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

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

robust dimension free isoperimetry in gaussian space we prove the first robust dimension free isoperimetric result for the standard gaussian measure n and the corresponding boundary measure n in rn the main result in the theory of gaussian isoperimetry proven in the 1970s by sudakov and tsirelson and independently by borell states that if n a 1 2 then the surface area of a is bounded by the surface area of a half space with the same measure n a 2 1 2 our results imply in particular that if a rn satisfies n a 1 2 and n a 2 1 2 then there exists a half space b rn such that n a b clog 1 2 1 for an absolute constant c since the gaussian isoperimetric result was established only recently a robust version of the gaussian isoperimetric result was obtained by cianchi et al who showed that n a b c n for some function c n with no effective bounds compared to the results of cianchi et al our results have optimal i e no dependence on the dimension but worse dependence on

geometric bounds on the ornstein uhlenbeck velocity process let x c n be the ornstein uhlenbeck velocity process in equilibrium and denote by a a x the first hitting time of a subseteq mathbb r n if a b n and x o a x n o a x n o b x n o b we prove that mathbb p tau a underset raise0 3em hbox smash scriptscriptstyle leqslant t underset raise0 3em hbox smash scriptscriptstyle geqslant mathbb p tau chi n underset raise0 3em hbox smash scriptscriptstyle leqslant a underset raise0 3em hbox smash scriptscriptstyle leqslant t and mathbb e left int limits 0 t wedge tau a 1 text b x text s d text s right underset raise0 3em hbox smash scriptscriptstyle leqslant mathbb e left int limits 0 t wedge tau left x n underset raise0 3em hbox smash scriptscriptstyle leqslant a right 1 left x n underset raise0 3em hbox smash scriptscriptstyle geqslant b right x text s d text s right here x n denotes the n th component of x

differentiability of relations and differential stability of perturbed optimization problems several new concepts of differentiability of multifunctions are introduced special attention is paid to relations given by perturbed inequalities applications are given to the study of the value or marginal function of a perturbed nonlinear program

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 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

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

the identity component of the isometry group of a compact lorentz manifold this result may be compared with a theorem of e ghys ghy see also bel asserting a similar conclusion but assuming that m has dimension 3 and that the action is just volume preserving and locally free the statement there is that the action of ag may be extended to an action of a finite cover of psl 2 r or to an action of the solvable 3 dimensional lie group sol here we have another motivation we want to understand the structure of lie groups acting isometrically on compact lorentz manifold the first results in the subject are due to zim and gro a final result is due to a s and zeg1 independently necessary and sufficient conditions were given in order that a lie group acts isometrically and locally faithfully on a compact lorentz manifold note however that if a group acts in such a fashion then its subgroups also act in the same way for instance all known examples of isometric actions of ag are obtained by viewing it as a subgroup of sl 2 r so a natural question is what are the maximal connected lie groups acting isometrically on a compact lorentz manifold equivalently

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

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

a quantitative version of the gibbard satterthwaite theorem for three alternatives the gibbard satterthwaite theorem states that every nondictatorial election rule among at least three alternatives can be strategically manipulated we prove a quantitative version of the gibbard satterthwaite theorem a random manipulation by a single random voter will succeed with a nonnegligible probability for any election rule among three alternatives that is far from being a dictatorship and from having only two alternatives in its range

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

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

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

quantum reverse hypercontractivity we develop reverse versions of hypercontractive inequalities for quantum channels by generalizing classical techniques we prove a reverse hypercontractive inequality for tensor products of qubit depolarizing channels we apply this to obtain a rapid mixing result for depolarizing noise applied to large subspaces and to prove bounds on a quantum generalization of non interactive correlation distillation

noncompact simple automorphism groups of lorentz manifolds and other geometric manifolds when such a g does act zimmer has proven that the full automorphism group of the lorentz manifold must be locally isomorphic to sl 2 r x k where k is a compact lie group m gromov has shown 2 that the universal cover m must then be lorentz isometric to a warped product sl 2 r f x s where sl 2 ir is the universal cover of sl 2 ii with the cartan killing metric s is a riemannian manifold and f s 0 o oo is a warping function this paper is mostly concerned with the following situation a noncompact simple r algebraic group g acting on a manifold m preserving a lorentz metric we do not assume m compact or finite volume or even complete in contrast the theorems above require compactness and use it in an essential way in their proofs but as we show strong analogues of these theorems hold even for m noncompact our first result is the following

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

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

the probability of intransitivity in dice and close elections we study the phenomenon of intransitivity in models of dice and voting first we follow a recent thread of research for n sided dice with pairwise ordering induced by the probability relative to 1 2 that a throw from one die is higher than the other we build on a recent result of polymath showing that three dice with i i d faces drawn from the uniform distribution on 1 ldots n and conditioned on the average of faces equal to n 1 2 are intransitive with asymptotic probability 1 4 we show that if dice faces are drawn from a non uniform continuous mean zero distribution conditioned on the average of faces equal to 0 then three dice are transitive with high probability we also extend our results to stationary gaussian dice whose faces for example can be the fractional brownian increments with hurst index h in 0 1 second we pose an analogous model in the context of condorcet voting we consider n voters who rank k alternatives independently and uniformly at random the winner between each two alternatives is decided by a majority vote based on the preferences we show that in this model if all pairwise elections are close to tied then the asymptotic probability of obtaining any tournament on the k alternatives is equal to 2 k k 1 2 which markedly differs from known results in the model without conditioning we also explore the condorcet voting model where methods other than simple majority are used for pairwise elections we investigate some natural definitions of close to tied for general functions and exhibit an example where the distribution over tournaments is not uniform under those definitions

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 invariance principle for polytopes let x be randomly chosen from 1 1 n and let y be randomly chosen from the standard spherical gaussian on n for any possibly unbounded polytope p formed by the intersection of k halfspaces we prove that pr x p pr y p log8 5k where is a parameter that is small for polytopes formed by the intersection of regular halfspaces i e halfspaces with low influence the novelty of our invariance principle is the polylogarithmic dependence on k previously only bounds that were at least linear in k were known the proof of the invariance principle is based on a generalization of the lindeberg method for proving central limit theorems and could be of use elsewhere we give two important applications of our invariance principle one from learning theory and the other from pseudorandomness 1 a bound of logo 1 k e1 6 on the boolean noise sensitivity of intersections of k regular halfspaces previous work gave bounds linear in k this gives a corresponding agnostic learning algorithm for intersections of regular halfspaces 2 a pseudorandom generator prg for estimating the gaussian volume of polytopes with k faces within error and seed length o log n poly log k 1 we also obtain prgs with similar parameters that fool polytopes formed by intersection of regular halfspaces over the hypercube using our prg constructions we obtain the first deterministic quasi polynomial time algorithms for approximately counting the number of solutions to a broad class of integer programs including dense covering problems and contingency tables

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

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

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

positivity improving operators and hypercontractivity here 1 1 is the famous nelson hypercontractivity inequality 14 which has proved to be extremely useful in euclidean quantum field theory see e g 16 like 1 1 the estimate 1 2 follows very easily from the it6 calculus of brownian motion 3 here in order to make the paper self contained and to introduce the reader to some general principles of interest we submit a new proof of 1 2 by mimicing a standard proof of 1 1 we should like to add that 1 2 has at least a formal resemblance to the familiar correlation inequalities from statistical mechanics see e g 11 12 16 using 1 1 and 1 2 it is simple to obtain similar estimates for the biquantization of a linear contraction on hilbert space sect 4 now suppose a is a self adjoint operator on l 2 of a probability space and set a a maxspec a sup spec a maxspec a where spec a denotes the spectrum of a below we will state conditions that guarantee that the gap a a is proper in fact a simplified version of our main result theorem 5 3 gives a positive numerical lower bound of a a if i a is positivity preserving ii sup ilzf i1 i f 12 1 k 2 and iii inf izfllp v i l f l l 1 f 0 k 0 some 0 p 1 here i iii imply that a has a unique ground state it shall be empha

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

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 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

on upper semicontinuity of global minima in constrained optimization problems abstract this paper studies stability properties of solutions for optimization problems subject to perturbations in constraints for problems formulated in a complete metric space sufficient conditions for topological upper semicontinuity of the solution multifunction are derived without any compactness assumptions when the feasible solution set is additionally assumed to be compact these conditions reduce to that of berge topological spaces macmillan co new york 1963 and in that case they turn out to be also necessary

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

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