrete space time discretizations ingredients in the convergence analysis include the proof of the analyticity with respect to the probabilistic parameters of the semidiscrete and fully discrete approximate solutions a numerical example is provided to illustrate the analyses

testing computability by width two obdds property testing is concerned with deciding whether an object e g a graph or a function has a certain property or is far for a prespecified distance measure from every object with that property in this work we consider the property of being computable by a read once width 2ordered binary decision diagram obdd also known as a branching program in two settings in the first setting the order of the variables is fixed and given to the algorithm while in the second setting it is not fixed that is while in the first setting we should accept a function f if it is computable by a width 2 obdd with a given order of the variables in the second setting we should accept a function f if there exists an order of the variables according to which a width 2 obdd can compute f width 2 obdds generalize two classes of functions that have been studied in the context of property testing linear functions over gf 2 and monomials in both these cases membership can be tested by performing a number of queries that is independent of the number of variables n and is linear in 1 e where e is the distance parameter in contrast we show that testing computability by width 2 obdds when the order of variables is fixed and known requires a number of queries that grows logarithmically with n for a constant e and we provide an algorithm that performs email protected logn e queries for the case where the order is not fixed we show that there is no testing algorithm that performs a number of queries that is sublinear in n

multigrid methods and sparse grid collocation techniques for parabolic optimal control problems with random coefficients an efficient computational framework to solve nonlinear parabolic optimal control problems with random coefficients is presented this framework allows us to investigate the influence of randomness or uncertainty of problem s parameters values on the control provided by the optimal control theory the proposed framework combines space time multigrid methods with sparse grid collocation techniques theoretical and numerical results of computation of stochastic optimal control solutions and formulation of mean control functions are presented

quantum locally testable codes we initiate the study of quantum locally testable codes text qltc s classical text ltc s are very important in computational complexity these codes are defined as the linear subspace satisfying a set of local constraints with the additional requirement that their soundness r delta which is the probability that a randomly chosen constraint is violated is proportional to the proximity delta where delta n is the distance of a word from the code excellent text ltc s exist in the classical world and they are tightly related to the celebrated text pcp probabilistically checkable proof theorem in quantum complexity quantum error correcting codes provide central examples in the study of the illusive behavior of multiparticle entanglement and they have played a crucial role in many computational complexity results we provide a definition of the quantum analogue of text ltc s and motivate it by connecting its central notions in the study of both entanglement and quantum ha

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

a randomized linear time algorithm to find minimum spanning trees we present a randomized linear time algorithm to find a minimum spanning tree in a connected graph with edge weights the algorithm uses random sampling in combination with a recently discovered linear time algorithm for verifying a minimum spanning tree our computational model is a unit cost random access machine with the restriction that the only operations allowed on edge weights are binary comparisons

a general classification rule for probability measures abstract we consider the problem of classifying an unknown probability distribution based on a sequence of random samples drawn according to this distribution specifically if a is a subset of the space of all probability measures m1 sigma over some compact polish space e we want to decide whether or not the unknown distribution belongs to a or its complement we propose an algorithm which leads a s to a correct decision for any a satisfying certain structural assumptions a refined decision procedure is also presented which given a countable collection ai c m1 sigma i 1 2 each satisfying the structural assumption will eventually determine a s the membership of the distribution in any finite number of the ai applications to density estimation and the problem of order determination of markov processes are discussed

fast monte carlo algorithms for finding low rank approximations we consider the problem of approximating a given m n matrix a by another matrix of specified rank k which is smaller than m and n the singular value decomposition svd can be used to find the best such approximation however it takes time polynomial in m n which is prohibitive for some modern applications in this article we develop an algorithm that is qualitatively faster provided we may sample the entries of the matrix in accordance with a natural probability distribution in many applications such sampling can be done efficiently our main result is a randomized algorithm to find the description of a matrix d of rank at most k so that holds with probability at least 1 where v vf is the frobenius norm the algorithm takes time polynomial in k 1 e log 1 only and is independent of m and n in particular this implies that in constant time it can be determined if a given matrix of arbitrary size has a good low rank approximation

on approximating the number of k cliques in sublinear time we study the problem of approximating the number of k cliques in a graph when given query access to the graph we consider the standard query model for general graphs via 1 degree queries 2

approximate consistency for transformations on words and trees we introduce approximate source consistency for transformations of words and trees the relaxed version of the source consistency problem a setting consists in an input schema an output schema and a relation t between input and output structures given an input structure i we want to decide if there is an output structure j in the output schema such that i j t we consider transducers of words and trees and prove the testability of this property for some edit distance on words and trees we exhibit randomized algorithms which distinguish between a consistent and an e far from consistent i by looking at a constant fraction of the large input i the main result on trees is based on a statistical representation of ordered unranked trees which may be used in other contexts

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

high order collocation methods for differential equations with random inputs recently there has been a growing interest in designing efficient methods for the solution of ordinary partial differential equations with random inputs to this end stochastic galerkin methods appear to be superior to other nonsampling methods and in many cases to several sampling methods however when the governing equations take complicated forms numerical implementations of stochastic galerkin methods can become nontrivial and care is needed to design robust and efficient solvers for the resulting equations on the other hand the traditional sampling methods e g monte carlo methods are straightforward to implement but they do not offer convergence as fast as stochastic galerkin methods in this paper a high order stochastic collocation approach is proposed similar to stochastic galerkin methods the collocation methods take advantage of an assumption of smoothness of the solution in random space to achieve fast convergence however the numerical implementation of stochastic collocation is trivial as it requires only repetitive runs of an existing deterministic solver similar to monte carlo methods the computational cost of the collocation methods depends on the choice of the collocation points and we present several feasible constructions one particular choice based on sparse grids depends weakly on the dimensionality of the random space and is more suitable for highly accurate computations of practical applications with large dimensional random inputs numerical examples are presented to demonstrate the accuracy and efficiency of the stochastic collocation methods

approximate membership for regular languages modulo the edit distance abstract we present an efficient probabilistic algorithm for testing approximate membership of words to regular languages modulo the edit distance our algorithm runs in polynomial time in the size of the input automaton and the inverse error precision in contrast to all previous approaches and independently of the size of the input word we also improve a previous approximate membership tester modulo the hamming distance such that it runs in polynomial complexity time but with larger polynomials than for the edit distance

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

the minimum consistent dfa problem cannot be approximated within any polynomial the minimum consistent dfa problem is that of finding a dfa with as few states as possible that is consistent with a given sample a finite collection of words each labeled as to whether the dfa found should accept or reject assuming that p np it is shown that for any constant k no polynomial time algorithm can be guaranteed to find a consistent dfa with fewer than opt k states where opt is the number of states in the minimum state dfa consistent with the sample this result holds even if the alphabet is of constant size two and if the algorithm is allowed to produce an nfa a regular expression or a regular grammar that is consistent with the sample a similar nonapproximability result is presented for the problem of finding small consistent linear grammars for the case of finding minimum consistent dfas when the alphabet is not of constant size but instead is allowed to vary with the problem specification the slightly stronger lower bound on approximability of opt 1 e log log opt is shown for any e 0

optimal control with stochastic pde constraints and uncertain controls abstract the optimal control of problems that are constrained by partial differential equations with uncertainties and with uncertain controls is addressed the lagrangian that defines the problem is postulated in terms of stochastic functions with the control function possibly decomposed into an unknown deterministic component and a known zero mean stochastic component the extra freedom provided by the stochastic dimension in defining cost functionals is explored demonstrating the scope for controlling statistical aspects of the system response one shot stochastic finite element methods are used to find approximate solutions to control problems it is shown that applying the stochastic collocation finite element method to the formulated problem leads to a coupling between stochastic collocation points when a deterministic optimal control is considered or when moments are included in the cost functional thereby forgoing the primary advantage of the collocation method over the stochastic galerkin method for the considered problem the application of the presented methods is demonstrated through a number of numerical examples the presented framework is sufficiently general to also consider a class of inverse problems and numerical examples of this type are also presented

every minor closed property of sparse graphs is testable abstract suppose g is a graph of bounded degree d and one needs to remove n of its edges in order to make it planar we show that in this case the statistics of local neighborhoods around vertices of g is far from the statistics of local neighborhoods around vertices of any planar graph g in fact a similar result is proved for any minor closed property of bounded degree graphs the main motivation of the above result comes from theoretical computer science using our main result we infer that for any minor closed property p there is a constant time algorithm for detecting if a graph is far from satisfying p this in particular answers an open problem of goldreich and ron stoc 1997 20 who asked if such an algorithm exists when p is the graph property of being planar the proof combines results from the theory of graph minors with results on convergent sequences of sparse graphs which rely on martingale arguments

a minimum spanning tree algorithm with inverse ackermann type complexity a deterministic algorithm for computing a minimum spanning tree of a connected graph is presented its running time is 0 m m n where is the classical functional inverse of ackermann s function and n respectively m is the number of vertices respectively edges the algorithm is comparison based it uses pointers not arrays and it makes no numeric assumptions on the edge costs

triangulation and embedding using small sets of beacons concurrent with recent theoretical interest in the problem of metric embedding a growing body of research in the networking community has studied the distance matrix defined by node to node latencies in the internet resulting in a number of recent approaches that approximately embed this distance matrix into low dimensional euclidean space there is a fundamental distinction however between the theoretical approaches to the embedding problem and this recent internet related work in addition to computational limitations internet measurement algorithms operate under the constraint that it is only feasible to measure distances for a linear or near linear number of node pairs and typically in a highly structured way indeed the most common framework for internet measurements of this type is a beacon based approach one chooses uniformly at random a constant number of nodes beacons in the network each node measures its distance to all beacons and one then has access to only these measurements for the remainder of the algorithm moreover beacon based algorithms are often designed not for embedding but for the more basic problem of triangulation in which one uses the triangle inequality to infer the distances that have not been measured here we give algorithms with provable performance guarantees for beacon based triangulation and embedding we show that in addition to multiplicative error in the distances performance guarantees for beacon based algorithms typically must include a notion of slack a certain fraction of all distances may be arbitrarily distorted for metric spaces of bounded doubling dimension which have been proposed as a reasonable abstraction of internet latencies we show that triangulation based distance reconstruction with a constant number of beacons can achieve multiplicative error 1 on a 1 e fraction of distances for arbitrarily small constants and e for this same class of metric spaces we give a beacon based embedding algorithm that achieves constant distortion on a 1 e fraction of distances this provides some theoretical justification for the success of the recent global network positioning algorithm of ng and zhang 2002 and it forms an interesting contrast with lower bounds showing that it is not possible to embed all distances in a doubling metric space with constant distortion we also give results for other classes of metric spaces as well as distributed algorithms that require only a sparse set of distances but do not place too much measurement load on any one node

approximating the minimum spanning tree weight in sublinear time we present a probabilistic algorithm that given a connected graph g represented by adjacency lists of average degree d with edge weights in the set 1 w and given a parameter 0 eps 1 2 estimates in time o dw varepsilon 2 log frac dw varepsilon the weight of the minimum spanning tree mst of g with a relative error of at most eps note that the running time does not depend on the number of vertices in g we also prove a nearly matching lower bound of omega dw varepsilon 2 on the probe and time complexity of any approximation algorithm for mst weight the essential component of our algorithm is a procedure for estimating in time o d eps 2 log frac d varepsilon the number of connected components of an unweighted graph to within an additive error of varepsilon n this becomes o eps 2 log frac 1 varepsilon for d o 1 the time bound is shown to be tight up to within the log frac d varepsilon factor our connected components algorithm picks o 1 varepsilon 2 vertices in the graph and then grows local spanning trees whose sizes are specified by a stochastic process from the local information collected in this way the algorithm is able to infer with high confidence an estimate of the number of connected components we then show how estimates on the number of components in various subgraphs of g can be used to estimate the weight of its mst

averaged control we analyze the problem of controlling parameter dependent systems we introduce the notion of averaged control according to which the quantity of interest is the average of the states with respect to the parameter first we consider the problem of controllability for linear finite dimensional systems and show that a necessary and sufficient condition for averaged controllability is an averaged rank condition in the spirit of the classical rank condition for linear control systems but involving averaged momenta of any order of the matrices generating the dynamics and representing the control action we also describe some open problems and directions of possible research in particular on the average controllability of evolution partial differential equations in this context we analyze also the averaged version of a classical optimal control problem for a parameter dependent elliptic equation and derive the corresponding optimality system

on lifting q difference operators in the askey scheme of basic hypergeometric polynomials we construct a q difference operator that lifts the continuous q hermite polynomials hn x q of rogers up to the continuous big q hermite polynomials hn x a q on the next level in the askey scheme of basic hypergeometric polynomials this operator is defined as exton s q exponential function eq aqdq in terms of the askey wilson divided q difference operator dq and it represents a particular q extension of the standard shift operator we next show that one can move two steps more upwards in order first to reach the al salam chihara family of polynomials qn x a b q and then the continuous dual q hahn polynomials pn x a b c q in both these cases lifting operators respectively turn out to be convolution type products of two and three one parameter q difference operators of the same type eq aqdq at the initial step at each step we also determine q difference operators that lift the weight function for the continuous q hermite polynomials hn x q successively up to the weight functions for hn x a q qn x a b q and pn x a b c q

efficient testing of bipartite graphs for forbidden induced subgraphs alon et al n alon e fischer m krivelevich and m szegedy combinatorica 20 2000 pp 451 476 showed that every property that is characterized by a finite collection of forbidden induced subgraphs is epsilon testable however the complexity of the test is double tower with respect to 1 epsilon as the only tool known to construct such tests uses a variant of szemeredi s regularity lemma here we show that any property of bipartite graphs that is characterized by a finite collection of forbidden induced subgraphs is epsilon testable with a number of queries that is polynomial in 1 epsilon our main tool is a new conditional version of the regularity lemma for binary matrices which may be interesting on its own

local monotonicity reconstruction we investigate the problem of monotonicity reconstruction as defined by ailon et al 2004 in a localized setting we have oracle access to a nonnegative real valued function f defined on the domain n d 1 dots n d where d is viewed as a constant we would like to closely approximate f by a monotone function g this should be done by a procedure a filter that given as input a point x in n d outputs the value of g x and runs in time that is polylogarithmic in n the procedure can indeed must be randomized but we require that all of the randomness be specified in advance by a single short random seed we construct such an implementation where the time and space per query is log n o 1 and the size of the seed is polynomial in log n and d furthermore with high probability the ratio of the hamming distance between g and f to the minimum possible hamming distance between a monotone function and f is bounded above by a function of d independent of n this allows for a local implementation one can initialize many copies of the filter with the same short random seed and they can autonomously handle queries while producing outputs that are consistent with the same approximating function g

testing eulerianity and connectivity in directed sparse graphs property testing problems are relaxations of decision problems a property testing algorithm referred to as a testing algorithm or tester has to decide if a given object has a prespecified property or is far from the property for a given distance parameter and for a prespecified distance measure the tester is given query access to the input and is required to run in sublinear time in this paper we focus on testing properties of directed graphs digraphs in particular we present the following results where n is the number of vertices in the graph d is the maximum degree and davg is the average degree we present a testing algorithm for the property of eulerianity in bounded degree digraphs which runs in time11the notation o g k stands for o g k polylog g k o 1 for unbounded degree digraphs we show a lower bound of n and give a testing algorithm that runs in time o n 3 2 we study the property of k vertex connectivity and give testing algorithms for both bounded degree and unbounded degree digraphs that run in time o ck d kd and o ck davg k 1 respectively where c 1 is a constant in addition we give a simpler analysis of the testing algorithm for k vertex connectivity in bounded degree undirected graphs that was shown by yoshida and ito y yoshida h ito property testing on k vertex connectivity of graphs in icalp 08 proceedings of the 35th international colloquium on automata languages and programming part i springer verlag berlin heidelberg 2008 pp 539 550 and extend the result to unbounded degree undirected graphs we consider the property of k edge connectivity in digraphs and simplify the analysis of the algorithm of yoshida and ito y yoshida h ito testing k edge connectivity of digraphs journal of system science and complexity 23 1 2010 91 101 for this property in addition we give a simpler analysis for the correctness of the testing algorithm for k edge connectivity in undirected graphs that was introduced by goldreich and ron o goldreich d ron property testing in bounded degree graphs algorithmica 2002 302 343

testing whether a digraph contains h free k induced subgraphs a subgraph induced by k vertices is called a k induced subgraph we prove that determining if a digraph g contains h free k induced subgraphs is w n 2 evasive then we construct an e tester to test this property an e tester for a property p is guaranteed to distinguish with probability at least 2 3 between the case of g satisfying p and the case of g being e far from satisfying p the query complexity of the e tester is independent of the size of the input digraph an e d tester for a property p is an e tester for p that is furthermore guaranteed to accept with probability at least 2 3 any input that is d close to satisfying p this paper presents an e d tester for whether a digraph contains h free k induced subgraphs

distribution free property testing we consider the problem of distribution free property testing of functions in this setting of property testing the distance between functions is measured with respect to a fixed but unknown distribution d on the domain the testing algorithms are given oracle access to random sampling from the domain according to this distribution d this notion of distribution free testing was previously defined but no distribution free property testing algorithm was known for any non trivial property we present the first such distribution free property testing algorithms for two of the central problems in this field the testers are obtained by extending some known results from standard uniform distribution property testing 1 a distribution free testing algorithm for low degree multivariate polynomials with query complexity o d 2 d cdot epsilon 1 where d is the total degree of the polynomial the same approach that is taken for the distribution free testing of low degree polynomials is shown to apply also to several other problems 2 a distribution free monotonicity testing algorithm for functions f n d rightarrow a for low dimensions e g when d is a constant with query complexity similar to the one achieved in the uniform setting on the negative side we prove an exponential gap between the query complexity required for uniform and distribution free monotonicity testing in the high dimensional case

on q analogues of the fourier and hankel transforms for h exton s q analogue of the bessel function going back to w hahn in a special case but different from f h jackson s q bessel functions we derive hansen lommel type orthogonality relations which by a symmetry turn out to be equivalent to orthogonality relations which are q analogues of the hankel integral transform pair these results are implicit in the context of quantum groups in a paper by vaksman and korogodskii as a specialization we get q cosines and q sines which admit q analogues of the fourier cosine and fourier sine transforms

learnability and the vapnik chervonenkis dimension valiant s learnability model is extended to learning classes of concepts defined by regions in euclidean space e n the methods in this paper lead to a unified treatment of some of valiant s results along with previous results on distribution free convergence of certain pattern recognition algorithms it is shown that the essential condition for distribution free learnability is finiteness of the vapnik chervonenkis dimension a simple combinatorial parameter of the class of concepts to be learned using this parameter the complexity and closure properties of learnable classes are analyzed and the necessary and sufficient conditions are provided for feasible learnability

the complexity of colouring problems on dense graphs abstract we consider the problem of deciding whether or not a graph has a vertex k colouring restricted to the class of graphs in which each vertex has degree at least n where is a fixed positive constant and n is the number of vertices of the graph we show that there is a deterministic polynomial time algorithm for this problem when k 3 k 2 and that this is best possible we also consider the corresponding enumeration problems

a new proof of the graph removal lemma let h be a xed graph with h vertices the graph removal lemma states that every graph on n vertices with o n h copies of h can be made h free by removing o n 2 edges we give a new proof which avoids szemer edi s regularity lemma and gives a better bound this approach also works to give improved bounds for the directed and multicolored analogues of the graph removal lemma this answers questions of alon and gowers

fourier gauss transforms of the al salam chihara polynomials we discuss classical fourier gauss transforms of a three parameter family of the continuous al salam chihara polynomials it is shown that they are related to both the continuous big q hermite and the q hermite polynomials

a characterization of low weight words that span generalized reed muller codes we consider the generalized reed muller code r sub fq spl rho m of order spl rho and length q sup m m 1 over the field f sub q where q p sup t for prime p and t spl ges 1 in particular we are interested in the case that t 1 so that q is not prime and the order spl rho is at least q as shown by ding and key under these conditions unless spl rho is very large i e spl rho m 1 q 1 p sup t 1 2 the code is not spanned by its minimum weight words furthermore there was no known characterization of words with small weight that span the code in this correspondence we characterize a set of words that span the code and show that their weight is upper bounded by q sup spl lceil m q 1 spl rho q q p spl rceil which is at most quadratic in the weight of the minimum weight words

algebraic testing and weight distributions of codes we study the testing problem that is the problem of determining maybe probabilistically if a function to which one has oracle access satisfies a given property we propose a framework in which to formulate and carry out the analysis of several known tests this framework establishes a connection between testing and the theory of weight distributions of codes we illustrate this connection by giving a coding theoretic interpretation of several tests that fall under the label of low degree tests we also show how the connection naturally suggests a new way of testing for linearity over finite fields we derive from the macwilliams theorems a general result the duality testing lemma and use it to analyze the simpler tests that fall into our framework in contrast to other analyses of tests the ones we present elicit the fact that a test s probability of rejecting a fuction depends on how far away the function is from every function that satisfies the property of interest

small bias probability spaces efficient constructions and applications it is shown how to efficiently construct a small probability space on n binary random variables such that for every subset its parity is either zero or one with almost equal probability they are called epsilon biased random variables the number of random bits needed to generate the random variables is o log n log frac 1 epsilon thus if epsilon is polynomially small then the size of the sample space is also polynomial random variables that are epsilon biased can be used to construct almost k wise independent random variables where epsilon is a function of k these probability spaces have various applications l derandomization of algorithms many randomized algorithms that require only k wise independence of their random bits where k is bounded by o log n can be derandomized by using epsilon biased random variables 2 reducing the number of random bits required by certain randomized algorithms e g verification of matrix multiplication 3 exhaustive tes

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

sublinear time algorithms for monomer dimer systems on bounded degree graphs for a graph g let z g be the partition function of the monomer dimer system defined by k m k g k where m k g is the number of matchings of size k in g we consider graphs of bounded degree and develop a sublinear time algorithm for estimating logz g at an arbitrary value 0 within additive error en with high probability the query complexity of our algorithm does not depend on the size of g and is polynomial in 1 e and we also provide a lower bound quadratic in 1 e for this problem this is the first analysis of a sublinear time approximation algorithm for a p complete problem our approach is based on the correlation decay of the gibbs distribution associated with z g we show that our algorithm approximates the probability for a vertex to be covered by a matching sampled according to this gibbs distribution in a near optimal sublinear time we extend our results to approximate the average size and the entropy of such a matching within an additive error with high probability where again the query complexity is polynomial in 1 e and the lower bound is quadratic in 1 e our algorithms are simple to implement and of practical use when dealing with massive datasets our results extend to other systems where the correlation decay is known to hold as for the independent set problem up to the critical activity

on the strength of comparisons in property testing an e test for a property p of functions from d 1 d to the positive integers is a randomized algorithm which makes queries on the value of an input function at specified locations and distinguishes with high probability between the case of the function satisfying p and the case that it has to be modified in more than ed places to make it satisfy p we prove that an e test for a property of integer sequences such as the property of the sequence being a monotone non decreasing sequence that depends in a strict sense only on the order relations between the sequence members cannot perform less queries in the worst case than the best e test which uses only comparisons between the queried values in addition we show that an adaptive algorithm for testing that a sequence is monotone non decreasing performs no better than the best non adaptive one with respect to query complexity from this follows a tight lower bound on tests for this property

symmetry techniques for the al salam chihara polynomials an algebraic interpretation for the al salam chihara q orthogonal polynomials is presented using an explicit realization in terms of divided difference operators their symmetry q algebra is determined and studied this algebraic structure allows one to derive a new expansion formula for these polynomials

constant round interactive proofs for delegating computation the celebrated sf ip sf pspace theorem lund fortnow karloff and nisan j acm 39 1992 pp 859 868 shamir j acm 39 1992 pp 869 877 allows an all powerful but untrusted pro

an anisotropic sparse grid stochastic collocation method for partial differential equations with random input data this work proposes and analyzes an anisotropic sparse grid stochastic collocation method for solving partial differential equations with random coefficients and forcing terms input data of the model the method consists of a galerkin approximation in the space variables and a collocation in probability space on sparse tensor product grids utilizing either clenshaw curtis or gaussian knots even in the presence of nonlinearities the collocation approach leads to the solution of uncoupled deterministic problems just as in the monte carlo method this work includes a priori and a posteriori procedures to adapt the anisotropy of the sparse grids to each given problem these procedures seem to be very effective for the problems under study the proposed method combines the advantages of isotropic sparse collocation with those of anisotropic full tensor product collocation the first approach is effective for problems depending on random variables which weigh approximately equally in the solution while the benefits of the latter approach become apparent when solving highly anisotropic problems depending on a relatively small number of random variables as in the case where input random variables are karhunen loeve truncations of smooth random fields this work also provides a rigorous convergence analysis of the fully discrete problem and demonstrates sub exponential convergence in the asymptotic regime and algebraic convergence in the preasymptotic regime with respect to the total number of collocation points it also shows that the anisotropic approximation breaks the curse of dimensionality for a wide set of problems numerical examples illustrate the theoretical results and are used to compare this approach with several others including the standard monte carlo in particular for moderately large dimensional problems the sparse grid approach with a properly chosen anisotropy seems to be very efficient and superior to all examined methods

approximating the minimum vertex cover in sublinear time and a connection to distributed algorithms for a given graph g over n vertices let opt g denote the size of an optimal solution in g of a particular minimization problem e g the size of a minimum vertex cover a randomized algorithm will be called an a approximation algorithm with an additive error for this minimization problem if for any given additive error parameter e 0 it computes a value opt such that with probability at least 2 3 it holds that opt g opt a opt g en assume that the maximum degree or average degree of g is bounded in this case we show a reduction from local distributed approximation algorithms for the vertex cover problem to sublinear approximation algorithms for this problem this reduction can be modified easily and applied to other optimization problems that have local distributed approximation algorithms such as the dominating set problem we also show that for the minimum vertex cover problem the query complexity of such approximation algorithms must grow at least linearly with the average degree d of the graph this lower bound holds for every multiplicative factor a and small constant e as long as d o n a in particular this means that for dense graphs it is not possible to design an algorithm whose complexity is o n

a sparse grid stochastic collocation method for partial differential equations with random input data this work proposes and analyzes a smolyak type sparse grid stochastic collocation method for the approximation of statistical quantities related to the solution of partial differential equations with random coefficients and forcing terms input data of the model to compute solution statistics the sparse grid stochastic collocation method uses approximate solutions produced here by finite elements corresponding to a deterministic set of points in the random input space this naturally requires solving uncoupled deterministic problems as in the monte carlo method if the number of random variables needed to describe the input data is moderately large full tensor product spaces are computationally expensive to use due to the curse of dimensionality in this case the sparse grid approach is still expected to be competitive with the classical monte carlo method therefore it is of major practical relevance to understand in which situations the sparse grid stochastic collocation method is more efficient than monte carlo this work provides error estimates for the fully discrete solution using l q norms and analyzes the computational efficiency of the proposed method in particular it demonstrates algebraic convergence with respect to the total number of collocation points and quantifies the effect of the dimension of the problem number of input random variables in the final estimates the derived estimates are then used to compare the method with monte carlo indicating for which problems the former is more efficient than the latter computational evidence complements the present theory and shows the effectiveness of the sparse grid stochastic collocation method compared to full tensor and monte carlo approaches

testing closeness of discrete distributions given samples from two distributions over an n element set we wish to test whether these distributions are statistically close we present an algorithm which uses sublinear in n specifically o n2 3e 8 3 log n independent samples from each distribution runs in time linear in the sample size makes no assumptions about the structure of the distributions and distinguishes the cases when the distance between the distributions is small less than e4 3n 1 3 32 en 1 2 4 or large more than e in e1 distance this result can be compared to the lower bound of n2 3e 2 3 for this problem given by valiant 2008 our algorithm has applications to the problem of testing whether a given markov process is rapidly mixing we present sublinear algorithms for several variants of this problem as well

the h laplace and q laplace transforms starting with a general definition of the laplace transform on arbitrary time scales we specify the particular concepts of the h laplace and q laplace transforms the convolution and inversion problems for these transforms are considered in some detail

a stochastic collocation method for elliptic partial differential equations with random input data this work proposes and analyzes a stochastic collocation method for solving elliptic partial differential equations with random coefficients and forcing terms these input data are assumed to depend on a finite number of random variables the method consists of a galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials gauss points in the probability space and naturally leads to the solution of uncoupled deterministic problems as in the monte carlo approach it treats easily a wide range of situations such as input data that depend nonlinearly on the random variables diffusivity coefficients with unbounded second moments and random variables that are correlated or even unbounded we provide a rigorous convergence analysis and demonstrate exponential convergence of the probability error with respect to the number of gauss points in each direction of the probability space under some regularity assumptions on the random input data numerical examples show the effectiveness of the method finally we include a section with developments posterior to the original publication of this work there we review sparse grid stochastic collocation methods which are effective collocation strategies for problems that depend on a moderately large number of random variables

min sum clustering with penalties abstract given a complete graph g v e a weight function w e n 0 on its edges and p n 0 a penalty function on its vertices the penalized k min sum problem is the problem of finding a partition of v to k 1 sets s 1 s k 1 minimizing i 1 k w s i p s k 1 where for s v w s e i j s w e and p s i s p i our main result is a randomized approximation scheme for the metric version of the penalized 1 min sum problem when the ratio between the minimal and maximal penalty is bounded for the metric penalized k min sum problem where k is a constant we offer a 2 approximation

a combinatorial characterization of the testable graph properties it s all about regularity a common thread in all of the recent results concerning the testing of dense graphs is the use of szemeredi s regularity lemma in this paper we show that in some sense this is not a coincidence our first result is that the property defined by having any given szemeredi partition is testable with a constant number of queries our second and main result is a purely combinatorial characterization of the graph properties that are testable with a constant number of queries this characterization roughly says that a graph property cal p can be tested with a constant number of queries if and only if testing cal p can be reduced to testing the property of satisfying one of finitely many szemeredi partitions this means that in some sense testing for szemeredi partitions is as hard as testing any testable graph property we thus resolve one of the main open problems in the area of property testing which was first raised by goldreich goldwasser and ron j acm 45 1998 pp 653 750 in the paper that initiated the study of graph property testing this characterization also gives an intuitive explanation as to what makes a graph property testable

information theory in property testing and monotonicity testing in higher dimension in property testing we are given oracle access to a function f and we wish to test if the function satisfies a given property p or it is e far from having that property in a more general setting the domain on which the function is defined is equipped with a probability distribution which assigns different weight to different elements in the domain this paper relates the complexity of testing the monotonicity of a function over the d dimensional cube to the shannon entropy of the underlying distribution we provide an improved upper bound on the query complexity of the property tester

approximate graph coloring by semidefinite programming we consider the problem of coloring k colorable graphs with the fewest possible colors we present a randomized polynomial time algorithm that colors a 3 colorable graph on n vertices with min o 1 3 log 1 2 log n o n 1 4 log 1 2 n colors where is the maximum degree of any vertex besides giving the best known approximation ratio in terms of n this marks the first nontrivial approximation result as a function of the maximum degree this result can be generalized to k colorable graphs to obtain a coloring using min o 1 2 k log 1 2 log n o n 1 3 k 1 log 1 2 n colors our results are inspired by the recent work of goemans and williamson who used an algorithm for semidefinite optimization problems which generalize linear programs to obtain improved approximations for the max cut and max 2 sat problems an intriguing outcome of our work is a duality relationship established between the value of the optimum solution to our semidefinite program and the lovasz t function we show lower bounds on the gap between the optimum solution of our semidefinite program and the actual chromatic number by duality this also demonstrates interesting new facts about the t function

a sublinear time approximation scheme for bin packing the bin packing problem is defined as follows given a set of n items with sizes 00 we present an algorithm a e that has sampling access to the input instance and outputs a value k such that c o p t k 1 e c o p t 1 where c o p t is the cost of an optimal solution it is clear that uniform sampling by itself will not allow a sublinear time algorithm in this setting a small number of items might constitute most of the total weight and uniform samples will not hit them in this work we use weighted samples where item i is sampled with probability proportional to its weight that is with probability w i iw i in the presence of weighted samples the approximation algorithm runs in o n poly 1 e g 1 e time where g x is an exponential function of x when both weighted sampling and uniform sampling are allowed o n 1 3 poly 1 e g 1 e time suffices in addition to an approximate value to c o p t our algorithm can also output a constant size template of a packing that can later be used to find a near optimal packing in linear time

elements of q harmonic analysis abstract measures that act as q analogs of lebesgue measure are introduced some q analogs of fourier and laplace transforms are studied last the connection between certain spaces of q exponential type and discrete q fourier transforms is discussed

testing metric properties finite metric spaces and in particular tree metrics play an important role in various disciplines such as evolutionary biology and statistics a natural family of problems concerning metrics is deciding given a matrix m whether or not it is a distance metric of a certain predetermined type here we consider the following relaxed version of such decision problems for any given matrix m and parameter e we are interested in determining by probing m whether m has a particular metric property p or whether it is e far from having the property in e far we mean that at least an e fraction of the entries of m must be modified so that it obtains the property the algorithm may query the matrix on entries m i j of its choice and is allowed a constant probability of error we describe algorithms for testing euclidean metrics tree metrics and ultrametrics furthermore we present an algorithm that tests whether a matrix m is an approximate ultrametric in all cases the query complexity and running time are polynomial in 1 e and independent of the size of the matrix finally our algorithms can be used to solve relaxed versions of the corresponding search problems in time that is sub linear in the size of the matrix

locally testable codes and pcps of almost linear length we initiate a systematic study of locally testable codes that is error correcting codes that admit very efficient membership tests specifically these are codes accompanied with tests that make a constant number of random queries into any given word and reject non codewords with probability proportional to their distance from the code locally testable codes are believed to be the combinatorial core of pcps however the relation is less immediate than commonly believed nevertheless we show that certain pcp systems can be modified to yield locally testable codes on the other hand we adapt techniques that we develop for the construction of the latter to yield new pcps our main results are locally testable codes and pcps of almost linear length specifically we prove the existence of the following constructs locally testable binary linear codes in which k information bits are encoded by a codeword of length k exp o log k this improves over previous results that either yield codewords of exponential length or obtained almost quadratic length codewords for sufficiently large nonbinary alphabet pcp systems of almost linear length for sat the length of the proof is n exp o log n and verification in performed by a constant number i e 19 of queries as opposed to previous results that used proof length n 1 p o 1 q for verification by q queries the novel techniques in use include a random projection of certain codewords and pcp oracles that preserves local testability an adaptation of pcp constructions to obtain linear pcp oracles for proving conjunctions of linear conditions and design of pcps with some new soundness properties a direct construction of locally testable linear codes of subexponential length

a polynomial regularity lemma for semialgebraic hypergraphs and its applications in geometry and property testing in this paper we prove several extremal results for geometrically defined hypergraphs in particular we establish an improved lower bound single exponentially decreasing in k on the best constant delta 0 such that the vertex classes p 1 ldots p k of every k partite k uniform semialgebraic hypergraph h p 1 cup cdots cup p k e with e ge varepsilon pi j 1 k p i have for 1 leq i leq k delta p i element subsets p i subseteq p i satisfying p 1 times cdots times p k subseteq e the best previously known lower bound on delta due to bukh and hubard decreased triple exponentially fast in k we give three geometric applications of our results in particular we establish the following strengthening of the so called same type lemma of barany and valtr any disjoint finite sets p 1 ldots p k subset mathbb r d k d have for 1 leq i leq k subsets p i of size at least 2 o d 3k log k p i with the property that every k tuple formed by taking on

testing probability distributions using conditional samples we study a new framework for property testing of probability distributions by considering distribution testing algorithms that have access to a conditional sampling oracle this is an oracle that takes as input a subset s subseteq n of the domain n of the unknown probability distribution cal d and returns a draw from the conditional probability distribution cal d restricted to s this new model allows considerable flexibility in the design of distribution testing algorithms in particular testing algorithms in this model can be adaptive we study a wide range of natural distribution testing problems in this new framework and some of its variants giving both upper and lower bounds on query complexity these problems include testing whether cal d is the uniform distribution cal u testing whether cal d cal d ast for an explicitly provided cal d ast testing whether two unknown distributions cal d 1 and cal d 2 are equivalent and estimating the va

optimality of adaptive galerkin methods for random parabolic partial differential equations galerkin discretizations of a class of parametric and random parabolic partial differential equations pdes are considered the parabolic pdes are assumed to depend on a vector y y 1 y 2 of possibly countably many parameters y j which are assumed to take values in 1 1 well posedness of weak formulations of these parametric equations in suitable bochner spaces is established adaptive galerkin discretizations of the equation based on a tensor product of a generalized polynomial chaos in the parameter domain 1 1 n and of suitable wavelet bases in the time interval i 0 t and the spatial domain d r d are proposed and their optimality is established

a quantitative arrow theorem arrow s impossibility theorem states that any constitution which satisfies independence of irrelevant alternatives iia and unanimity and is not a dictator has to be non transitive in this paper we study quantitative versions of arrow theorem consider n voters who vote independently at random each following the uniform distribution over the six rankings of three alternatives arrow s theorem implies that any constitution which satisfies iia and unanimity and is not a dictator has a probability of at least 6 n for a non transitive outcome when n is large 6 n is a very small probability and the question arises if for large number of voters it is possible to avoid paradoxes with probability close to 1 here we give a negative answer to this question by proving that for every epsilon 0 there exists a delta delta epsilon 0 which depends on epsilon only such that for all n and all constitutions on three alternatives if the constitution satisfies the iia condition for every pair of alternatives a b the probability that the constitution ranks a above b is at least epsilon for every voter i the probability that the social choice function agrees with a dictatorship on i at most 1 epsilon then the probability of a non transitive outcome is at least our results generalize to any number k 3 of alternatives and to other distributions over the alternatives we further derive a quantitative characterization of all social choice functions satisfying the iia condition whose outcome is transitive with probability at least 1 our results provide a quantitative statement of arrow theorem and its generalizations and strengthen results of kalai and keller who proved quantitative arrow theorems for k 3 and for balanced constitutions only i e for constitutions which satisfy for every pair of alternatives a b that the probability that the constitution ranks a above b is exactly 1 2 the main novel technical ingredient of our proof is the use of inverse hypercontractivity to show that if the outcome is transitive with high probability then there are no two different voters who are pivotal with for two different pairwise preferences with non negligible probability another important ingredient of the proof is the application of non linear invariance to lower bound the probability of a paradox for constitutions where all voters have small probability for being pivotal

fourier gauss transforms of the continuous big q hermite polynomials we examine the fourier gauss transformation properties of a two parameter family of the continuous big q hermite polynomials

a polynomial approximation scheme for scheduling on uniform processors using the dual approximation approach in this paper we present a polynomial approximation scheme for the minimum makespan problem on uniform parallel processors more specifically the problem is to find a schedule for a set of independent jobs on a collection of machines of different speeds so that the last job to finish is completed as quickly as possible we give a family of polynomial time algorithms a such that a delivers a solution that is within a relative error of e of the optimum the technique employed is the dual approximation approach where infeasible but superoptimal solutions for a related dual problem are converted to the desired feasible but possibly suboptimal solution

stochastic optimal robin boundary control problems of advection dominated elliptic equations in this work we deal with a stochastic optimal robin boundary control problem constrained by an advection diffusion reaction elliptic equation with advection dominated term we assume that the uncertainty comes from the advection field and consider a stochastic robin boundary condition as control function a stochastic saddle point system is formulated and proved to be equivalent to the first order optimality system for the optimal control problem based on which we provide the existence and uniqueness of the optimal solution as well as some results on stochastic regularity with respect to the random variables stabilized finite element approximations in physical space and collocation approximations in stochastic space are applied to discretize the optimality system a global error estimate in the product of physical space and stochastic space for the numerical approximation is derived illustrative numerical experiments are provided

testing membership in languages that have small width branching programs combinatorial property testing initiated formally by goldreich goldwasser and ron in j acm 45 1998 pp 653 750 and inspired by rubinfeld and sudan siam j comput 25 1996 pp 252 271 deals with the following relaxation of decision problems given a fixed property and an input x one wants to decide whether x has the property or is far from having the property the main result here is that if cal g g n 0 1 n rightarrow 0 1 is a family of boolean functions which have oblivious read once branching programs of width w then for every n and epsilon 0 there is a randomized algorithm that always accepts every x in 0 1 n if g n x 1 and rejects it with high probability if at least epsilon n bits of x should be modified in order for it to be in gn 1 1 the algorithm makes frac 2 w epsilon o w queries in particular for constant epsilon and w the query complexity is o 1 this generalizes the results of alon et al proceedings of the40th ieee symposium on foundations of computer science ieee computer society 1999 pp 645 655 asserting that regular languages are epsilon testable for every epsilon 0

robust optimal robin boundary control for the transient heat equation with random input data the problem of robust optimal robin boundary control for a parabolic partial dierential equation with uncertain input data is considered as a measure of robustness the variance of the random system response is included in two dierent cost functionals uncertainties in both the underlying state equation and the control variable are quantied through random elds the paper is mainly concerned with the numerical resolution of the problem to this end a gradient method is proposed a stochastic collocation nite element method is used for the numerical resolution of the associated state and adjoint state equations two numerical experiments illustrate the performance of the algorithm

ω regular languages are testable with a constant number of queries we continue the study of combinatorial property testing for a property an e test for for 0 0 we describe an algorithm that gets as input an infinite lasso shape word of the form x y for finite words x and y samples only a constant number of letters in x and y returns yes if w l a and returns no with probability 2 3 if w is e far from l a we also discuss the applicability of property testing to formal verification where regular languages are used for the specification of the behavior of nonterminating reactive systems and computations correspond to lasso shape words

a q analogue of hierarchy and q painleve vi a q analogue of the drinfel d sokolov hierarchy is proposed as a reduction of the q kp hierarchy applying a similarity reduction and a q laplace transformation to the hierarchy one can obtain the q painleve vi equation proposed by jimbo and sakai

a stochastic collocation method for elliptic partial differential equations with random input data in this paper we propose and analyze a stochastic collocation method to solve elliptic partial differential equations with random coefficients and forcing terms input data of the model the input data are assumed to depend on a finite number of random variables the method consists in a galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials gauss points in the probability space and naturally leads to the solution of uncoupled deterministic problems as in the monte carlo approach it can be seen as a generalization of the stochastic galerkin method proposed in i babuscka r tempone and g e zouraris siam j numer anal 42 2004 pp 800 825 and allows one to treat easily a wider range of situations such as input data that depend nonlinearly on the random variables diffusivity coefficients with unbounded second moments and random variables that are correlated or even unbounded we provide a rigorous convergence analysis and demonstrate exponential convergence of the probability error with respect to the number of gauss points in each direction in the probability space under some regularity assumptions on the random input data numerical examples show the effectiveness of the method

free bits pcps and nonapproximability towards tight results this paper continues the investigation of the connection between probabilistically checkable proofs pcps and the approximability of np optimization problems the emphasis is on proving tight nonapproximability results via consideration of measures such as the free bit complexity and the amortized free bit complexity of proof systems the first part of the paper presents a collection of new proof systems based on a new error correcting code called the long code we provide a proof system that has amortized free bit complexity of 2 epsilon implying that approximating maxclique within n frac13 e and approximating the chromatic number within n frac15 e are hard assuming np neq corp for any e 0 we also derive the first explicit and reasonable constant hardness factors for min vertex cover msat 2 and max cut and we improve the hardness factor for msat 3 we note that our nonapproximability factors for maxsnp problems are appreciably close to the values known to be achievable by polynomial time algorithms finally we note a general approach to the derivation of strong nonapproximability results under which the problem reduces to the construction of certain gadgets the increasing strength of nonapproximability results obtained via the pcp connection motivates us to ask how far this can go and whether pcps are inherent in any way the second part of the paper addresses this the main result is a reversal of the connection due to feige et al fglss connection j acm 43 1996 pp 268 292 where the latter had shown how to translate proof systems for np into np hardness of approximation results for maxclique we show how any np hardness of approximation result for maxclique yields a proof system for np roughly our result says that for any constant f if maxclique is np hard to approximate within n 1 1 f then np subseteq overline fpcp log f the latter being the class of languages possessing proofs of logarithmic randomness and amortized free bit complexity f this suggests that pcps are inherent to obtaining nonapproximability results furthermore the tight relation suggests that reducing the amortized free bit complexity is necessary for improving the nonapproximability results for maxclique the third part of our paper initiates a systematic investigation of the properties of pcp and fpcp free pcp as a function of the following various parameters randomness query complexity free bit complexity amortized free bit complexity proof size etc we are particularly interested in triviality results which indicate which classes are not powerful enough to capture np we also distill the role of randomized reductions in this area and provide a variety of useful transformations between proof checking complexity classes

testing hereditary properties of nonexpanding bounded degree graphs we study graph properties that are testable for bounded degree graphs in time independent of the input size our goal is to distinguish between graphs having a predetermined graph property and graphs that are far from every graph having that property it is well known that in the bounded degree graph model where two graphs are considered far if they differ in varepsilon n edges for a positive constant varepsilon many graph properties cannot be tested even with a constant or even with a polylogarithmic number of queries therefore in this paper we focus our attention on testing graph properties for special classes of graphs specifically we show that every hereditary graph property is testable with a constant number of queries provided that every sufficiently large induced subgraph of the input graph has poor expansion this result implies that for example any hereditary property e g k colorability h freeness etc is testable in the bounded degree graph model for planar graphs graphs with bounded genus interval graphs etc no such results have been known before and prior to our work very few graph properties have been known to be testable with a constant number of queries for general graph classes in the bounded degree graph model

computational limitations on learning from examples the computational complexity of learning boolean concepts from examples is investigated it is shown for various classes of concept representations that these cannot be learned feasibly in a distribution free sense unless r np these classes include a disjunctions of two monomials b boolean threshold functions and c boolean formulas in which each variable occurs at most once relationships between learning of heuristics and finding approximate solutions to np hard optimization problems are given

testing the s t disconnectivity of graphs and digraphs property testing is concerned with constant time algorithms for deciding whether a given object satisfies a predetermined property or is far from satisfying it in this paper we consider testing properties related to the connectivity of two vertices in sparse graphs we present one sided error testers for s t disconnectivity with query complexity 2 o 1 e for digraphs and o 1 e 2 for graphs where e is an error parameter furthermore we show that these algorithms are the best possible in view of query complexity i e we give matching lower bounds for two sided error testers for both cases we also give a constant time algorithm for testing the s t disconnectivity of a directed bounded degree hypergraph which can be used to test the satisfiability of horn sat