extended formulations nonnegative factorizations and randomized communication protocols an extended formulation of a polyhedron p p is a linear description of a polyhedron q q together with a linear map pi such that pi q p q p these objects are of fundamental importance in polyhedral combinatorics and optimization theory and the subject of a number of studies yannakakis factorization theorem yannakakis in j comput syst sci 43 3 441 466 1991 provides a surprising connection between extended formulations and communication complexity showing that the smallest size of an extended formulation of p p equals the nonnegative rank of its slack matrix s s moreover yannakakis also shows that the nonnegative rank of s s is at most 2 c 2c where c c is the complexity of any deterministic protocol computing s s in this paper we show that the latter result can be strengthened when we allow protocols to be randomized in particular we prove that the base 2 2 logarithm of the nonnegative rank of any nonnegative matrix equals the minimum complexity of a randomized communication protocol computing the matrix in expectation using yannakakis factorization theorem this implies that the base 2 2 logarithm of the smallest size of an extended formulation of a polytope p p equals the minimum complexity of a randomized communication protocol computing the slack matrix of p p in expectation we show that allowing randomization in the protocol can be crucial for obtaining small extended formulations specifically we prove that for the spanning tree and perfect matching polytopes small variance in the protocol forces large size in the extended formulation

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

the matching problem has no fully polynomial size linear programming relaxation schemes recently rothvos established that every linear program lp expressing the matching polytope has an exponential number of inequalities formally the matching polytope has exponential extension complexity we generalize this result by deriving strong bounds on the lp inapproximability of the matching problem for fixed 0 every 1 varepsilon n approximating lp requires an exponential number of inequalities where n is the number of vertices this is sharp given the well known rho approximation of size o binom n 1 1 rho provided by the odd sets of size up to 1 1 rho thus matching is the first problem in p which does not admit a fully polynomial size lp relaxation scheme the lp equivalent of an fully polynomial time approximation scheme which provides a sharp separation from the polynomial size lp relaxation scheme obtained e g through constant sized odd sets mentioned above analyzing the size of lp formulations is equivalent to examining the nonnegative rank of matrices we study the nonnegative rank through an information theoretic approach while it reuses key ideas from rothvos the main lower bounding technique is different we employ the information theoretic notion of wyner s common information used for studying lp formulations this allows us to analyze the nonnegative rank of perturbations of slack matrices e g the approximations of the matching polytope it turns out that the high extension complexity for the matching problem stems from the same source of hardness as in the case of the correlation polytope a direct sum structure

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

new bounds on classical and quantum one way communication complexity in this paper we provide new bounds on classical and quantum distributional communication complexity in the two party one way model of communication in the classical one way model our bound extends the well known upper bound of kremer nisan and ron i kremer n nisan d ron on randomized one round communication complexity in proceedings of the 27th acm symposium on theory of computing stoc 1995 pp 596 605 to include non product distributions let email protected 0 1 2 be a constant we show that for a boolean function f xxy 0 1 and a non product distribution m on xxy d e 1 m f o i x y 1 vc f where d e 1 m f represents the one way distributional communication complexity of f with error at most e under m vc f represents the vapnik chervonenkis dimension of f and i x y represents the mutual information under m between the random inputs of the two parties for a non boolean function f xxy 1 k k 2 an integer we show a similar upper bound on d e 1 m f in terms of k i x y and the pseudo dimension of f deffk a generalization of the vc dimension for non boolean functions in the quantum one way model we provide a lower bound on the distributional communication complexity under product distributions of a function f in terms of the well studied complexity measure of f referred to as the rectangle bound or the corruption bound of f we show for a non boolean total function f xxy z and a product distribution m on xxy q e 3 8 1 m f email protected rec e 1 m f where q e 3 8 1 m f represents the quantum one way distributional communication complexity of f with error at most e 3 8 under m and rec e 1 m f represents the one way rectangle bound of f with error at most e under m similarly for a non boolean partial function f xxy email protected and a product distribution m on xxy we show q e 6 2 1 5 4 1 m f email protected rec e 1 m f

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

mixed states in one spatial dimension decompositions and correspondence with nonnegative matrices we study six natural decompositions of mixed states in one spatial dimension the matrix product density operator form the local purification form the separable decomposition for separable states and their three translational invariant analogs for bipartite states diagonal in the computational basis we show that these decompositions correspond to well studied factorizations of an associated nonnegative matrix specifically the first three decompositions correspond to the minimal factorization the nonnegative factorization and the positive semidefinite factorization we also show that a symmetric version of these decompositions corresponds to the symmetric factorization the completely positive factorization and the completely positive semidefinite transposed factorization we leverage this correspondence to characterize the six decompositions of mixed states

an equivalent relationship in discrete dynamical systems an equivalent relationship between the generalized ergodicity almost everywhere e chain transitivity and the domain of influence of an invariant set for a discrete dynamical system on a compact metric space is explored a necessary and sufficient condition for an invariant set to be a periodic orbit is given

computing maximally separated sets in the plane let s be a set of n points in reals 2 given an integer 1 le k le n we wish to find a maximally separated subset i subseteq s of size k this is a subset for which the minimum among the k choose 2 pairwise distances between its points is as large as possible the decision problem associated with this problem is to determine whether there exists i subseteq s i k so that all k choose 2 pairwise distances in i are at least 2 this problem can also be formulated in terms of disk intersection graphs let d be the set of unit disks centered at the points of s the disk intersection graph g of d has as edges all pairs of disks with nonempty intersection any set i with the above properties is then the set of centers of disks that form an independent set in the graph g this problem is known to be np complete if k is part of the input in this paper we first present a linear time eps approximation algorithm for any constant k next we give exact algorithms for the cases k 3 and k 4 that run in time o n 4 3 polylog n we also present a simpler n o sqrt k time exact algorithm as compared with the recent algorithm in j alber and j fiala j algorithms 52 2004 pp 134 151 for arbitrary values of k

efficient protocols for generating bipartite classical distributions and quantum states we investigate the fundamental problem of generating bipartite classical distributions or quantum states by designing efficient communication protocols and proving their optimality we establish a number of intriguing connections to fundamental measures in optimization convex geometry and information theory 1 to generate a classical distribution p x y we tightly characterize the minimum amount of quantum communication needed by the psd rank of p as a matrix a measure recently proposed by fiorini et al proc 44th acm symp theory comput pp 95 106 2012 in studies of the minimum size of extended formulations of optimization problems such as tsp this echos the previous characterization for the optimal classical communication cost by the nonnegative rank of p the result is obtained via investigating the more general case of bipartite quantum state generation and designing an optimal protocol for it 2 when an approximation is allowed to generate a distribution x y p we present a classical protocol of the communication cost o c x y 1 where c x y is common information a well studied measure in information theory introduced by wyner ieee trans inf theory 21 2 163 179 1975 this also links nonnegative rank and common information two seemingly unrelated quantities in different fields 3 for approximately generating a quantum pure state we completely characterize the minimum cost by a corresponding approximate rank closing a possibly exponential gap left in ambainis etal siam j comput 32 6 1570 1585 2003

on the distributional complexity of disjointness we prove that the distributional communication complexity of the predicate disjointness with respect to a very simple measure on inputs is n

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

exponential lower bounds for polytopes in combinatorial optimization we solve a 20 year old problem posed by yannakakis and prove that no polynomial size linear program lp exists whose associated polytope projects to the traveling salesman polytope even if the lp is not required to be symmetric moreover we prove that this holds also for the cut polytope and the stable set polytope these results were discovered through a new connection that we make between one way quantum communication protocols and semidefinite programming reformulations of lps

quantum game players can have advantage without discord the last two decades have witnessed a rapid development of quantum information processing a new paradigm which studies the power and limit of quantum advantages in various information processing tasks problems such as when quantum advantage exists and if existing how much it could be are at a central position of these studies in a broad class of scenarios there are implicitly or explicitly at least two parties involved who share a state and the correlation in this shared state is the key factor to the efficiency under concern in these scenarios the shared entanglement or discord is usually what accounts for quantum advantage in this paper we examine a fundamental problem of this nature from the perspective of game theory a branch of applied mathematics studying selfish behaviors of two or more players we exhibit a natural zero sum game in which the chance for any player to win the game depends only on the ending correlation we show that in a certain classical equilibrium a situation in which no player can further increase her payoff by any local classical operation whoever first uses a quantum computer has a big advantage over its classical opponent the equilibrium is fair to both players and as a shared correlation it does not contain any discord yet a quantum advantage still exists this indicates that at least in game theory the previous notion of discord as a measure of non classical correlation needs to be reexamined when there are two players with different objectives

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

average case polyhedral complexity of the maximum stable set problem we study the minimum number of constraints needed to formulate random instances of the maximum stable set problem via linear programs lps in two distinct models in the uniform model the constraints of the lp are not allowed to depend on the input graph which should be encoded solely in the objective function there we prove a 2 omega n log n 2 n logn lower bound with probability at least 1 2 2 n 1 2 2n for every lp that is exact for a randomly selected set of instances each graph on at most n vertices being selected independently with probability p geqslant 2 left begin array c n 4 2 end array right n p 2 n 42 n in the non uniform model the constraints of the lp may depend on the input graph but we allow weights on the vertices the input graph is sampled according to the g n p model there we obtain upper and lower bounds holding with high probability for various ranges of p we obtain a super polynomial lower bound all the way from p omega left frac log 6 varepsilon n n right p log6 nn to p o left frac 1 log n right p o1logn our upper bound is close to this as there is only an essentially quadratic gap in the exponent which currently also exists in the worst case model finally we state a conjecture that would close this gap both in the average case and worst case models

lieb thirring trace inequalities and a question of bourin in this paper we propose some conjectures in order to solve one of the questions posed by bourin regarding a special type of inequalities referred to as subadditivity inequalities in the case of the hilbert schmidt norm some of these conjectures are settled affirmatively for special cases in an algorithmic way by using some number theory tools

positive semidefinite rank let m in mathbb r p times q m rp q be a nonnegative matrix the positive semidefinite rank psd rank of m is the smallest integer k for which there exist positive semidefinite matrices a i b j ai bj of size k times k k k such that m ij mathrm trace a i b j mij trace aibj the psd rank has many appealing geometric interpretations including semidefinite representations of polyhedra and information theoretic applications in this paper we develop and survey the main mathematical properties of psd rank including its geometry relationships with other rank notions and computational and algorithmic aspects

communication lower bounds using directional derivatives we study the set disjointness problem in the most powerful model of bounded error communication the k party randomized number on the forehead model we show that set disjointness requires n 2kk bits of communication where n is the size of the universe our lower bound generalizes to quantum communication where it is essentially optimal proving this bound was a longstanding open problem even in restricted settings such as one way classical protocols with k 4 parties wigderson 1997 the proof contributes a novel technique for lower bounds on multiparty communication based on directional derivatives of protocols over the reals

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

approximation of boolean functions by combinatorial rectangles this paper deals with the number of monochromatic combinatorial rectangles required to approximate a boolean function on a constant fraction of all inputs where each rectangle may use its own partition of the input variables the main result of the paper is that the number of rectangles required for the approximation of boolean functions in this model is very sensitive to the allowed error there is an explicitly defined sequence of boolean functions fn on n variables such that fn has rectangle approximations with a constant number of rectangles and one sided error 13 o 1 or two sided error 14 o 1 but on the other hand fn requires exponentially many rectangles if the error bounds are decreased by an arbitrarily small constant as applications of this result the following separation results for read once branching programs are obtained the functions from the main result require only linear size for nondeterministic read once branching programs and randomized read once branching programs with two sided error 13 o 1 while randomized read once branching programs with constant two sided error smaller than 13 and unambiguous nondeterministic read once branching programs require exponential size

finding quasi core with simulated stacked neural networks studying networks is promising for diverse applications we are often interested in exploring significant substructures in different types of real life networks finding cliques which denote a complete subgraph of a graph is one such important problem in network analysis interestingly many real life networks often contain a significant number of almost quasi complete subgraphs which are not entirely complete due to the presence of noise considering these networks as weighted adds further challenges to the problem finding quasi complete subgraphs in weighted graphs has never been formally addressed in this paper we propose a stacked neural network model for finding out the largest quasi complete module core in weighted graphs we show the effectiveness of the proposed approach on dimacs graphs we also highlight its utility in analyzing scientific collaboration networks social networks and biological networks

on the complexity of approximating the independent set problem we show that for some positive constant c it is not feasible to approximate independent set for graphs of n nodes within a factor of n c provided maximum 2 satisfiability does not have a randomized polynomial time approximation scheme we also study reductions preserving the quality of approximations and exhibit complete problems

all linear and integer programs are slim 3 way transportation programs we show that any rational convex polytope is polynomial time representable as a 3 way line sum transportation polytope of slim r c 3 format this universality theorem has important consequences for linear and integer programming and for confidential statistical data disclosure we provide a polynomial time embedding of arbitrary linear programs and integer programs in such slim transportation programs and in bitransportation programs our construction resolves several standing problems on 3 way transportation polytopes for example it demonstrates that unlike the case of 2 way contingency tables the range of values an entry can attain in any slim 3 way contingency table with specified 2 margins can contain arbitrary gaps our smallest such example has format 6 4 3 our construction provides a powerful automatic tool for studying concrete questions about transportation polytopes and contingency tables for example it automatically provides new proofs for some classical results including a well known real feasible but integer infeasible 6 4 3 transportation polytope of m vlach and bitransportation programs where any feasible bitransportation must have an arbitrarily large prescribed denominator

some 0 1 polytopes need exponential size extended formulations we prove that there are 0 1 polytopes p subseteq mathbb r n that do not admit a compact lp formulation more precisely we show that for every n there is a set x subseteq 0 1 n such that conv x must have extension complexity at least 2 n 2 cdot 1 o 1 in other words every polyhedron q that can be linearly projected on conv x must have exponentially many facets in fact the same result also applies if conv x is restricted to be a matroid polytope conditioning on mathbf np not subseteq mathbf p poly our result rules out the existence of a compact formulation for any mathbf np hard optimization problem even if the formulation may contain arbitrary real numbers

nondeterministic quantum query and communication complexities we study nondeterministic quantum algorithms for boolean functions f such algorithms have positive acceptance probability on input x iff f x 1 in the setting of query complexity we show that the nondeterministic quantum complexity of a boolean function is equal to its nondeterministic polynomial degree we also prove a quantum vs classical gap of 1 vs n for nondeterministic query complexity for a total function in the setting of communication complexity we show that the nondeterministic quantum complexity of a two party function is equal to the logarithm of the rank of a nondeterministic version of the communication matrix this implies that the quantum communication complexities of the equality and disjointness functions are n 1 if we do not allow any error probability we also exhibit a total function in which the nondeterministic quantum communication complexity is exponentially smaller than its classical counterpart

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

self improved gaps almost everywhere for the agnostic approximation of monomials given a learning sample we focus on the hardness of finding monomials having low error inside the interval bounded below by the smallest error achieved by a monomial the best rule and bounded above by the error of the default class the poorest rule it is well known that when its lower bound is zero it is an easy task to find in linear time a monomial with zero error what we prove is that when this bound is not zero regardless of the location of the default class in 0 1 2 it becomes a huge complexity burden to beat significantly the default class in fact under some complexity theoretical assumptions it may already be hard to beat the trivial approximation ratios even when relaxing the time complexity constraint to be quasi polynomial or sub exponential our results also hold with uniform weights over the examples

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

approximation limits of linear programs beyond hierarchies we develop a framework for proving approximation limits of polynomial size linear programs lps from lower bounds on the nonnegative ranks of suitably defined matrices this framework yields unconditional impossibility results that are applicable to any lp as opposed to only programs generated by hierarchies using our framework we prove that o n1 2 approximations for clique require lps of size 2n this lower bound applies to lps using a certain encoding of clique as a linear optimization problem moreover we establish a similar result for approximations of semidefinite programs by lps our main technical ingredient is a quantitative improvement of razborov s 38 rectangle corruption lemma for the high error regime which gives strong lower bounds on the nonnegative rank of shifts of the unique disjointness matrix

computer assisted proof of chaos in the lorenz equations abstract in this paper we prove with computer assistance the existence of chaos in a suitable poincare map generated by the lorenz system of equations by chaos we mean the existence of symbolic dynamics with infinite number of periodic trajectories the proof combines abstract results based on the fixed point index and finite rigorous computer calculations discussion concerning numerical algorithms is also included

trellises formed by stable and unstable manifolds in the plane a trellis is the figure formed by the stable and unstable manifolds of a hyperbolic periodic point of a diseomorphism of a 2 manifold this paper describes and classifies some trellises the set of homoclinic points is linearly ordered as a subset of the stable manifold and again as a subset of the unstable manifold each homoclinic point is assigned a type number which is constant on its orbit combinatorial properties of trellises are studied using type numbers and the pair of linear orderings trellises are important because their closures in some cases are strange attractors and in other cases are ergodic zones introduction in chapter 33 of new methods of celestial mechanics poincare describes the figure formed by the stable and unstable manifolds of a hyperbolic fixed point of a transformation of the upper half plane as follows when we try to represent the figure formed by these two curves and their intersections each of which corresponds to a doubly asymptotic solution these intersections form a type of trellis tissue or grid with infinitely serrated mesh neither of the two curves must ever 6ut across itself again but must bend back upon itself in a very complex manner in order to cut across all of the meshes in the grid an infinite number of times the complexity of this figure will be striking and i shall not even try to draw it nothing is more suitable for providing us with an idea of the complex nature of the three body problem and of all the problems of dynamics in general the aim of this paper is to understand figures of the type poincare described so well if p is a hyperbolic periodic point of a diffeomorphism of a 2 manifold then the figure formed by its stable and unstable manifolds will be called the trellis of p whenever these manifolds have nonempty intersection figure 1 illustrates the beginning development of such a trellis similarly the figure formed by the stable and unstable manifolds of a cycle of hyperbolic periodic points is called the trellis of the cycle a cycle of hyperbolic periodic points of a diffeomorphism is a finite collection p of periodic points having a cyclic permutation p p such that the unstable manifold of p intersects the stable manifold of p for each p g p it is natural to conjecture that in some cases ergodic zones are the closures of trellises anosov transformations of the torus provide examples where this is the case 1 one may also conjecture that strange attractors are sometimes the closures of the unstable manifolds of cycles of hyperbolic periodic points for certain values of the parameters a and b the henon map of the plane defined by t x y 1 y ax by may provide examples received by the editors may 7 1985 1980 mathematics subject classification primary 58f13 secondary 58f05 58f11 1986 american mathematical society 0002 9947 86 1 00 25 per page

approximating the maximum weight clique using replicator dynamics given an undirected graph with weights on the vertices the maximum weight clique problem mwcp is to find a subset of mutually adjacent vertices a clique having the largest total weight this is a generalization of the problem of finding the maximum cardinality clique of an unweighted graph which is the special case of the mwcp when all vertex weights are equal the problem is np hard for arbitrary graphs and so is the problem of approximating it within a constant factor we present a parallel distributed heuristic for approximating the mwcp based on dynamics principles it centers around a continuous characterization of the mwcp a purely combinatorial problem and lets it be formulated in terms of continuous quadratic programming one drawback is the presence of spurious solutions and we present their characterizations to avoid them we introduce a regularized continuous formulation of the mwcp and show how it completely solves the problem the formulation naturally maps onto a parallel distributed computational network whose dynamical behavior is governed by the replicator equations these are dynamical systems introduced in evolutionary game theory and population genetics to model evolutionary processes on a macroscopic scale we present theoretical results which guarantee that the solutions provided by our clique finding replicator network are actually those sought experimental results confirm the effectiveness of the proposed approach

inequalities related to bourin and heinz means with a complex parameter abstract a conjecture posed by s hayajneh and f kittaneh claims that given a b positive matrices 0 t 1 and any unitarily invariant norm the following inequality holds a t b 1 t b t a 1 t a t b 1 t a 1 t b t recently r bhatia proved the inequality for the case of the frobenius norm and for t 1 4 3 4 in this paper using complex methods we extend this result to complex values of the parameter t z in the strip z c re z 1 4 3 4 we give an elementary proof of the fact that equality holds for some z in the strip if and only if a and b commute we also show a counterexample to the general conjecture by exhibiting a pair of positive matrices such that the claim does not hold for the uniform norm finally we give a counterexample for a related singular value inequality given by s j a t b 1 t b t a 1 t s j a b answering in the negative a question made by k audenaert and f kittaneh the methods of proof and examples can be adapted with no modifications to operator algebras infinite dimensional setting for instance it follows that the inequality above holds for hilbert schmidt operators with their banach algebra norm derived from the infinite trace of b h

a fast network decomposition algorithm and its applications to constant time distributed computation abstract a partition c 1 c 2 c q of g v e into clusters of strong respectively weak diameter d such that the supergraph obtained by contracting each c i is l colorable is called a strong resp weak d l network decomposition network decompositions were introduced in a seminal paper by awerbuch goldberg luby and plotkin in 1989 awerbuch et al showed that strong d l network decompositions with d l exp o log n log log n can be computed in distributed deterministic time o d even more importantly they demonstrated that network decompositions can be used for a great variety of applications in the message passing model of distributed computing the result of awerbuch et al was improved by panconesi and srinivasan in 1992 in the latter result d l exp o log n and the running time is o d as well in another remarkable breakthrough linial and saks in 1992 showed that weak o log n o log n network decompositions can be computed in distributed randomized time o log 2 n much more recently barenboim 2012 devised a distributed randomized constant time algorithm for computing strong network decompositions with d o 1 however the parameter l in his result is o n 1 2 in this paper we drastically improve the result of barenboim and devise a distributed randomized constant time algorithm for computing strong o 1 o n network decompositions as a corollary we derive a constant time randomized o n approximation algorithm for the distributed minimum coloring problem improving the previously best known o n 1 2 approximation guarantee we also derive other improved distributed algorithms for a variety of problems most notably for the extremely well studied distributed minimum dominating set problem currently there is no known deterministic polylogarithmic time algorithm we devise a deterministic polylogarithmic time approximation algorithm for this problem addressing an open problem of lenzen and wattenhofer 2010

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

some upper and lower bounds on psd rank positive semidefinite rank psd rank is a relatively new complexity measure on matrices with applications to combinatorial optimization and communication complexity we first study several basic properties of psd rank and then develop new techniques for showing lower bounds on the psd rank all of these bounds are based on viewing a positive semidefinite factorization of a matrix m as a quantum communication protocol these lower bounds depend on the entries of the matrix and not only on its support the zero nonzero pattern overcoming a limitation of some previous techniques we compare these new lower bounds with known bounds and give examples where the new ones are better as an application we determine the psd rank of approximations of some common matrices

conley index for random dynamical systems conley index theory is a very powerful tool in the study of dynamical systems in this paper we generalize conley index theory to discrete random dynamical systems our constructions are basically the random version of franks and richeson in j franks d richeson shift equivalence and the conley index trans amer math soc 352 2000 3305 3322 for maps and the relations of isolated invariant sets between time continuous random dynamical systems and corresponding time h maps are discussed two examples are presented to illustrate results in this paper

new strong direct product results in communication complexity we show two new direct product results in two different models of communication complexity our first result is in the one way public coin model let f x y z be a relation and e 0 be a constant let r1 pube f represent the communication complexity of f with worst case error e in this model we show that if for computing fk k independent copies of f in this model o k r1 pub1 3 f communication is used then the success is exponentially small in k we show a new tight characterization of communication complexity in this model which strengthens the tight characterization shown in jain et al 2008 we use this new characterization to show our direct product result and this characterization may also be of independent interest our second direct product result is in the model of two way public coin communication complexity we show a direct product result for all relations in this model in terms of a new complexity measure that we define our new measure is a generalization to nonproduct distributions of the two way product subdistribution bound of jain et al 2008 our direct product result therefore generalizes to nonproduct distributions their direct product result in terms of the two way product subdistribution bound as an application of our new direct product result we reproduce via completely different arguments strong direct product result for the set disjointness problem which was previously shown by klauck 2010 we show this by proving that our new complexity measure gives a tight lower bound of n for the set disjointness problem on n bit inputs this strengthens the linear lower bound on the rectangle corruption bound for set disjointness shown by razborov 1992 in addition we show that many previously known direct product results in this model are uniformly implied and often strengthened by our result

the sign rank of ac 0 the sign rank of a matrix a a ij with pm1 entries is the least rank of a real matrix b b ij with a ij b ij 0 for all i j we obtain the first exponential lower bound on the sign rank of a function in mathsf ac 0 namely let f x y bigwedge i 1 dots m bigvee j 1 dots m 2 x ij wedge y ij we show that the matrix f x y x y has sign rank exp omega m this in particular implies that sigma 2 cc not subseteq mathsf upp cc which solves a longstanding open problem in communication complexity posed by babai frankl and simon proceedings of the 27th symposium on foundations of computer science focs 1986 pp 337 347 our result additionally implies a lower bound in learning theory specifically let phi 1 dots phi r 0 1 n to mathbb r be functions such that every dnf formula f 0 1 n to 1 1 of polynomial size has the representation f equiv mathrm sgn a 1 phi 1 dots a r phi r for some reals a 1 dots a r we prove that then r geqslant exp omega n 1 3 which essentially matches an upper bound of exp tilde o n 1 3 due to klivans and servedio j comput system sci 68 2004 pp 303 318 finally our work yields the first exponential lower bound on the size of threshold of majority circuits computing a function in mathsf ac 0 this substantially generalizes and strengthens the results of krause and pudlak theoret comput sci 174 1997 pp 137 156

isolating blocks and epsilon chains for maps abstract the relationship between the outputs and computer simulations of dynamical systems and their true orbit structures is explored epsilon chains are used as a mathematical representation of computer generated pseudo orbits the question when is a pseudo orbit shadowed by a true orbit is discussed isolated invariant sets are defined and are shown to be contained in special neighborhoods called isolating blocks the conley index is extended and applied to maps

the log approximate rank conjecture is false we construct a simple and total boolean function f e f c xor on 2n variables that has only o sn spectral norm o n2 approximate rank and o n2 5 approximate nonnegative rank we show it has polynomially large randomized bounded error communication complexity of sn this yields the first exponential gap between the logarithm of the approximate rank and randomized communication complexity for total functions thus f witnesses a refutation of the log approximate rank conjecture that was posed by lee and shraibman as a very natural analogue for randomized communication of the still unresolved log rank conjecture for deterministic communication the best known previous gap for any total function between the two measures is a recent 4th power separation by goos et al additionally our function f refutes grolmusz s conjecture and a variant of the log approximate nonnegative rank conjecture suggested recently by kol et al both of which are implied by the log approximate rank conjecture the complement of f has exponentially large approximate nonnegative rank this answers a question of lee 32 showing that approximate nonnegative rank can be exponentially larger than approximate rank the inner function f also falsifies a conjecture about parity measures of boolean functions made by tsang et al the latter conjecture implied the log rank conjecture for xor functions we are pleased to note that shortly after we published our results two independent groups of researchers anshu et al and sinha and de wolf used our function f to prove that the quantum log rank conjecture is also false by showing that f has n1 6 quantum communication complexity

on exact and renyi common informations recently two extensions of wyner s common information exact and renyi common informations were introduced respectively by kumar li and el gamal kle and the present authors the class of common information problems involves determining the minimum rate of the common input to two independent processors needed to exactly or approximately generate a target joint distribution for the exact common information problem exact generation of the target distribution is required while for wyner s and alpha renyi common informations the relative entropy and renyi divergence with order alpha were respectively used to quantify the discrepancy between the synthesized and target distributions the exact common information is larger than or equal to wyner s common information however it was hitherto unknown whether the former is strictly larger than the latter for some joint distributions in this paper we first establish the equivalence between the exact and infty renyi common informations and then provide single letter upper and lower bounds for these two quantities for doubly symmetric binary sources we show that the upper and lower bounds coincide which implies that for such sources the exact and infty renyi common informations are completely characterized interestingly we observe that for such sources these two common informations are strictly larger than wyner s this answers an open problem posed by kle furthermore we extend wyner s infty renyi and exact common informations to sources with countably infinite or continuous alphabets including gaussian sources

a review on algorithms for maximum clique problems the maximum clique problem mcp is to determine in a graph a clique i e a complete subgraph of maximum cardinality the mcp is notable for its capability of modeling other combinatorial problems and real world applications as one of the most studied np hard problems many algorithms are available in the literature and new methods are continually being proposed given that the two existing surveys on the mcp date back to 1994 and 1999 respectively one primary goal of this paper is to provide an updated and comprehensive review on both exact and heuristic mcp algorithms with a special focus on recent developments to be informative we identify the general framework followed by these algorithms and pinpoint the key ingredients that make them successful by classifying the main search strategies and putting forward the critical elements of the most relevant clique methods this review intends to encourage future development of more powerful methods and motivate new applications of the clique approaches

quantum communication and complexity in the setting of communication complexity two distributed parties want to compute a function depending on both their inputs using as little communication as possible the required communication can sometimes be significantly lowered if we allow the parties the use of quantum communication we survey the main results of the young area of quantum communication complexity its relation to teleportation and dense coding the main examples of fast quantum communication protocols lower bounds and some applications

geometric packing under nonuniform constraints we study the problem of discrete geometric packing here given weighted regions say in the plane and points with capacities one has to pick a maximum weight subset of the regions such that no point is covered more than its capacity we provide a general framework and an algorithm for approximating the optimal solution for packing in hypergraphs arising out of such geometric settings using this framework we get a flotilla of results on this problem and also on its dual where one wants to pick a maximum weight subset of the points when the regions have capacities for example for the case of fat triangles of similar size we show an o 1 approximation and prove that no ptas is possible

a simple simulated annealing algorithm for the maximum clique problem abstract simulated annealing technique has mostly been used to solve various optimization and learning problems and it is well known that the maximum clique problem is one of the most studied np hard optimization problems owing to its numerous applications in this note a simple simulated annealing algorithm for the maximum clique problem is proposed and tested on all 80 dimacs maximum clique instances although it is simple the proposed simulated annealing algorithm is efficient on most of the dimacs maximum clique instances the simulation results show that the proposed simulated annealing algorithm outperforms a recent efficient simulated annealing algorithm proposed by xu and ma and the solutions obtained by the proposed simulated annealing algorithm have the equal quality with those obtained by a recent trust region heuristic algorithm of stanislav busygin

completely positive semidefinite rank an n times n matrix x is called completely positive semidefinite cpsd if there exist d times d hermitian positive semidefinite matrices p i i 1 n for some d ge 1 such that x ij mathrm tr p ip j for all i j in 1 ldots n the cpsd rank of a cpsd matrix is the smallest d ge 1 for which such a representation is possible in this work we initiate the study of the cpsd rank which we motivate in two ways first the cpsd rank is a natural non commutative analogue of the completely positive rank of a completely positive matrix second we show that the cpsd rank is physically motivated as it can be used to upper and lower bound the size of a quantum system needed to generate a quantum behavior in this work we present several properties of the cpsd rank unlike the completely positive rank which is at most quadratic in the size of the matrix no general upper bound is known on the cpsd rank of a cpsd matrix in fact we show that the cpsd rank can be sub exponential in terms of the size specifically for any n ge 1 we construct a cpsd matrix of size 2n whose cpsd rank is 2 varomega sqrt n our construction is based on gram matrices of lorentz cone vectors which we show are cpsd the proof relies crucially on the connection between the cpsd rank and quantum behaviors in particular we use a known lower bound on the size of matrix representations of extremal quantum correlations which we apply to high rank extreme points of the n dimensional elliptope lastly we study cpsd graphs i e graphs g with the property that every doubly nonnegative matrix whose support is given by g is cpsd we show that a graph is cpsd if and only if it has no odd cycle of length at least 5 as a subgraph this coincides with the characterization of cp graphs

the quantum communication complexity of sampling sampling is an important primitive in probabilistic and quantum algorithms in the spirit of communication complexity given a function f x times y rightarrow 0 1 and a probability distribution cal d over x times y we define the sampling complexity of f cal d as the minimum number of bits that alice and bob must communicate for alice to pick x in x and bob to pick y in y as well as a value z such that the resulting distribution of x y z is close to the distribution cal d f cal d in this paper we initiate the study of sampling complexity in both the classical and quantum models we give several variants of a definition we completely characterize some of these variants and give upper and lower bounds on others in particular this allows us to establish an exponential gap between quantum and classical sampling complexity for the set disjointness function

on the extension complexity of combinatorial polytopes in this paper we extend recent results of fiorini et al on the extension complexity of the cut polytope and related polyhedra we first describe a lifting argument to show exponential extension complexity for a number of np complete problems including subset sum and three dimensional matching we then obtain a relationship between the extension complexity of the cut polytope of a graph and that of its graph minors using this we are able to show exponential extension complexity for the cut polytope of a large number of graphs including those used in quantum information and suspensions of cubic planar graphs

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

trace inequalities for products of positive definite matrices it is shown that for positive definite matrices a and b the inequality apbq bpaq 2 apbq aqbp 2 holds for all positive real number p and q for which p p q is in 1 4 3 4 this is a significant improvement on a recent result of hayajneh and kittaneh lieb thirring trace inequalities and a question of bourin j math phys 54 033504 2013

quantum entanglement and communication complexity we consider a variation of the communication complexity scenario where the parties are supplied with an extra resource particles in an entangled quantum state we note that quantum nonlocality can be naturally expressed in the language of communication complexity these are communication complexity problems where the output is embodied in the correlations between the outputs of the individual parties without entanglement the parties must communicate to produce the required correlations whereas with entanglement no communication is necessary to produce the correlations in this sense nonlocality proofs can also be viewed as communication complexity problems where the presence of quantum entanglement reduces the amount of necessary communication we show how to transform examples of nonlocality into more traditional communication complexity problems where the output is explicitly determined by each individual party the resulting problems require communication with or without entanglement but the required communication is less when entanglement is available all these results are a noteworthy contrast to the well known fact that entanglement cannot be used to actually simulate or compress classical communication between remote parties

the communication complexity of correlation let x and y be finite nonempty sets and x y a pair of random variables taking values in x y we consider communication protocols between two parties alice and bob for generating x and y alice is provided an x x generated according to the distribution of x and is required to send a message to bob in order to enable him to generate y y whose distribution is the same as that of y x x both parties have access to a shared random string generated in advance let t x y be the minimum over all protocols of the expected number of bits alice needs to transmit to achieve this we show that i x y t x y i x y 2 log2 i x y o 1 we also consider the worst case communication required for this problem where we seek to minimize the average number of bits alice must transmit for the worst case x x we show that the communication required in this case is related to the capacity c e of the channel e derived from x y that maps x x to the distribution of y x x we also show that the required communication t e satisfies c e t e c e 2 log2 c e 1 o 1 using the first result we derive a direct sum theorem in communication complexity that substantially improves the previous such result shown by jain radhakrishnan and sen in proc 30th international colloquium of automata languages and programming icalp ser lecture notes in computer science vol 2719 2003 pp 300 315 these results are obtained by employing a rejection sampling procedure that relates the relative entropy between two distributions to the communication complexity of generating one distribution from the other

on unapproximable versions of np complete problems we prove that all of karp s 21 original np complete problems have a version that is hard to approximate these versions are obtained from the original problems by adding essentially the same simple constraint we further show that these problems are absurdly hard to approximate in fact no polynomial time algorithm can even approximate log k of the magnitude of these problems to within any constant factor where logk denotes the logarithm iterated k times unless np is recognized by slightly superpolynomial randomized machines we use the same technique to improve the constant epsilon such that max clique is hard to approximate to within a factor of n epsilon finally we show that it is even harder to approximate two counting problems counting the number of satisfying assignments to a monotone 2sat formula and computing the permanent of 1 0 1 matrices

on the power of randomized multicounter machines one way two counter machines represent a universal model of computation here we consider the polynomial time classes of multicounter machines with a constant number of reversals and separate the computational power of nondeterminism randomization and determinism for instance we show that polynomial time one way multicounter machines with error probability tending to zero with growing input length can recognize languages that cannot be accepted by polynomial time nondeterministic two way multicounter machines with a bounded number of reversals a similar result holds for the comparison of determinism and one sided error randomization and of determinism and las vegas randomization

a copositive formulation for the stability number of infinite graphs in the last decade copositive formulations have been proposed for a variety of combinatorial optimization problems for example the stability number independence number in this paper we generalize this approach to infinite graphs and show that the stability number of an infinite graph is the optimal solution of some infinite dimensional copositive program for this we develop a duality theory between the primal convex cone of copositive kernels and the dual convex cone of completely positive measures we determine the extreme rays of the latter cone and we illustrate this theory with the help of the kissing number problem

the pattern matrix method we develop a novel technique for communication lower bounds the pattern matrix method specifically fix an arbitrary function f 0 1 n to 0 1 and let a f be the matrix whose columns are each an application of f to some subset of the variables x 1 x 2 ldots x 4n we prove that a f has bounded error communication complexity omega d where d is the approximate degree of f this result remains valid in the quantum model regardless of prior entanglement in particular it gives a new and simple proof of razborov s breakthrough quantum lower bounds for disjointness and other symmetric predicates we further characterize the discrepancy approximate rank and approximate trace norm of a f in terms of well studied analytic properties of f broadly generalizing several recent results on small bias communication and agnostic learning the method of this paper has also enabled important progress in multiparty communication complexity

graph distances in the data stream model we explore problems related to computing graph distances in the data stream model the goal is to design algorithms that can process the edges of a graph in an arbitrary order given only a limited amount of working memory we are motivated by both the practical challenge of processing massive graphs such as the web graph and the desire for a better theoretical understanding of the data stream model in particular we are interested in the trade offs between model parameters such as per data item processing time total space and the number of passes that may be taken over the stream these trade offs are more apparent when considering graph problems than they were in previous streaming work that solved problems of a statistical nature our results include the following 1 spanner construction there exists a single pass tilde o tn 1 1 t space tilde o t 2n 1 t time per edge algorithm that constructs a 2t 1 spanner for t omega log n log log n the algorithm satisfies the semistreaming space restriction of o n operatorname polylog n and has per edge processing time o operatorname polylog n this resolves an open question from j feigenbaum et al theoret comput sci 348 2005 pp 207 216 2 breadth first search bfs trees for any even constant k we show that any algorithm that computes the first k layers of a bfs tree from a prescribed node with probability at least 2 3 requires either greater than k 2 passes or tilde omega n 1 1 k space since constructing bfs trees is an important subroutine in many traditional graph algorithms this demonstrates the need for new algorithmic techniques when processing graphs in the data stream model 3 graph distance lower bounds any t approximation of the distance between two nodes requires omega n 1 1 t space we also prove lower bounds for determining the length of the shortest cycle and other graph properties 4 techniques for decreasing per edge processing we discuss two general techniques for speeding up the per edge computation time of streaming algorithms while increasing the space by only a small factor

lower bounds on nonnegative rank via nonnegative nuclear norms the nonnegative rank of an entrywise nonnegative matrix a in mathbb r m times n a r m n is the smallest integer r r such that a a can be written as a uv a uv where u in mathbb r m times r u r m r and v in mathbb r r times n v r r n are both nonnegative the nonnegative rank arises in different areas such as combinatorial optimization and communication complexity computing this quantity is np hard in general and it is thus important to find efficient bounding techniques especially in the context of the aforementioned applications in this paper we propose a new lower bound on the nonnegative rank which unlike most existing lower bounds does not solely rely on the matrix sparsity pattern and applies to nonnegative matrices with arbitrary support the idea involves computing a certain nuclear norm with nonnegativity constraints which allows to lower bound the nonnegative rank in the same way the standard nuclear norm gives lower bounds on the standard rank our lower bound is expressed as the solution of a copositive programming problem and can be relaxed to obtain polynomial time computable lower bounds using semidefinite programming we compare our lower bound with existing ones and we show examples of matrices where our lower bound performs better than currently known ones

upper and lower bounds on the power of advice proving superpolylogarithmic lower bounds for dynamic data structures has remained an open problem despite years of research p trascu proposed an exciting approach for breaking this barrier via a two player communication model in which one player gets private advice at the beginning of the protocol he gave reductions from the problem of solving an asymmetric version of set disjointness in his model to a diverse collection of natural dynamic data structure problems in the cell probe model he also conjectured that for any hard problem in the standard two party communication model the asymmetric version of the problem is hard in his model provided not too much advice is given in this paper we prove several surprising results about his model we show that there exist boolean functions requiring linear randomized communication complexity in the two party model for which the asymmetric versions in his model have deterministic protocols with exponentially smaller complexity for set disjointness which

orbits of unbounded energy in quasi periodic perturbations of geodesic flows abstract we show that certain mechanical systems including a geodesic flow in any dimension plus a quasi periodic perturbation by a potential have orbits of unbounded energy the assumptions we make in the case of geodesic flows are a the metric and the external perturbation are smooth enough b the geodesic flow has a hyperbolic periodic orbit such that its stable and unstable manifolds have a tranverse homoclinic intersection c the frequency of the external perturbation is diophantine d the external potential satisfies a generic condition depending on the periodic orbit considered in b the assumptions on the metric are c 2 open and are known to be dense on many manifolds the assumptions on the potential fail only in infinite codimension spaces of potentials the proof is based on geometric considerations of invariant manifolds and their intersections the main tools include the scattering map of normally hyperbolic invariant manifolds as well as standard perturbation theories averaging kam and melnikov techniques we do not need to assume that the metric is riemannian and we obtain results for finsler or lorentz metrics indeed there is a formulation for hamiltonian systems satisfying scaling hypotheses we do not need to make assumptions on the global topology of the manifold nor on its dimension

a measure conquer approach for the analysis of exact algorithms for more than 40 years branch reduce exponential time backtracking algorithms have been among the most common tools used for finding exact solutions of np hard problems despite that the way to analyze such recursive algorithms is still far from producing tight worst case running time bounds motivated by this we use an approach that we call measure conquer as an attempt to step beyond such limitations the approach is based on the careful design of a nonstandard measure of the subproblem size this measure is then used to lower bound the progress made by the algorithm at each branching step the idea is that a smarter measure may capture behaviors of the algorithm that a standard measure might not be able to exploit and hence lead to a significantly better worst case time analysis in order to show the potentialities of measure conquer we consider two well studied np hard problems minimum dominating set and maximum independent set for the first problem we consider the current best algorithm and prove thanks to a better measure a much tighter running time bound for it for the second problem we describe a new simple algorithm and show that its running time is competitive with the current best time bounds achieved with far more complicated algorithms and standard analysis our examples show that a good choice of the measure made in the very first stages of exact algorithms design can have a tremendous impact on the running time bounds achievable

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

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

on the parameterized complexity of approximating dominating set we study the parameterized complexity of approximating the k dominating set domset problem where an integer k and a graph g on n vertices are given as input and the goal is to find a dominating set of size at most f k k whenever the graph g has a dominating set of size k when such an algorithm runs in time t k poly n i e fpt time for some computable function t it is said to be an f k fpt approximation algorithm for k domset whether such an algorithm exists is listed in the seminal book of downey and fellows 2013 as one of the most infamous open problems in parameterized complexity this work gives an almost complete answer to this question by showing the non existence of such an algorithm under w 1 fpt and further providing tighter running time lower bounds under stronger hypotheses specifically we prove the following for every computable functions t f and every constant e 0 assuming w 1 fpt there is no f k fpt approximation algorithm for k domset assuming the exponential time hypothesis eth there is no f k approximation algorithm for k domset that runs in t k no k time assuming the strong exponential time hypothesis seth for every integer k 2 there is no f k approximation algorithm for k domset that runs in t k nk e time assuming the k sum hypothesis for every integer k 3 there is no f k approximation algorithm for k domset that runs in t k n k 2 e time previously only constant ratio fpt approximation algorithms were ruled out under sf w 1 fpt and log1 4 m k fpt approximation algorithms were ruled out under eth chen and lin focs 2016 recently the non existence of an f k fpt approximation algorithm for any function f was shown under gap eth chalermsook et al focs 2017 note that to the best of our knowledge no running time lower bound of the form nd this allows us to easily apply known techniques to solve them

real rank versus nonnegative rank abstract we consider the set of m n nonnegative real matrices and define the nonnegative rank of a matrix a to be the minimum k such that a bc where b is m k and c is k n given that the real rank of a is j for some j we give bounds on the nonnegative rank of a and a 2

amplifying lower bounds by means of self reducibility we observe that many important computational problems in nc1 share a simple self reducibility property we then show that for any problem a having this self reducibility property a has polynomial size tc0 circuits if and only if it has tc0 circuits of size n1 for every 0 counting the number of wires in a circuit as the size of the circuit as an example of what this observation yields consider the boolean formula evaluation problem bfe which is complete for nc1 and has the self reducibility property it follows from a lower bound of impagliazzo paturi and saks that bfe requires depth d tc0 circuits of size n1 d if one were able to improve this lower bound to show that there is some constant 0 independent of the depth d such that every tc0 circuit family recognizing bfe has size at least n1 then it would follow that tc0 nc1 we show that proving lower bounds of the form n1 is not ruled out by the natural proof framework of razborov and rudich and hence there is currently no known barrier for separating classes such as acc0 tc0 and nc1 via existing natural approaches to proving circuit lower bounds we also show that problems with small uniform constant depth circuits have algorithms that simultaneously have small space and time bounds we then make use of known time space tradeoff lower bounds to show that sat requires uniform depth d tc0 and ac0 6 circuits of size n1 c for some constant c depending on d