coreness of cooperative games with truncated submodular profit functions coreness represents solution concepts related to core in cooperative games which captures the stability of players motivated by the scale effect in social networks economics and other scenario we study the coreness of cooperative game with truncated submodular profit functions specifically the profit function f cdot is defined by a truncation of a submodular function sigma cdot f cdot sigma cdot if sigma cdot ge eta and f cdot 0 otherwise where eta is a given threshold in this paper we study the core and three core related concepts of truncated submodular profit cooperative game we first prove that whether core is empty can be decided in polynomial time and an allocation in core also can be found in polynomial time when core is not empty when core is empty we show hardness results and approximation algorithms for computing other core related concepts including relative least core value absolute least core value and least average dissatisfaction value

a tight sqrt 2 approximation for linear 3 cut we investigate the approximability of the linear 3 cut problem in directed graphs the input here is a directed graph d v e with node weights and three specified terminal nodes s r t in v and the goal is to find a minimum weight subset of non terminal nodes whose removal ensures that s cannot reach r and t and r cannot reach t the precise approximability of linear 3 cut has been wide open until now the best known lower bound under the unique games conjecture ugc was 4 3 while the best known upper bound was 2 using a trivial algorithm in this work we completely close this gap we present a sqrt 2 approximation algorithm and show that this factor is tight under ugc our contributions are twofold 1 we analyze a natural two step deterministic rounding scheme through the lens of a single step randomized rounding scheme with non trivial distributions and 2 we construct integrality gap instances that meet the upper bound of sqrt 2 our gap instances can be viewed as a weighted graph sequence converging to a graph limit structure we complement our results by showing connections between the linear 3 cut problem and other fundamental cut problems in directed graphs

hardness of reconstructing multivariate polynomials over finite fields we study the polynomial reconstruction problem for low degree multivariate polynomials over finite field mathbb f 2 in this problem we are given a set of points mathbf x in 0 1 n and target values f mathbf x in 0 1 for each of these points with the promise that there is a polynomial over mathbb f 2 of degree at most d that agrees with f at 1 varepsilon fraction of the points our goal is to find a degree d polynomial that has good agreement with f we show that it is np hard to find a polynomial that agrees with f on more than 1 2 d delta fraction of the points for any epsilon delta 0 this holds even with the stronger promise that the polynomial that fits the data is in fact linear whereas the algorithm is allowed to find a polynomial of degree d previously the only known hardness of approximation or even np completeness was for the case when d 1 which follows from a celebrated result of hastad j acm 48 2001 pp 798 859 in the setting of computational learning our result shows the hardness of nonproper agnostic learning of parities where the learner is allowed a low degree polynomial over mathbb f 2 as a hypothesis this is the first nonproper hardness result for this central problem in computational learning our results can be extended to multivariate polynomial reconstruction over any finite field

subexponential algorithms for unique games and related problems subexponential time approximation algorithms are presented for the u nique g ames and s mall s et e xpansion problems specifically for some absolute constant c the following two algorithms are presented 1 an exp kne time algorithm that given as input a k alphabet unique game on n variables that has an assignment satisfying 1 ec fraction of its constraints outputs an assignment satisfying 1 e fraction of the constraints 2 an exp ne time algorithm that given as input an n vertex regular graph that has a set s of n vertices with edge expansion at most ec outputs a set s of at most n vertices with edge expansion at most e subexponential algorithm is also presented with improved approximation to m ax c ut s parsest c ut and v ertex c over on some interesting subclasses of instances these instances are graphs with low threshold rank an interesting new graph parameter highlighted by this work khot s unique games conjecture ugc states that it is np hard to achieve approximation guarantees such as ours for u nique g ames while the results here stop short of refuting the ugc they do suggest that u nique g ames are significantly easier than np hard problems such as m ax 3 s at m ax 3 lin l abel c over and more which are believed not to have a subexponential algorithm achieving a nontrivial approximation ratio of special interest in these algorithms is a new notion of graph decomposition that may have other applications namely it is shown for every e 0 and every regular n vertex graph g by changing at most fraction of g s edges one can break g into disjoint parts so that the stochastic adjacency matrix of the induced graph on each part has at most ne eigenvalues larger than 1 where depends polynomially on e the subexponential algorithm combines this decomposition with previous algorithms for u nique g ames on graphs with few large eigenvalues kolla and tulsiani 2007 kolla 2010

hard constraint satisfaction problems have hard gaps at location 1 an instance of the maximum constraint satisfaction problem max csp is a finite collection of constraints on a set of variables and the goal is to assign values to the variables that maximises the number of satisfied constraints max csp captures many well known problems such as maxk sat and max cut and is consequently np hard thus it is natural to study how restrictions on the allowed constraint types or constraint language affect the complexity and approximability of max csp the pcp theorem is equivalent to the existence of a constraint language for which max csp has a hard gap at location 1 i e it is np hard to distinguish between satisfiable instances and instances where at most some constant fraction of the constraints are satisfiable all constraint languages for which the csp problem i e the problem of deciding whether all constraints can be satisfied is currently known to be np hard have a certain algebraic property we prove that any constraint language with this algebraic property makes max csp have a hard gap at location 1 which in particular implies that such problems cannot have a ptas unless p np we then apply this result to max csp restricted to a single constraint type this class of problems contains for instance max cut and max dicut assuming p np we show that such problems do not admit ptas except in some trivial cases our results hold even if the number of occurrences of each variable is bounded by a constant finally we give some applications of our results

frames and pseudo inverses abstract we point out some connections between the existing theories for frames and pseudo inverses in particular using the pseudo inverse of an unbounded operator we show how to make frame like decompositions with respect to some sequences which do not satisfy the upper frame condition as a special case this approach immediately gives nontrivial results for usual frames

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

combination can be hard approximability of the unique coverage problem we prove semilogarithmic inapproximability for a maximization problem called unique coverage given a collection of sets find a subcollection that maximizes the number of elements covered exactly once specifically assuming that mathrm np not subseteq operatorname bptime 2 n varepsilon for an arbitrary varepsilon 0 we prove o 1 log sigma n inapproximability for some constant sigma sigma varepsilon we also prove o 1 log 1 3 varepsilon n inapproximability for any varepsilon 0 assuming that refuting random instances of 3sat is hard on average and we prove o 1 log n inapproximability under a plausible hypothesis concerning the hardness of another problem balanced bipartite independent set we establish an omega 1 log n approximation algorithm even for a more general budgeted setting and obtain an omega 1 log b approximation algorithm when every set has at most b elements we also show that our inapproximability results extend to envy free pricing an important problem in computational economics we describe how the budgeted unique coverage problem motivated by real world applications has close connections to other theoretical problems including max cut maximum coverage and radio broadcasting

grothendieck type inequalities in combinatorial optimization we survey connections of the grothendieck inequality and its variants to combinatorial optimization and computational complexity 2011 wiley periodicals inc

optimal inapproximability results for max cut and other 2 variable csps in this paper we show a reduction from the unique games problem to the problem of approximating max cut to within a factor of alpha text tiny gw epsilon for all epsilon 0 here alpha text tiny gw approx 878567 denotes the approximation ratio achieved by the algorithm of goemans and williamson in j assoc comput mach 42 1995 pp 1115 1145 this implies that if the unique games conjecture of khot in proceedings of the 34th annual acm symposium on theory of computing 2002 pp 767 775 holds then the goemans williamson approximation algorithm is optimal our result indicates that the geometric nature of the goemans williamson algorithm might be intrinsic to the max cut problem our reduction relies on a theorem we call majority is stablest this was introduced as a conjecture in the original version of this paper and was subsequently confirmed in e mossel r o donnell and k oleszkiewicz proceedings of the 46th annual ieee symposium on foundations of computer science 2005 pp 21 30 a stronger version of this conjecture called plurality is stablest is still open although e mossel r o donnell and k oleszkiewicz proceedings of the 46th annual ieee symposium on foundations of computer science 2005 pp 21 30 contains a proof of an asymptotic version of it our techniques extend to several other two variable constraint satisfaction problems in particular subject to the unique games conjecture we show tight or nearly tight hardness results for max 2sat max q cut and max 2lin q for max 2sat we show approximation hardness up to a factor of roughly 943 this nearly matches the 940 approximation algorithm of lewin livnat and zwick in proceedings of the 9th annual conference on integer programming and combinatorial optimization springer verlag berlin 2002 pp 67 82 furthermore we show that our 943 factor is actually tight for a slightly restricted version of max 2sat for max q cut we show a hardness factor which asymptotically for large q matches the approximation factor achieved by frieze and jerrum improved approximation algorithms for max k cut and max bisection in integer programming and combinatorial optimization springer verlag berlin pp 1 13 namely 1 1 q 2 rm ln q q 2 for max 2lin q we show hardness of distinguishing between instances which are 1 epsilon satisfiable and those which are not even roughly q epsilon 2 satisfiable these parameters almost match those achieved by the recent algorithm of charikar makarychev and makarychev proceedings of the 38th annual acm symposium on theory of computing 2006 pp 205 214 the hardness result holds even for instances in which all equations are of the form x i x j c at a more qualitative level this result also implies that 1 epsilon vs e hardness for max 2lin q is equivalent to the unique games conjecture

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

on languages reducible to algorithmically randomlanguages in this paper languages bounded reducible to algorithmically random languages are studied these are the languages whose characteristic sequences are algorithmically random as defined by martin lof inform and control 9 1966 pp 602 619 here rand denotes the class of algorithmically random languages the reducibilities le cal r are very general but are defined so that if a in r b then there is a machine m with the properties that l m b a and every computation of m relative to any oracle halts book lutz and wagner math systems theory 27 1994 pp 201 209 studied sf almost r defined to be a mid for almost every b a le cal r b they showed that sf almost r r rand cap rec where rec denotes the class of recursive languages so that sf almost r is the recursive part of r rand in this paper this characterization is strengthened by showing that for every b in rand sf almost r r b cap rec a pair a b of languages is an independent pair of algorithmically random languages if a oplus b in rand in this paper it is shown that for every r and for every independent pair a b sf almost r r a cap r b

u cross gram matrices and their invertibility abstract the gram matrix is defined for bessel sequences by combining synthesis with subsequent analysis operators if different sequences are used and an operator u is inserted we reach so called u cross gram matrices this can be seen as reinterpretation of the matrix representation of operators using frames in this paper we investigate some necessary or sufficient conditions for schatten p class properties and the invertibility of u cross gram matrices in particular we show that under mild conditions the pseudo inverse of a u cross gram matrix can always be represented as a u cross gram matrix with dual frames of the given ones we link some properties of u cross gram matrices to approximate duals finally we state several stability results more precisely it is shown that the invertibility of u cross gram matrices is preserved under small perturbations

mapping dynamic bayesian networks to alpha shapes application to human faces identification across ages we propose to map a dynamic bayesian network dbn to an ordered family of shapes to improve dbns classification power this mission is achieved by 1 embedding a dbn into a topological manifold and 2 applying the shape geometric constructor to build hierarchical structures assigned to the dbn this continuous representation of traditional dbns as shapes allows more information to be obtained about the objects to be classified these latter are viewed as hierarchies of geometrical objects with different levels of detail topological signatures are therefore unraveled and classification accuracy is enhanced we have applied the proposed formalism to the task of facial identification across ages preliminary results demonstrate that the proposed formalism is a powerful tool since it has outperformed some dbn models the k nn classifier and some recent approaches

relative to a random oracle a bf p a ne bf np a ne text co bf np a with probability 1 let a be a language chosen randomly by tossing a fair coin for each string x to determine whether x belongs to a with probability 1 each of the relativized classes textbf logspace a bf p a bf np a bf pp a and textbf pspace a is properly contained in the next also bf np a ne text co bf np a with probability 1 by contrast with probability 1 the class bf p a coincides with the class bf bpp a of languages recognized by probabilistic oracle machines with error probability uniformly bounded below tfrac 1 2 bf np a is shown with probability 1 to contain a bf p a immune set i e a set having no infinite subset in bf p a the relationship of bf p a immunity to p sparseness and bf np a completeness is briefly discussed bf p a immune sets in bf np a can be sparse or moderately dense but not co sparse relativization with respect to a random length preserving permutation pi instead of a rand

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

approximating graph constrained max cut an instance of the graph constrained max cut mathsf gcmc problem consists of i an undirected graph g v e and ii edge weights c v atopwithdelims 2 rightarrow mathbb r on a complete undirected graph the objective is to find a subset s subseteq v of vertices satisfying some graph based constraint in g that maximizes the weight sum u in s v not in s c uv of edges in the cut s v setminus s the types of graph constraints we can handle include independent set vertex cover dominating set and connectivity our main results are for the case when g is a graph with bounded treewidth where we obtain a frac 1 2 approximation algorithm our algorithm uses an lp relaxation based on the sherali adams hierarchy it can handle any graph constraint for which there is a dynamic program of a specific form using known decomposition results these imply essentially the same approximation ratio for mathsf gcmc under constraints such as independent set dominating set and connectivity on a planar graph g

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

some applications of hypercontractive inequalities in quantum information theory hypercontractive inequalities have become important tools in theoretical computer science and have recently found applications in quantum computation in this note we discuss how hypercontractive inequalities in various settings can be used to obtain fairly concise proofs of several results in quantum information theory a recent lower bound of lancien and winter on the bias achievable by local measurements which are 4 designs spectral concentration bounds for k local hamiltonians and a recent result of pellegrino and seoane sepulveda giving general lower bounds on the classical bias obtainable in multiplayer xor games

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 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

nullspaces and frames in this paper we give new characterizations of riesz and conditional riesz frames in terms of the properties of the nullspace of their synthesis operators on the other hand we also study the oblique dual frames whose coefficients in the reconstruction formula minimize different weighted norms

an improved zero one law for algorithmically random sequences abstract results on random oracles typically involve showing that a class ifx p x has lebesgue measure one i e that some property p x holds for almost every x a potentially more informative approach is to show that p x is true for every x in some explicitly defined class of random sequences or languages in this note we consider the algorithmically random sequences originally defined by martin lof and their generalizations the n random sequences our result is an effective form of the classical zero one law for each n 1 if a class ifx p x is closed under finite variation and has arithmetical complexity 0 n 1 or 0 n 1 roughly the property p can be expressed with n 1 alternations of quantifiers then either p holds for every n random sequence or else holds for none of them this result has been used by book and mayordomo to give new characterizations of complexity classes of the form almost or the languages which can be or reduced to almost every oracle where or is a reducibility

comparing apples and oranges query trade off in submodular maximization fast algorithms for submodular maximization problems have a vast potential use in applicative settings such as machine learning social networks and economics though fast algorithms were known for some special cases only recently such algorithms were considered in the general case of maximizing a monotone submodular function subject to a matroid independence constraint the known fast algorithm matches the best possible approximation guarantee while trying to reduce the number of value oracle queries the algorithm performs our main result is a new algorithm for this general case that establishes a surprising trade off between two seemingly unrelated quantities the number of value oracle queries and the number of matroid independence queries performed by the algorithm specifically one can decrease the former by increasing the latter and vice versa while maintaining the best possible approximation guarantee such a trade off is very useful since various applications might incur significantly diff

approximation algorithms for connected maximum cut and related problems an instance of the connected maximum cut problem consists of an undirected graph g v e and the goal is to find a subset of vertices s v that maximizes the number of edges in the cut s such that the induced graph g s is connected we present the first non trivial omega frac 1 log n approximation algorithm for the connected maximum cut problem in general graphs using novel techniques we then extend our algorithm to an edge weighted case and obtain a poly logarithmic approximation algorithm interestingly in stark contrast to the classical max cut problem we show that the connected maximum cut problem remains np hard even on unweighted planar graphs on the positive side we obtain a polynomial time approximation scheme for the connected maximum cut problem on planar graphs and more generally on graphs with bounded genus

existence of parseval oblique duals of a frame sequence abstract we present a collection of necessary and sufficient conditions for the existence of a parseval oblique dual in an appropriate subspace of a frame sequence this result extends the previous results by han and larson antezana et al and han our proof shows that the problem is closely related with that of extending an operator with norm less than or equal to 1 to an isometry on a subspace