algebraic methods for interactive proof systems a new algebraic technique for the construction of interactive proof systems is presented our technique is used to prove that every language in the polynomial time hierarchy has an interactive proof system this technique played a pivotal role in the recent proofs that ip pspace 28 and that mip nexp 4

on the np hardness of max not 2 we prove that for any epsilon 0 given a satisfiable instance of max ntw not 2 it is np hard to find an assignment that satisfies a fraction frac 58 epsilon of the constraints this up to the existence of epsilon matches the approximation ratio obtained by the trivial algorithm that just picks an assignment at random and thus the result is tight said equivalently the result proves that max ntw is approximation resistant on satisfiable instances and this makes complete our understanding of arity three maximum constraint satisfaction problems with regards to approximation resistance

interactive proofs and the hardness of approximating cliques the contribution of this paper is two fold first a connection is established between approximating the size of the largest clique in a graph and multi prover interactive proofs second an efficient multi prover interactive proof for np languages is constructed where the verifier uses very few random bits and communication bits last the connection between cliques and efficient multi prover interaction proofs is shown to yield hardness results on the complexity of approximating the size of the largest clique in a graph of independent interest is our proof of correctness for the multilinearity test of functions

approximate constraint satisfaction requires large lp relaxations we prove super polynomial lower bounds on the size of linear programming relaxations for approximation versions of constraint satisfaction problems we show that for these problems polynomial sized linear programs are no more powerful than programs arising from a constant number of rounds of the sherali adams hierarchy in particular any polynomial sized linear program for m ax c ut has an integrality gap of and any such linear program for m ax 3 s at has an integrality gap of f

minimal ideals of jordan systems we show that minimal ideals of jordan systems algebras triple systems and pairs are either simple or trivial so answering the question posed by nam and mccrimmon in 1983

probability laws related to the jacobi theta and riemann zeta functions and brownian excursions this paper reviews known results which connect riemann s integral representations of his zeta function involving jacobi s theta function and its derivatives to some particular probability laws governing sums of independent exponential variables these laws are related to one dimensional brownian motion and to higher dimensional bessel processes we present some characterizations of these probability laws and some approximations of riemann s zeta function which are related to these laws

on generalized harmonic number sums we evaluate generalized harmonic number sums with parameter in terms of values of polylogarithm functions and several examples are given instances of such sums occur in diverse areas including analytic number theory and in calculations of high energy nuclear and atomic physics

trivial minimal ideals of jordan systems abstract we show that in jordan systems algebras triple systems and pairs monomials containing two elements of a trivial minimal ideal vanish so improving the answer given by anquela and cortes in j a anquela t cortes minimal ideals of jordan systems invent math 168 2007 83 90 to the problem of nam and mccrimmon in n s nam k mccrimmon minimal ideals in quadratic jordan algebras proc amer math soc 88 4 1983 579 583 inspired by the problem posed in 1971 by zhevlakov in the dniester notebook unsolved problems in the theory of rings and modules second ed 1976 see filippov et al 1993 5

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

local and subquotient inheritance of simplicity in jordan systems abstract in this paper we prove that the local algebras of a simple jordan pair are simple jordan pairs all of which local algebras are simple are also studied showing that they have a nonzero simple heart which is described in terms of powers of the original pair similar results are given for jordan triple systems and algebras finally we characterize the inner ideals of a simple pair which determine simple subquotients answering the question posed by o loos and e neher 1994 j algebra166 255 295

on multivariate lagrange interpolation lagrange interpolation by polynomials in several variables is studied through a finite difference approach we establish an interpolation formula analogous to that of newton and a remainder formula both of them in terms of finite differences we prove that the finite difference admits an integral representation involving simplex spline functions in particular this provides a remainder formula for lagrange interpolation of degree n of a function f which is a sum of integrals of certain n 1 st directional derivatives of f multiplied by simplex spline functions we also provide two algorithms for the computation of lagrange interpolants which use only addition scalar multiplication and point evaluation of polynomials

statistical algorithms and a lower bound for detecting planted cliques we introduce a framework for proving lower bounds on computational problems over distributions against algorithms that can be implemented using access to a statistical query oracle for such algorithms access to the input distribution is limited to obtaining an estimate of the expectation of any given function on a sample drawn randomly from the input distribution rather than directly accessing samples most natural algorithms of interest in theory and in practice for example moments based methods local search standard iterative methods for convex optimization mcmc and simulated annealing can be implemented in this framework our framework is based on and generalizes the statistical query model in learning theory kearns 1998 our main application is a nearly optimal lower bound on the complexity of any statistical query algorithm for detecting planted bipartite clique distributions or planted dense subgraph distributions when the planted clique has size o n1 2 for any constant 0 the assumed hardness of variants of these problems has been used to prove hardness of several other problems and as a guarantee for security in cryptographic applications our lower bounds provide concrete evidence of hardness thus supporting these assumptions

from electrostatics to almost optimal nodal sets for polynomial interpolation in a simplex the electrostatic interpretation of the jacobi gauss quadrature points is exploited to obtain interpolation points suitable for approximation of smooth functions defined on a simplex moreover several new estimates based on extensive numerical studies for approximation along the line using jacobi gauss lobatto quadrature points as the nodal sets are presented the electrostatic analogy is extended to the two dimensional case with the emphasis being on nodal sets inside a triangle for which two very good matrices of nodal sets are presented the matrices are evaluated by computing the lebesgue constants and they share the property that the nodes along the edges of the simplex are the gauss lobatto quadrature points of the chebyshev and legendre polynomials respectively this makes the resulting nodal sets particularly well suited for integration with conventional spectral methods and supplies a new nodal basis for h p finite element methods

on the power of membership queries in agnostic learning we study the properties of the agnostic learning framework of haussler 1992 and kearns schapire and sellie 1994 in particular we address the question is there any situation in which membership queries are useful in agnostic learning our results show that the answer is negative for distribution independent agnostic learning and positive for agnostic learning with respect to a specific marginal distribution namely we give a simple proof that any concept class learnable agnostically by a distribution independent algorithm with access to membership queries is also learnable agnostically without membership queries this resolves an open problem posed by kearns et al 1994 for agnostic learning with respect to the uniform distribution over 0 1 n we show a concept class that is learnable with membership queries but computationally hard to learn from random examples alone assuming that one way functions exist

a threshold of ln n for approximating set cover given a collection f of subsets of s 1 n setcover is the problem of selecting as few as possiblesubsets from f such that their union covers s and maxk cover is the problem of selecting k subsets from f such that their union has maximum cardinality both these problems arenp hard we prove that 1 o 1 ln n is a threshold below which setcover cannot be approximated efficiently unless np has slightlysuperpolynomial time algorithms this closes the gap up to low orderterms between the ratio of approximation achievable by the greedyalogorithm which is 1 o 1 lnn and provious results of lund and yanakakis that showed hardness ofapproximation within a ratio of log 2 n 2s0 72 ln n for max k cover we show an approximationthreshold of 1 1 e up tolow order terms under assumption that p np

the approximability of non boolean satisfiability problems and restricted integer programming in this paper we present improved approximation algorithms for two classes of maximization problems defined in barland et al j comput system sci 57 2 1998 144 our factors of approximation substantially improve the previous known results and are close to the best possible on the other hand we show that the approximation results in the framework of barland et al hold also in the parallel setting and thus we have a new common framework for both computational settings we prove almost tight non approximability results thus solving a main open question of barland et al we obtain the results through the constraint satisfaction problem over multi valued domains for which we develop approximation algorithms and show non approximability results our parallel approximation algorithms are based on linear programming and random rounding they are better than previously known sequential algorithms the non approximability results are based on new recent progress in the fields of probabilistically checkable proofs and multi prover one round proof systems

on the complexity of optimization over the standard simplex we review complexity results for minimizing polynomials over the standard simplex and unit hypercube in addition we derive new results on the computational complexity of approximating the minimum of some classes of functions including lipschitz continuous functions on the standard simplex the main tools used in the analysis are bernstein approximation and lagrange interpolation on the simplex combined with an earlier result by de klerk et al a ptas for the minimization of polynomials of fixed degree over the simplex theoretical computer science 361 2 3 2006 210 225

non approximability of weighted multiple sequence alignment we consider a weighted generalization of multiple sequence alignment msa with sum of pair score msa without weights is known to be np complete and can be approximated within a constant factor but it is unknown whether it has a polynomial time approximation scheme weighted multiple sequence alignment wmsa can be approximated within a factor of o log2n where n is the number of sequences we prove that wmsa alignment is max jnp hard and establish a numerical lower bound on its approximability namely 324 323 e this lower bound is obtained already for the simple binary weighted case where the weights are restricted to 0 and 1 furthermore we show that wmsa and its restriction to binary weights can be approximated to the same degree

the sum of like powers of the zeros of the riemann zeta function on analyse une methode de calcul de la somme r p r ou r est un entier superieur a 1 la somme etant prise sur tous les zeros complexes de la fonction zeta de riemann

some optimal inapproximability results we prove optimal up to an arbitrary e 0 inapproximability results for max e k sat for k 3 maximizing the number of satisfied linear equations in an over determined system of linear equations modulo a prime p and set splitting as a consequence of these results we get improved lower bounds for the efficient approximability of many optimization problems studied previously in particular for max e2 sat max cut max di cut and vertex cover

probabilistic checking of proofs a new characterization of np we give a new characterization of np the class np contains exactly those languages l for which membership proofs a proof that an input x is in l can be verified probabilistically in polynomial time using logarithmic number of random bits and by reading sublogarithmic number of bits from the proof we discuss implications of this characterization specifically we show that approximating clique and independent set even in a very weak sense is np hard

gadgets approximation and linear programming we present a linear programming based method for finding gadgets i e combinatorial structures reducing constraints of one optimization problem to constraints of another a key step in this method is a simple observation which limits the search space to a finite one using this new method we present a number of new computer constructed gadgets for several different reductions this method also answers a question posed by bellare goldreich and sudan siam j comput 27 1998 pp 804 915 of how to prove the optimality of gadgets linear programming duality gives such proofs the new gadgets when combined with recent results of ha stad proceedings of the 29th acm symposium on theory of computing 1997 pp 1 10 improve the known inapproximability results for max cut and max dicut showing that approximating these problems to within factors of 16 17 epsilon and 12 13 epsilon respectively is np hard for every epsilon 0 prior to this work the best known inapproximability thresholds for both problems were 71 72 m bellare o goldreich and m sudan siam j comput 27 1998 pp 804 915 without using the gadgets from this paper the best possible hardness that would follow from bellare goldreich and sudan and ha s tad is 18 19 we also use the gadgets to obtain an improved approximation algorithm for max3 sat which guarantees an approximation ratio of 801 this improves upon the previous best bound implicit from m x goemans and d p williamson j acm 42 1995 pp 1115 1145 u feige and m x goemans proceedings of the third israel symposium on theory of computing and systems 1995 pp 182 189 of 7704

on approximating string selection problems with outliers many problems in bioinformatics are about finding strings that approximately represent a collection of given strings we look at more general problems where some input strings can be classified as outliers the close to most strings problem is given a set s of the same length strings and a parameter d find a string x that maximizes the number of non outliers within hamming distance d of x we prove that this problem has no polynomial time approximation scheme ptas unless np has randomized polynomial time algorithms correcting a decade old erroneous proof made previously in the literature the most strings with few bad columns problem is to find a maximum size subset of input strings so that the number of non identical positions is at most k we show it has no ptas unless p np we also observe closest to k strings has no efficient ptas eptas unless a parameterized complexity hierarchy collapses in sum outliers help model problems associated with using biological data but we show the problem of finding an approximate solution is computationally difficult

minimal achievable approximation ratio for max mq in finite fields given a multivariate quadratic polynomial system in a finite field f q the problem max mq is to find a solution satisfying the maximal number of equations we prove that the probability of a random assignment satisfying a non degenerate quadratic equation is at least 1q o q n 2 where n is the number of the variables in the equation consequently the random assignment provides a polynomial time approximation algorithm with approximation ratio q o q n 2 for non degenerate max mq for large n the ratio is close to q according to a result by hastad it is np hard to approximate max mq with an approximation ratio of email protected for a small positive number e therefore the minimal approximation ratio that can be achieved in polynomial time for max mq is q

aggregating inconsistent information ranking and clustering we address optimization problems in which we are given contradictory pieces of input information and the goal is to find a globally consistent solution that minimizes the extent of disagreement with the respective inputs specifically the problems we address are rank aggregation the feedback arc set problem on tournaments and correlation and consensus clustering we show that for all these problems and various weighted versions of them we can obtain improved approximation factors using essentially the same remarkably simple algorithm additionally we almost settle a long standing conjecture of bang jensen and thomassen and show that unless np bpp there is no polynomial time algorithm for the problem of minimum feedback arc set in tournaments

complexity of approximating bounded variants of optimization problems we study low degree graph problems such as maximum independent set and minimum vertex cover the goal is to improve approximation lower bounds for them and for a number of related problems like max b set packing min b set cover and max b dimensional matching b 3 we prove for example that it is np hard to achieve an approximation factor of 95 94 for max 3 dm and a factor of 48 47 for max 4 dm in both cases the hardness result applies even to instances with exactly two occurrences of each element

ip pspace in this paper it is proven that when both randomization and interaction are allowed the proofs that can be verified in polynomial time are exactly those proofs that can be generated with polynomial space

new results on the stieltjes constants asymptotic and exact evaluation abstract the stieltjes constants k a are the expansion coefficients in the laurent series for the hurwitz zeta function about s 1 we present new asymptotic summatory and other exact expressions for these and related constants

quadratic jordan systems of hermitian type abstract in this paper we extend zel manov s classification of linear jordan systems triple systems and pairs of hermitian type to quadratic jordan systems over an arbitrary ring of scalars using associative triple systems of the first kind for what we believe to be a more natural tool to describe special jordan triples we show that an i special prime quadratic jordan triple system the ideals of which remain semiprime with a nonzero hermitian part lies between an ample subspace of hermitian elements in a prime associative triple system and those in its martindale system of symmetric quotients modulo some extra definitions the structure of strongly prime jordan pairs of hermitian type follows almost immediately from the jordan triple result

a fourier analytic approach to reed muller decoding we present a fourier analytic approach to list decoding reed muller codes over arbitrary finite fields we use this to show that quadratic forms over any field are locally list decodable up to their minimum distance the analogous statement for linear polynomials was proved in the celebrated works of goldreich previously tight bounds for quadratic polynomials were known only for q 2 and 3 the best bound known for other fields was the johnson radius departing from previous work on reed muller decoding which relies on some form of self corrector our work applies ideas from fourier analysis of boolean functions to low degree polynomials over finite fields in conjunction with results about the weight distribution we believe that the techniques used here could find other applications we present some applications to testing and learning

error bounds for some semidefinite programming approaches to polynomial minimization on the hypercube we consider the problem of minimizing a polynomial on the hypercube 0 1 n and derive new error bounds for the hierarchy of semidefinite programming approximations to this problem corresponding to the positivstellensatz of schmudgen math ann 289 1991 pp 203 206 the main tool we employ is bernstein approximations of polynomials which also gives constructive proofs and degree bounds for positivity certificates on the hypercube

quantum computing postselection and probabilistic polynomial time i study the class of problems efficiently solvable by a quantum computer given the ability to postselect on the outcomes of measurements i prove that this class coincides with a classical complexity class called pp or probabilistic polynomial time using this result i show that several simple changes to the axioms of quantum mechanics would let us solve pp complete problems efficiently the result also implies as an easy corollary a celebrated theorem of beigel reingold and spielman that pp is closed under intersection as well as a generalization of that theorem due to fortnow and reingold this illustrates that quantum computing can yield new and simpler proofs of major results about classical computation

a fast algorithm for maximizing a non monotone dr submodular integer lattice function abstract in this paper we consider the problem of maximizing a non monotone and non negative dr submodular function on a bounded integer lattice b x 1 x n z n 0 x k b k 1 k n without any constraint where b b 1 b n z n we design an algorithm for the problem and measure its performance by its approximation ratio and the number of value oracle queries it needs where the latter one is the dominating term in the running time of an algorithm it has been showed that for the problem considered any algorithm achieving an approximation ratio greater than 1 2 requires an exponential number of value oracle queries in the literature there are two algorithms that reach 1 2 approximation guarantee the first algorithm needs o n b oracle queries the second one reduces its number of oracle queries to o n max 1 log b but it needs large storage in this paper we present a randomized approximation algorithm that has an approximation guarantee of 1 2 calls o n max 1 log b oracle queries and does not need large storage improving the results of the literature

vertex cover in conflict graphs abstract we study a graph class called conflict graph which is a union of a finite number of given forests of complete multipartite graph it is interesting that conflict graph can model many natural problems such as in database application and others we show that this property is non trivial if limiting the number of forests of complete multipartite graph then study the problem of vertex cover on conflict graph in this paper the results list as follows if the number of forests of complete multipartite graph is fixed conflict graph is non trivial property but finding 1 36 approximation algorithms is np hard also given 2 forests of complete multipartite graph and maximum degree less than 7 vertex cover problem of conflict graph is np complete moreover it is shown to be np hard to find an algorithm for vertex cover of conflict graph within 41 40 e for any e 0 if there is no the degree restriction over the graph at last we design an approximation algorithm for the conflict graph consisting of r forests of complete multipartite graph and show that the approximation ratio can be bounded by 2 1 2 r and it s near optimal

new summation relations for the stieltjes constants the stieltjes constants have been of interest for over a century yet their detailed behaviour remains under investigation these constants appear in the laurent expansion of the hurwitz zeta funct

combinatorial network abstraction by trees and distances we draw attention to combinatorial network abstraction problems these are specified by a class p of pattern graphs and a real valued similarity measure r that is based on certain graph properties for a fixed pattern p and similarity measure r the optimization task on a given graph g is to find a subgraph g g which belongs to p and minimizes r g g in this work we consider this problem for the natural and somewhat general case of trees and distance based similarity measures in particular we systematically study spanning trees of graphs that minimize distances approximate distances and approximate closeness centrality with respect to standard vector and matrix norms complexity analysis within a unifying framework shows that all considered variants of the problem are np complete except for the case of distance minimization with respect to the norm l if a subset of edges can be forced into the spanning tree no polynomial time constant factor approximation algorithmexists for the distance approximation problems unless p np

linearity testing in characteristic two let dist f g pr sub u f u spl ne g u denote the relative distance between functions f g mapping from a group g to a group h and let dist f denote the minimum over all linear functions homomorphisms g of dist f g given a function f g spl rarr h we let err f pr sub u spl upsi f u f spl upsi spl ne f u spl upsi denote the rejection probability of the blum luby rubinfeld 1993 linearity test linearity testing is the study of the relationship between err f and dist f and in particular lower bounds on err f in terms of dist f we discuss when the underlying groups are g gf 2 sup n and h gf 2 in this case the collection of linear functions describe a hadamard code of block length 2 sup n and for an arbitrary function f mapping gf 2 sup n to gf 2 the distance dist l measures its distance to a hadamard code err f is a parameter that is easy to measure and linearity testing studies the relationship of this parameter to the distance of f the code and corresponding test are used in the construction of efficient probabilistically checkable proofs and thence in the derivation of hardness of approximation improved analyses translate into better nonapproximability results we present a description of the relationship between err f and dist f which is nearly complete in all its aspects and entirely complete in some we present functions l u 0 1 spl rarr 0 1 such that for all x spl isin 0 1 we have l x spl les err f spl les u x whenever dist f x with the upper bound being tight on the whole range and the lower bound tight on a large part of the range and close on the rest part of our strengthening is obtained by showing a new connection between linearity testing and fourier analysis

approximating the radii of point sets we consider the problem of computing the outer radii of point sets in this problem we are given integers n d and k where k le d and a set p of n points in re d the goal is to compute the outer k radius of p denoted by cal r k p which is the minimum over all d k dimensional flats f of max p in p d p f where d p f is the euclidean distance between the point p and flat f computing the radii of point sets is a fundamental problem in computational convexity with many significant applications the problem admits a polynomial time algorithm when the dimension d is constant u faigle w kern and m streng math program 73 1996 pp 1 5 here we are interested in the general case in which the dimension d is not fixed and can be as large as n where the problem becomes np hard even for k 1 it is known that r k p can be approximated in polynomial time by a factor of 1 varepsilon for any varepsilon 0 when d k is a fixed constant m babdoiu s har peled and p indyk in proceedings of the acm symposium on the theory of computing 2002 s har peled and k varadarajan in proceedings of the acm symposium on computing geometry 2002 a polynomial time algorithm that guarantees a factor of o sqrt log n approximation for r 1 p the width of the point set p is implied by the results of nemirovski roos and terlaky math program 86 1999 pp 463 473 and nesterov handbook of semidefinite programming theory algorithms kluwer academic publishers norwell ma 2000 in this paper we show that r k p can be approximated by a ratio of o sqrt log n for any 1 leq k leq d thus matching the previously best known ratio for approximating the special case r 1 p the width of point set p our algorithm is based on semidefinite programming relaxation with a new mixed deterministic and randomized rounding procedure we also prove an inapproximability result that gives evidence that our approximation algorithm is doing well for a large range of k we show that there exists a constant delta 0 such that the following holds for any 0 eps 1 there is no polynomial time algorithm that approximates r k p within log n delta for all k such that k leq d d varepsilon unless np subseteq dtime 2 log m o 1 our inapproximability result for r k p extends a previously known hardness result of brieden discrete comput geom 28 2002 pp 201 209 and is proved by modifying brieden s construction using basic ideas from probabilistically checkable proofs pcp theory

on certain sums over the non trivial zeta zeroes let z denote the riemann zeta function and x s s 2 s 1 p s 2 g s 2 z s the classical completed zeta function where g is the gamma function riemann 1859 1860 edwards 1974 ivic 1985 titchmarsh 1986 within the critical strip s 0 re s 1 the complex zeroes of z and x coincide and we denote them by r the x function is entire of order 1 and of maximal type herein we mainly investigate certain sums bn over complex zeta function zeroes we provide various representations and properties of these sums with the main specific results stated in 2 we also supply some remarks on the li criterion li 1997 for the riemann hypothesis rh we recall that the li equivalence for the rh results as a necessary and sufficient condition that the logarithmic derivative of the function x 1 1 z be analytic in the unit disc this obtains from a conformal map of the critical strip to this disc the equivalence li 1997 states that a necessary and sufficient condition for the non trivial zeroes of the z function to lie on the critical line re s 1 is that constants lk 1 are non negative for every integer k the sequence ln 1

hardness of learning halfspaces with noise learning an unknown halfspace also called a perceptron from labeled examples is one of the classic problems in machine learning in the noise free case when a halfspace consistent with all the training examples exists the problem can be solved in polynomial time using linear programming however under the promise that a halfspace consistent with a fraction 1 varepsilon of the examples exists for some small constant varepsilon 0 it was not known how to efficiently find a halfspace that is correct on even 51 of the examples nor was a hardness result that ruled out getting agreement on more than 99 9 of the examples known in this work we close this gap in our understanding and prove that even a tiny amount of worst case noise makes the problem of learning halfspaces intractable in a strong sense specifically for arbitrary epsilon delta 0 we prove that given a set of examples label pairs from the hypercube a fraction 1 varepsilon of which can be explained by a halfspace it is np hard to find a halfspace that correctly labels a fraction 1 2 delta of the examples the hardness result is tight since it is trivial to get agreement on 1 2 the examples in learning theory parlance we prove that weak proper agnostic learning of halfspaces is hard this settles a question that was raised by blum et al in their work on learning halfspaces in the presence of random classification noise algorithmica 22 1998 pp 35 52 and raised by authors of some more recent works as well along the way we also obtain a strong hardness result for another basic computational problem solving a linear system over the rationals

inapproximability results for equations over infinite groups an equation over a group g is an expression of form w 1 w k 1 g where each w i is either a variable an inverted variable or a group constant and 1 g denotes the identity element such an equation is satisfiable if there is a setting of the variables to values in g such that the equality is realized engebretsen et al 2002 10 in this paper we study the problem of simultaneously satisfying a family of equations over an infinite group g let eq g k denote the problem of determining the maximum number of simultaneously satisfiable equations in which each equation has occurrences of exactly k different variables when g is an infinite cyclic group we show that it is np hard to approximate eq 1 g 3 to within 48 47 e where eq 1 g 3 denotes the special case of eq g 3 in which a variable may only appear once in each equation it is np hard to approximate eq 1 g 2 to within 30 29 e it is np hard to approximate the maximum number of simultaneously satisfiable equations of degree at most d to within d e for any e for any k 4 it is np hard to approximate eq g k within any constant factor these results extend hastad s results hastad 2001 17 and results of engebretsen et al 2002 10 who established the inapproximability results for equations over finite abelian groups and any finite groups respectively

on the maximum satisfiability of random formulas say that a k cnf a formula is p satisfiable if there exists a truth assignment satisfying a fraction 1 2 k pp 2 k of its clauses note that every k cnf formula is 0 satisfiable let fk n m denote a random k cnf formula on n variables with m clauses for every k 2 and every r 0 we determine p and e k eo k2 k 2 such that with probability tending to 1 as n a random k cnf formula fk n rn is p satisfiable but not pp satisfiable

an alternative proof of a ptas for fixed degree polynomial optimization over the simplex the problem of minimizing a polynomial over the standard simplex is one of the basic np hard nonlinear optimization problems it contains the maximum clique problem in graphs as a special case it is known that the problem allows a polynomial time approximation scheme ptas for polynomials of fixed degree which is based on polynomial evaluations at the points of a sequence of regular grids in this paper we provide an alternative proof of the ptas property the proof relies on the properties of bernstein approximation on the simplex we also refine a known error bound for the scheme for polynomials of degree three the main contribution of the paper is to provide new insight into the ptas by establishing precise links with bernstein approximation and the multinomial distribution

on the maximum feasible subsystem problem iiss and iis hypergraphs we consider the max fs problem for a given infeasible linear system ax b determine a feasible subsystem containing as many inequalities as possible this problem which is np hard and also difficult to approximate has a number of interesting applications in a wide range of fields in this paper we examine structural and algorithmic properties of max fs and of irreducible infeasible subsystems iiss which are intrinsically related since one must delete at least one constraint from each iis to attain feasibility first we provide a new simplex decomposition characterization of iiss and prove that finding a smallest cardinality iis is very difficult to approximate then we discuss structural properties of iis hypergraphs i e hypergraphs in which each edge corresponds to an iis and show that recognizing iis hypergraphs subsumes the steinitz problem for polytopes and hence is np hard finally we investigate rank facets of the feasible subsystem polytope whose vertices are incidence vectors of feasible subsystems of a given infeasible system in particular using the iis hypergraph structural result we show that only two very specific types of rank inequalities induced by generalized antiwebs which generalize cliques odd holes and antiholes to general independence systems can arise as facets

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

recognizing frozen variables in constraint satisfaction problems in constraint satisfaction problems over finite domains some variables can be frozen that is they take the same value in all possible solutions we study the complexity of the problem of recognizing frozen variables with restricted sets of constraint relations allowed in the instances we show that the complexity of such problems is determined by certain algebraic properties of these relations under the assumption that np conp and consequently ptime np we characterize all tractable problems and describe large classes of np complete conp complete and dp complete problems as an application of these results we completely classify the complexity of the problem in two cases 1 with domain size 2 and 2 when all unary relations are present we also give a rough classification for domain size 3

relations and positivity results for the derivatives of the riemann ξ function we present and evaluate the integer order derivatives of the riemann xi function these derivatives contain logarithmic integrals of powers multiplying a specific jacobi theta function and as such can be alternatively viewed as certain mellin transforms at integer argument we describe how the derivatives at s 0 s and s 1 can be evaluated exactly we further show based upon a novel representation that the even order derivatives at s are all positive as are all derivatives at s 1 an expression is presented for the derivatives on the critical line which may be useful in studying the zeros of the function t it

a parallel repetition theorem we show that a parallel repetition of any two prover one round proof system mip 2 1 decreases the probability of error at an exponential rate no constructive bound was previously known the constant in the exponent in our analysis depends only on the original probability of error and on the total number of possible answers of the two provers the dependency on the total number of possible answers is logarithmic which was recently proved to be almost the best possible u feige and o verbitsky proc 11th annual ieee conference on computational complexity ieee computer society press los alamitos ca 1996 pp 70 76

solving integer programs over monotone inequalities in three variables a framework for half integrality and good approximations abstract we define a class of monotone integer programs with constraints that involve up to three variables each a generic constraint in such integer program is of the form ax by z c where a and b are nonnegative and the variable z appears only in that constraint we devise an algorithm solving such problems in time polynomial in the length of the input and the range of variables u the solution is obtained from a minimum cut on a graph with o nu nodes and o mu arcs where n is the number of variables of the types x and y and m is the number of constraints our algorithm is also valid for nonlinear objective functions nonmonotone integer programs are optimization problems with constraints of the type ax by z c without restriction on the signs of a and b such problems are in general np hard we devise here an algorithm relying on a transformation to the monotone case that delivers half integral superoptimal solutions in polynomial time such solutions provide bounds on the optimum value that can only be superior to bounds provided by linear programming relaxation when the half integral solution can be rounded to an integer feasible solution this is a 2 approximate solution in that the technique is a unified 2 approximation technique for a large class of problems the results apply also for general integer programming problems with worse approximation factors that depend on a quantifier measuring how far the problem is from the class of problems we describe the algorithm described here has a wide array of problem applications an additional important consequence of our results is that nonmonotone problems in the framework are max snp hard and at least as hard to approximate as vertex cover problems that are amenable to the analysis provided here are easily recognized the analysis itself is entirely technical and involves manipulating the constraints and transforming them to a totally unimodular system while losing no more than a factor of 2 in the integrality

beating the random ordering is hard every ordering csp is approximation resistant we prove that assuming the unique games conjecture ugc every problem in the class of ordering constraint satisfaction problems ocsps where each constraint has constant arity is approximation resistant in other words we show that if rho is the expected fraction of constraints satisfied by a random ordering then obtaining a rho approximation for any rho rho is ug hard for the simplest ocsp the maximum acyclic subgraph mas problem this implies that obtaining a rho approximation for any constant rho 1 2 is ug hard specifically for every constant varepsilon 0 the following holds given a directed graph g that has an acyclic subgraph consisting of a fraction 1 varepsilon of its edges it is ug hard to find one with more than 1 2 varepsilon of its edges note that it is trivial to find an acyclic subgraph with 1 2 the edges by taking either the forward or backward edges in an arbitrary ordering of the vertices of g the mas problem has been well studied and beating the random ordering for mas has been a basic open problem an ocsp of arity k is specified by a subset pi subseteq s k of permutations on 1 2 dots k an instance of such an ocsp is a set v and a collection of constraints each of which is an ordered k tuple of v the objective is to find a global linear ordering of v while maximizing the number of constraints ordered as in pi a random ordering of v is expected to satisfy a rho frac pi k fraction we show that for any fixed k it is hard to obtain a rho approximation for pi ocsp for any rho rho the result is in fact stronger we show that for every lambda subseteq pi subseteq s k and an arbitrarily small varepsilon it is hard to distinguish instances where a 1 varepsilon fraction of the constraints can be ordered according to lambda from instances where at most a rho varepsilon fraction can be ordered as in pi a special case of our result is that the betweenness problem is hard to approximate beyond a factor 1 3 the results naturally generalize to ocsps which assign a payoff to the different permutations finally our results imply unconditionally that a simple semidefinite relaxation for mas does not suffice to obtain a better approximation

the structure of strongly prime quadratic jordan algebras abstract we extend the structure theory for strongly prime jordan algebras of arbitrary dimension to quadratic algebras of characteristic 2 we show that such j are either 1 hermitian forms h 0 a j h q a lying between an ample subspace of hermitian elements in a prime associative algebra a and those in its martindale ring of quotients q a 2 clifford forms with a scalar extension j j which is an algebra j j q 1 of a nondegenerate quadratic from q with basepoint 1 or 3 albert forms with a scalar extension j j which is a 27 dimensional split albert algebra j h 3 k of 3 3 hermitian matrices over an 8 dimensional split octonion algebra k as a consequence the simple jordan algebras of arbitrary dimension are either hermitian h 0 a for simple a or clifford forms in j q 1 or are albert algebras in particular our methods give an idempotent free classification of all simple finite dimensional algebras which are i special the key to this characterization is the existence of zel manov polynomials in the free special jordan algebra which are both hermitian polynomials whose values always look like hermitian elements and at the same time clifford polynomials nontrivial on 3 3 hermitian matrices hence on special algebras with 3 interconnected idempotents if a zel manov polynomial does not vanish on j then j is of hermitian type and if it does vanish on j then j is of clifford type one example of such a polynomial is q48 p16 x1 y1 z1 w1 p16 x2 y2 z2 w2 p16 x3 y3 z3 w3 for p16 d2x y z 2 dx y w dx y w dx y z x y z thus the structure of jordan algebras depends very directly on polynomial identities and commutators

new results concerning power series expansions of the riemann xi function and the li keiper constants the riemann hypothesis is equivalent to the li criterion governing a sequence of real constants that are certain logarithmic derivatives of the riemann xi function evaluated at unity a new represe

new limits of treewidth based tractability in optimization sparse structures are frequently sought when pursuing tractability in optimization problems they are exploited from both theoretical and computational perspectives to handle complex problems that become manageable when sparsity is present an example of this type of structure is given by treewidth a graph theoretical parameter that measures how tree like a graph is this parameter has been used for decades for analyzing the complexity of various optimization problems and for obtaining tractable algorithms for problems where this parameter is bounded the goal of this work is to contribute to the understanding of the limits of the treewidth based tractability in optimization our results are as follows first we prove that in a certain sense the already known positive results on extension complexity based on low treewidth are the best possible secondly under mild assumptions we prove that treewidth is the only graph theoretical parameter that yields tractability for a wide class of optimization problems a fact well known in graphical models in machine learning and in constraint satisfaction problems which here we extend to an approximation setting in optimization

affine reductions for lps and sdps we define a reduction mechanism for lp and sdp formulations that degrades approximation factors in a controlled fashion our reduction mechanism is a minor restriction of classical hardness reductions requiring an additional independence assumption and it allows for reusing many hardness reductions that have been used to show inapproximability in the context of pcp theorems as a consequence we establish strong linear programming inapproximability for lps with a polynomial number of constraints for many problems in particular we obtain a frac 3 2 varepsilon inapproximability for answering an open question in chan et al proceedings of focs pp 350 359 2013 https doi org 10 1109 focs 2013 45 and prove an inapproximability factor of frac 1 2 varepsilon for bounded degree in the case of sdps we obtain inapproximability results for these problems relative to the sdp inapproximability of textsf maxcut moreover using our reduction framework we are able to reproduce various results for csps from chan et al proceedings of focs pp 350 359 2013 https doi org 10 1109 focs 2013 45 via simple reductions from max 2 xor

the complexity and approximability of finding maximum feasible subsystems of linear relations we study the combinatorial problem which consists given a system of linear relations of finding a maximum feasible subsystem that is a solution satisfying as many relations as possible the computational complexity of this general problem named max fls is investigated for the four types of relations and various constrained versions of max fls where a subset of relations must be satisfied or where the variables take bounded discrete values are also considered we establish the complexity of solving these problems optimally and whenever they are intractable we determine their degree of approximability max fls with or relations is np hard even when restricted to homogeneous systems with bipolar coefficients whereas it can be solved in polynomial time for relations with real coefficients the various np hard versions of max fls belong to different approximability classes depending on the type of relations and the additional constraints we show that the range of approximability stretches from apx complete problems which can be approximated within a constant but not within every constant unless p np to npo pb complete ones that are as hard to approximate as all np optimization problems with polynomially bounded objective functions while max fls with equations and integer coefficients cannot be approximated within pe for some e 0 where p is the number of relations the same problem over gf q for a prime q can be approximated within q but not within qe for some e 0 max fls with strict or nonstrict inequalities can be approximated within 2 but not within every constant factor our results also provide strong bounds on the approximability of two variants of max fls with and relations that arise when training perceptrons which are the building blocks of artificial neural networks and when designing linear classifiers

2 e sat is np hard we prove the following hardness result for a natural promise variant of the classical cnf satisfiability problem given a cnf formula where each clause has width w and the guarantee that there exis

the approximability of max csp with fixed value constraints in the maximum constraint satisfaction problem max csp one is given a finite collection of possibly weighted constraints on overlapping sets of variables and the goal is to assign values from a given finite domain to the variables so as to maximize the number or the total weight for the weighted case of satisfied constraints this problem is np hard in general and therefore it is natural to study how restricting the allowed types of constraints affects the approximability of the problem in this article we show that any max csp problem with a finite set of allowed constraint types which includes all fixed value constraints i e constraints of the form x e a is either solvable exactly in polynomial time or else is apx complete even if the number of occurrences of variables in instances is bounded moreover we present a simple description of all polynomial time solvable cases of our problem this description relies on the well known algebraic combinatorial property of supermodularity

convergence analysis for lasserre s measure based hierarchy of upper bounds for polynomial optimization we consider the problem of minimizing a continuous function f over a compact set mathbf k k we analyze a hierarchy of upper bounds proposed by lasserre siam j optim 21 3 864 885 2011 obtained by searching for an optimal probability density function h on mathbf k k which is a sum of squares of polynomials so that the expectation int mathbf k f x h x dx kf x h x dx is minimized we show that the rate of convergence is no worse than o 1 sqrt r o 1 r where 2r is the degree bound on the density function this analysis applies to the case when f is lipschitz continuous and mathbf k k is a full dimensional compact set satisfying some boundary condition which is satisfied e g for convex bodies the rth upper bound in the hierarchy may be computed using semidefinite programming if f is a polynomial of degree d and if all moments of order up to 2r d 2r d of the lebesgue measure on mathbf k k are known which holds for example if mathbf k k is a simplex hypercube or a euclidean ball

a fault containing self stabilizing 3 2δ 1 approximation algorithm for vertex cover in anonymous networks abstract the non computability of many distributed tasks in anonymous networks is well known this paper presents a deterministic self stabilizing algorithm to compute a 3 2 1 approximation of a minimum vertex cover in anonymous networks the algorithm operates under the distributed unfair scheduler stabilizes after o n m moves respectively o rounds and requires o log n storage per node recovery from a single fault is reached within a constant time and the contamination number is o for trees the algorithm computes a 2 approximation of a minimum vertex cover

techniques from combinatorial approximation algorithms yield efficient algorithms for random 2 k sat we apply techniques from the theory of approximation algorithms to the problem of deciding whether a random k sat formula is satisfiable let form n k m denote a random k sat instance with n variables and m clauses using known approximation algorithms for max cut or min bisection we show how to certify that formn 4 m is unsatisfiable efficiently provided that m cn2 for a sufficiently large constant c 0 in addition we present an algorithm based on the lovasz function that decides within polynomial expected time whether formn k m is satisfiable provided that k is even and m c 4knk 2 finally we present an algorithm that approximates random max 2 sat on input formn 2 m within a factor of 1 o n m 1 2 in expected polynomial time for m cn

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

matrix methods for the simplicial bernstein representation and for the evaluation of multivariate polynomials abstract in this paper multivariate polynomials in the bernstein basis over a simplex simplicial bernstein representation are considered two matrix methods for the computation of the polynomial coefficients with respect to the bernstein basis the so called bernstein coefficients are presented also matrix methods for the calculation of the bernstein coefficients over subsimplices generated by subdivision of the standard simplex are proposed and compared with the use of the de casteljau algorithm the evaluation of a multivariate polynomial in the power and in the bernstein basis is considered as well all the methods solely use matrix operations such as multiplication transposition and reshaping some of them rely also on the bidiagonal factorization of the lower triangular pascal matrix or the factorization of this matrix by a toeplitz matrix the latter one enables the use of the fast fourier transform hereby reducing the amount of arithmetic operations

on the power of multi prover interactive protocols abstract we look at complexity issues of interactive proof systems with multiple provers separated from each other this model developed by ben or et al 1988 allows the verifier to play the provers off each other we show this model equivalent to an alternative interactive proof system model using oracles as provers we also show that every language accepted by these models lies in nondeterministic exponential time we exhibit a relativized world where a co np language does not have multiple prover interactive proofs finally we show a simple example that one cannot parallelize multiple prover protocols as easily as the single prover model

hedging uncertainty approximation algorithms for stochastic optimization problems we study two stage finite scenario stochastic versions of several combinatorial optimization problems and provide nearly tight approximation algorithms for them our problems range from the graph theoretic shortest path vertex cover facility location to set theoretic set cover bin packing and contain representatives with different approximation ratios the approximation ratio of the stochastic variant of a typical problem is found to be of the same order of magnitude as its deterministic counterpart furthermore we show that common techniques for designing approximation algorithms such as lp rounding the primal dual method and the greedy algorithm can be adapted to obtain these results

the complexity of approximating a nonlinear program we consider the problem of finding the maximum of a multivariate polynomial inside a convex polytope we show that there is no polynomial time approximation algorithm for this problem even one with a very poor guarantee unless p np we show that even when the polynomial is quadratic i e quadratic programming there is no polynomial time approximation unless np is contained in quasi polynomial time