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

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

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

improved non approximability results for minimum vertex cover with density constraints we provide new non approximability results for the restrictions of the min vertex cover problem to bounded degree sparse and dense graphs we show that for a sufficiently large b the recent 16 15 lower bound proved by bellare et al 3 extends with negligible loss to graphs with bounded degree b then we consider sparse graphs with no dense components i e everywhere sparse graphs and we show a similar result but with a better trade off between non approximability and sparsity finally we observe that the min vertex cover problem remains apx complete when restricted to dense graph and thus recent techniques developed by arora et al 1 for several max snp problems restricted to dense instances cannot be applied

on dinur s proof of the pcp theorem probabilistically checkable proofs are proofs that can checked probabilistically by reading very few bits of the proof in the early 1990 s it was shown that proofs could be transformed into probabilistically checkable ones with only a modest increase in their size the initial transformations though elementary were a little too complex a recent work due to irit dinur gives a dramatically simple and radically new construction of probabilistically checkable proofs this article explains the notion of a probabilistically checkable proof presents the formal definition and then introduces the reader to dinur s work along with some of the context

the approximability of constraint satisfaction problems we study optimization problems that may be expressed as boolean constraint satisfaction problems an instance of a boolean constraint satisfaction problem is given by m constraints applied to n boolean variables different computational problems arise from constraint satisfaction problems depending on the nature of the underlying constraints as well as on the goal of the optimization task here we consider four possible goals max csp min csp is the class of problems where the goal is to find an assignment maximizing the number of satisfied constraints minimizing the number of unsatisfied constraints max ones min ones is the class of optimization problems where the goal is to find an assignment satisfying all constraints with maximum minimum number of variables set to 1 each class consists of infinitely many problems and a problem within a class is specified by a finite collection of finite boolean functions that describe the possible constraints that may be used tight bounds on the approximability of every problem in max csp were obtained by creignou j comput system sci 51 1995 pp 511 522 in this work we determine tight bounds on the approximability i e the ratio to within which each problem may be approximated in polynomial time of every problem in max ones min csp and min ones combined with the result of creignou this completely classifies all optimization problems derived from boolean constraint satisfaction our results capture a diverse collection of optimization problems such as max 3 sat max cut max clique min cut nearest codeword etc our results unify recent results on the in approximability of these optimization problems and yield a compact presentation of most known results moreover these results provide a formal basis to many statements on the behavior of natural optimization problems that have so far been observed only empirically

solving the maximum duo preservation string mapping problem with linear programming abstract in this paper we introduce the maximum duo preservation string mapping problem mpsm which is complementary to the minimum common string partition problem mcsp when each letter occurs at most k times in any input string the version of mpsm is called k mpsm in order to design approximation algorithms for mpsm we also introduce the constrained maximum induced subgraph problem cmis and the constrained minimum induced subgraph cnis problem we show that both cmis and cnis are np complete we also study the approximation algorithms for the restricted version of cmis which is called k cmis k 2 using linear programming method we propose an approximation algorithm for 2 cmis with approximation ratio 2 and an approximation algorithm for k cmis k 3 with approximation ratio k 2 based on approximation algorithms for k cmis we get approximation algorithms for k mpsm with the same approximation ratio

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

approximating the independence number via the j function we describe an approximation algorithm for the independence number of a graph if a graph onn vertices has an independence numbern k m for some fixed integerk 3 and somem 0 the algorithm finds in random polynomial time an independent set of size tilde omega m 3 mathord left vphantom 3 k 1 right kern nulldelimiterspace k 1 improving the best known previous algorithm of boppana and halldorsson that finds an independent set of size m1 k 1 in such a graph the algorithm is based on semi definite programming some properties of the lovasz function of a graph and the recent algorithm of karger motwani and sudan for approximating the chromatic number of a graph if the function of ann vertex graph is at leastmn1 2 k for some absolute constantm we describe another related efficient algorithm that finds an independent set of sizek several examples show the limitations of the approach and the analysis together with some related arguments supply new results on the problem of estimating the largest possible ratio between the function and the independence number of a graph onn vertices 1998 the mathematical programming society inc published by elsevier science b v

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

on bounded queries and approximation this paper investigates the computational complexity of approximating several np optimization problems using the number of queries to an np oracle as a complexity measure the results show a tradeoff between the closeness of the approximation and the number of queries required for an approximation factor k n log log k n n queries to an np oracle can be used to approximate the maximum clique size of a graph within a factor of k n however this approximation cannot be achieved using fewer than log log k n n c queries to any oracle unless pe np where c is a constant that does not depend on k these results hold for approximation factors k n geq 2 that belong to a class of functions which includes any integer constant function log n log a n and n 1 a similar results are obtained for graph coloring set cover and other np optimization problems

bounded queries approximations and the boolean hierarchy this paper investigates nondeterministic bounded query classes in relation to the complexity of np hard approximation problems and the boolean hierarchy nondeterministic bounded query classes turn out be rather suitable for describing the complexity of np hard approximation problems the results in this paper take advantage of this machine based model to prove that in many cases np approximation problems have the upward collapse property that is a reduction between np approximation problems of apparently different complexity at a lower level results in a similar reduction at a higher level for example if maxclique reduces to log n approximating maxclique using many one reductions then the traveling salesman problem tsp is equivalent to maxclique under many one reductions several upward collapse theorems are presented in this paper the proofs of these theorems rely heavily on the machinery provided by the nondeterministic bounded query classes in fact these results depend on a surprising connection between the boolean hierarchy and nondeterministic bounded query classes

expander graphs and their applications a major consideration we had in writing this survey was to make it accessible to mathematicians as well as to computer scientists since expander graphs the protagonists of our story come up in numerous and often surprising contexts in both fields but perhaps we should start with a few words about graphs in general they are of course one of the prime objects of study in discrete mathematics however graphs are among the most ubiquitous models of both natural and human made structures in the natural and social sciences they model relations among species societies companies etc in computer science they represent networks of communication data organization computational devices as well as the flow of computation and more in mathematics cayley graphs are useful in group theory graphs carry a natural metric and are therefore useful in geometry and though they are just one dimensional complexes they are useful in certain parts of topology e g knot theory in statistical physics graphs can represent local connections between interacting parts of a system as well as the dynamics of a physical process on such systems the study of these models calls then for the comprehension of the significant structural properties of the relevant graphs but are there nontrivial structural properties which are universally important expansion of a graph requires that it is simultaneously sparse and highly connected expander graphs were first defined by bassalygo and pinsker and their existence first proved by pinsker in the early 70s the property of being an expander seems significant in many of these mathematical computational and physical contexts it is not surprising that expanders are useful in the design and analysis of communication networks what is less obvious is that expanders have surprising utility in other computational settings such as in the theory of error correcting codes and the theory of pseudorandomness in mathematics we will encounter e g their role in the study of metric embeddings and in particular in work around the baum connes conjecture expansion is closely related to the convergence rates of markov chains and so they play a key role in the study of monte carlo algorithms in statistical mechanics and in a host of practical computational applications the list of such interesting and fruitful connections goes on and on with so many applications we will not even

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

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

quantum de finetti theorems under local measurements with applications quantum de finetti theorems are a useful tool in the study of correlations in quantum multipartite states in this paper we prove two new quantum de finetti theorems both showing that under tests formed by local measurements in each of the subsystems one can get an exponential improvement in the error dependence on the dimension of the subsystems we also obtain similar results for non signaling probability distributions we give several applications of the results to quantum complexity theory polynomial optimization and quantum information theory the proofs of the new quantum de finetti theorems are based on information theory in particular on the chain rule of mutual information the results constitute improvements and generalizations of a recent de finetti theorem due to brandao christandl and yard

sensory constraints on auditory identification of the material and geometric properties of struck bars a computational formula is derived for estimating the constraints limited auditory sensitivity imposes on auditory identification of the material and geometric properties of struck bars the formula combines a model of the transverse motion of the bar with empirical psychometric functions to map out null regions in the bar s physical parameter space where changes in the frequency amplitude and decay of partials are likely below threshold for detection parameters of the physical space include bar density young s modulus fluid and viscoelastic damping factors bar length and bar cross sectional area as related to bar shape and hollowness the formula is used to estimate the possible effect of limited sensitivity in past studies on the auditory identification of bar attributes the results suggest that sensitivity may indeed have played a role in some studies and that apparent discrepancies in results may be understood based on whether the predominant source of damping was internal or external to the bar the formula identifies conditions representing an expected bound on identification performance and thereby may be used to aid in the design of future studies for which the struck bar is the sound source of choice

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

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

stopping functionals for gaussian quadrature formulas abstract gaussian formulas are among the most often used quadrature formulas in practice in this survey an overview is given on stopping functionals for gaussian formulas which are of the same type as quadrature formulas i e linear combinations of function evaluations in particular methods based on extended formulas like the important gauss kronrod and patterson schemes and methods which are based on gaussian nodes are presented and compared

a simple deterministic reduction for the gap minimum distance of code problem we present a simple deterministic gap preserving reduction from sat to the minimum distance of code problem over f 2 we also show how to extend the reduction to work over any fixed finite field previously a randomized reduction was known due to dumer micciancio and sudan which was recently derandomized by cheng and wan these reductions rely on highly nontrivial coding theoretic constructions whereas our reduction is elementary as an additional feature our reduction gives hardness within a constant factor even for asymptotically good codes i e having constant positive rate and relative distance previously it was not known how to achieve a deterministic reduction for such codes

on the hardness of approximating minimization problems we prove results indicating that it is hard to compute efficiently good approximate solutions to the graph coloring set covering and other related minimization problems specifically there is an e 0 such that graph coloring cannot be approximated with ratio n e unless p np set covering cannot be approximated with ratio c log n for any c 1 4 unless np is contained in dtime n poly log n similar results follow for related problems such as clique cover fractional chromatic number dominating set and others

new hardness results for congestion minimization and machine scheduling we study the approximability of two natural np hard problems the first problem is congestion minimization in directed networks in this problem we are given a directed graph and a set of source sink pairs the goal is to route all the pairs with minimum congestion on the network edges the second problem is machine scheduling where we are given a set of jobs and for each job there is a list of intervals on which it can be scheduled the goal is to find the smallest number of machines on which all jobs can be scheduled such that no two jobs overlap in their execution on any machine both problems are known to be o log n log log n approximable via the randomized rounding technique of raghavan and thompson 1987 however until recently only max snp hardness was known for each problem we make progress in closing this gap by showing that both problems are o log log n hard to approximate unless np dtime no log log log n

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

hardness of approximate hypergraph coloring we introduce the notion of covering complexity of a verifier for probabilistically checkable proofs pcps such a verifier is given an input a claimed theorem and an oracle representing a purported proof of the theorem the verifier is also given a random string and decides whether to accept the proof or not based on the given random string we define the covering complexity of such a verifier on a given input to be the minimum number of proofs needed to satisfy the verifier on every random string i e on every random string at least one of the given proofs must be accepted by the verifier the covering complexity of pcp verifiers offers a promising route to getting stronger inapproximability results for some minimization problems and in particular hyper graph coloring problems we present a pcp verifier for np statements that queries only four bits and yet has a covering complexity of one for true statements and a superconstant covering complexity for statements not in the language moreover the acceptance predicate of this verifier is a simple not all equal check on the four bits it reads this enables us to prove that for any constant c it is np hard to color a 2 colorable 4 uniform hypergraph using just c colors and also yields a superconstant inapproximability result under a stronger hardness assumption

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

shorter arithmetization of nondeterministic computations arithmetizing computation is a crucial component of many fundamental results in complexity theory including results that gave insight into the power of interactive proofs multi prover interactive proofs and probabilistically checkable proofs informally an arithmetization is a way to encode a machine s computation so that its correctness can be easily verified via few probabilistic algebraic checks we study the problem of arithmetizing nondeterministic computations for the purpose of constructing short probabilistically checkable proofs pcps with polylogarithmic query complexity in such a setting a pcp s proof length depends at least linearly on the length in bits of the encoded computation thus minimizing the number of bits in the encoding is crucial for minimizing pcp proof length in this paper we show how to arithmetize any t step computation on a nondeterministic turing machine by using a polynomial encoding of length o t log t 2 previously the best known length was t log t 4 for nondeterministic random access machines our length is o t log t 2 o 1 while prior work only achieved t log t 5 the polynomial encoding that we use is the reed solomon code when combined with the best pcps of proximity for this code our result yields quasilinear size pcps with polylogarithmic query complexity that are shorter by at least two logarithmic factors than in all prior work our arithmetization also enjoys additional properties first it is succinct i e the encoding of the computation can be probabilistically checked in log t o 1 time this property is necessary for constructing short pcps with a polylogarithmic time verifier furthermore our techniques extend in a certain well defined sense to the arithmetization of yet other nexp complete languages

testing reed muller codes a code is locally testable if there is a way to indicate with high probability that a vector is far enough from any codeword by accessing only a very small number of the vector s bits we show that the reed muller codes of constant order are locally testable specifically we describe an efficient randomized algorithm to test if a given vector of length n 2 sup m is a word in the rth order reed muller code r r m of length n 2 sup m for a given integer r spl ges 1 and real spl epsi 0 the algorithm queries the input vector spl upsi at o 1 spl epsi r2 sup 2r positions on the one hand if spl upsi is at distance at least spl epsi n from the closest codeword then the algorithm discovers it with probability at least 2 3 on the other hand if spl upsi is a codeword then it always passes the test our result is almost tight any algorithm for testing r r m must perform spl omega 1 spl epsi 2 sup r queries

bounding queries in the analytic polynomial time hierarchy abstract in a previous paper the present authors baier and wagner 1996 investigated an hierarchy over p using word quantifiers as well as two types of set quantifiers the so called analytic polynomial time hierarchy the fact that some constructions there result in a bounded number of oracle queries and the recent pcp results which can be expressed by set quantifiers with a bounded number of queries motivated us to examine a hierarchy which extends the analytic polynomial time hierarchy by considering restrictions on the number of oracle queries this hierarchy is called bounded analytic polynomial time hierarchy we show that every class from this hierarchy having a certain normal form coincides with one of the classes np conp pspace kexp or kexp k 1 all these characterizations remain valid if the queries are asked in a nonadaptive form i e in parallel

pspace has constant round quantum interactive proof systems in this paper we introduce quantum interactive proof systems which are interactive proof systems in which the prover and verifier may perform quantum computations and exchange quantum messages it is proved that every language in pspace has a quantum interactive proof system that requires a total of only three messages to be sent between the prover and verifier and has exponentially small one sided probability of error it follows that quantum interactive proof systems are strictly more powerful than classical interactive proof systems in the constant round case unless the polynomial time hierarchy collapses to the second level

on agnostic learning of parities monomials and halfspaces we study the learnability of several fundamental concept classes in the agnostic learning framework of d haussler inform and comput 100 1992 pp 78 150 and m kearns r schapire and l sellie machine learning 17 1994 pp 115 141 we show that under the uniform distribution agnostically learning parities reduce to learning parities with random classification noise commonly referred to as the noisy parity problem together with the parity learning algorithm of a blum a kalai and h wasserman j acm 50 2003 pp 506 519 this gives the first nontrivial algorithm for agnostic learning of parities we use similar techniques to reduce learning of two other fundamental concept classes under the uniform distribution to learning of noisy parities namely we show that learning of disjunctive normal form dnf expressions reduces to learning noisy parities of just logarithmic number of variables and learning of k juntas reduces to learning noisy parities of k variables we give essentially optimal hardness results for agnostic learning of monomials over 0 1 n and halfspaces over mathbb q n we show that for any constant epsilon finding a monomial halfspace that agrees with an unknown function on 1 2 epsilon fraction of the examples is np hard even when there exists a monomial halfspace that agrees with the unknown function on 1 epsilon fraction of the examples this resolves an open question due to blum and significantly improves on a number of previous hardness results for these problems we extend these results to epsilon 2 log 1 lambda n epsilon 2 sqrt log n in the case of halfspaces for any constant lambda 0 under stronger complexity assumptions

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

primality and identity testing via chinese remaindering we give a simple and new randomized primality testing algorithm by reducing primality testing for number n to testing if a specific univariate identity over zn holds we also give new randomized algorithms for testing if a multivariate polynomial over a finite field or over rationals is identically zero the first of these algorithms also works over zn for any n the running time of the algorithms is polynomial in the size of arithmetic circuit representing the input polynomial and the error parameter these algorithms use fewer random bits and work for a larger class of polynomials than all the previously known methods for example the schwartz zippel test schwartz 1980 zippel 1979 chen kao and lewin vadhan tests chen and kao 1997 lewin and vadhan 1998

checking approximate computations of polynomials and functional equations a majority of the results on self testing and correcting deal with programs which purport to compute the correct results precisely we relax this notion of correctness and show how to check programs that compute only a numerical approximation to the correct answer the types of programs that we deal with are those computing polynomials and functions defined by certain types of functional equations we present results showing how to perform approximate checking self testing and self correcting of polynomials settling in the affirmative a question raised by p gemmell et al proceedings of the 23rd acm symposium on theory of computing 1991 pp 32 42 r rubinfeld and m sudan proceedings of the third annual acm siam symposium on discrete algorithms orlando fl 1992 pp 23 43 r rubinfeld and m sudan siam j comput 25 1996 pp 252 271 we obtain this by first building approximate self testers for linear and multilinear functions we then show how to perform approximate checking self testing and self correcting for those functions that satisfy addition theorems settling a question raised by r rubinfeld siam j comput 28 1999 pp 1972 1997 in both cases we show that the properties used to test programs for these functions are both robust in the approximate sense and stable finally we explore the use of reductions between functional equations in the context of approximate self testing our results have implications for the stability theory of functional equations

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

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

formation potential field for trajectory tracking control of multi agents in constrained space abstractin this paper a novel formation potential field method is proposed for the multi agent formation control the objective is to control a group of agents to automatically generate and maintain a specific formation while avoiding internal collisions and collisions with spatial constraints a formation potential field is designed combining multiple local attractive potential fields with multiple local repulsive potential fields to further relax requirements of agents initial positions and enhance the robustness a global attractive potential field is added outside the influence range of the local formation potential field the optimality of the proposed scheme in formation time is analysed as well as illustrated by contrast simulation results following the design of the formation potential field two controllers are proposed accordingly to achieve a stable dynamic formation during the process of trajectory tracking while the saturation effect of input is taken into account furthermore a collisi

algebraic attacks against random local functions and their countermeasures suppose that you have n truly random bits x x 1 ldots x n and you wish to use them to generate m gg n pseudorandom bits y y 1 ldots y m using a local mapping i e each y i should depend on at most d o 1 bits of x in the polynomial regime of m n s s 1 the only known solution originating from goldreich electronic colloquium on computational complexity eccc 2000 is based on random local functions compute y i by applying some fixed public d ary predicate p to a random public tuple of distinct input indices x i 1 ldots x i d our goal in this paper is to understand for any value of s how the pseudorandomness of the resulting sequence depends on the choice of the underlying predicate we derive the following results 1 we show that pseudorandomness against mathbb f 2 linear adversaries i e the distribution y has small bias is achieved if the predicate is a k omega s resilient i e uncorrelated with any k subset of its inputs a

the complexity of counting in sparse regular and planar graphs we show that a number of graph theoretic counting problems remain cal np hard indeed cal p complete in very restricted classes of graphs in particular we prove that the problems of counting matchings vertex covers independent sets and extremal variants of these all remain hard when restricted to planar bipartite graphs of bounded degree or regular graphs of constant degree we obtain corollaries about counting cliques in restricted classes of graphs and counting satisfying assignments to restricted classes of monotone 2 cnf formulae to achieve these results a new interpolation based reduction technique which preserves properties such as constant degree is introduced

asymmetric k center is log n hard to approximate in the asymmetric k center problem the input is an integer k and a complete digraph over n points together with a distance function obeying the directed triangle inequality the goal is to choose a set of k points to serve as centers and to assign all the points to the centers so that the maximum distance of any point from its center is as small as possible we show that the asymmetric k center problem is hard to approximate up to a factor of log n o 1 unless np dtime nlog log n since an o log n approximation algorithm is known for this problem this resolves the asymptotic approximability of asymmetric k center this is the first natural problem whose approximability threshold does not polynomially relate to the known approximation classes we also resolve the approximability threshold of the metric symmetric k center problem with costs

the primal dual method for approximation algorithms in this survey we give an overview of a technique used to design and analyze algorithms that provide approximate solutions to np hard problems in combinatorial optimization because of parallels with the primal dual method commonly used in combinatorial optimization we call it the primal dual method for approximation algorithms we show how this technique can be used to derive approximation algorithms for a number of different problems including network design problems feedback vertex set problems and facility location problems

the inapproximability of non np hard optimization problems the inapproximability of non np hard optimization problems is investigated techniques are given to show that the problems log dominating set and log hypergraph vertex cover cannot be approximated to a constant ratio in polynomial time unless the corresponding np hard versions are also approximable in deterministic subexponential time a direct connection is established between non np hard problems and a pcp characterization of np reductions from the pcp characterization show that log clique is not approximable in polynomial time and max sparse sat does not have a fptas under the assumption that np dtime 2o n log n a number of nontrivial approximation preserving reductions are also presented making it possible to extend inapproximability results to more natural non np hard problems such as tournament dominating set and rich hypergraph vertex cover

towards optimal lower bounds for clique and chromatic number it was previously known that neither max clique nor min chromatic number can be approximated in polynomial time within n1 e for any constant e 0 unless np zpp in this paper we extend the reductions used to prove these results and combine the extended reductions with a recent result of samorodnitsky and trevisan to show that unless np zptime 2o log n log log n 3 2 neither max clique nor min chromatic number can be approximated in polynomial time within n1 e n where e o log log n 1 2 since there exists polynomial time algorithms approximating both problems within n1 e n where e n log log n log n our result shows that the best possible ratio we can hope for is of the form n1 o 1 for some yet unknown value of o 1 between o log log n 1 2 and log log n log n

constant rate pcps for circuit sat with sublinear query complexity the pcp theorem arora et al 1998 arora and safra 1998 says that every np proof can be encoded to another proof namely a probabilistically checkable proof pcp which can be tested by a verifier that queries only a small part of the pcp a natural question is how large is the blow up incurred by this encoding that is how long is the pcp compared to the original np proof the state of the art work of ben sasson and sudan 2008 and dinur 2007 shows that one can encode proofs of length n by pcps of length n poly log n that can be verified using a constant number of queries in this work we show that if the query complexity is relaxed to ne then one can construct pcps of length o n for circuit sat and pcps of length o tlog t for any language in ntime t more specifically for any e 0 we present nonuniform probabilistically checkable proofs pcps of length 2o 1 e n that can be checked using ne queries for circuit sat instances of size n our pcps have perfect completeness and constant soundness this is the first constant rate pcp construction that achieves constant soundness with nontrivial query complexity o n our proof replaces the low degree polynomials in algebraic pcp constructions with tensors of transitive algebraic geometry ag codes we show that the automorphisms of an ag code can be used to simulate the role of affine transformations that are crucial in earlier high rate algebraic pcp constructions using this observation we conclude that any asymptotically good family of transitive ag codes over a constant sized alphabet leads to a family of constant rate pcps with polynomially small query complexity such codes are constructed in the appendix to this article for the first time for every message length building on an earlier construction for infinitely many message lengths by stichtenoth 2006

new hardness results for routing on disjoint paths in the classical node disjoint paths sf ndp problem the input consists of an undirected n vertex graph g and a collection mathcal m s 1 t 1 ldots s k t k of pairs of its vert

best order streaming model we study a new model of computation called best order stream for graph problems roughly it is a proof system where a space limited verifier has to verify a proof sequentially i e it reads the proof as a stream moreover the proof itself is just a specific ordering of the input data this model is closely related to many models of computation in other areas such as data streams communication complexity and proof checking and could be used in applications such as cloud computing in this paper we focus on graph problems where the input is a sequence of edges we show that even under this model checking some basic graph properties deterministically requires linear space in the number of nodes we also show that in contrast with this randomized verifiers are powerful enough to check many graph properties in polylogarithmic space

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

inapproximability results for maximum edge biclique minimum linear arrangement and sparsest cut we consider the minimum linear arrangement problem and the uniform sparsest cut problem so far these two notorious np hard graph problems have resisted all attempts to prove inapproximability results we show that they have no polynomial time approximation scheme unless np complete problems can be solved in randomized subexponential time furthermore we show that the same techniques can be used for the maximum edge biclique problem for which we obtain a hardness factor similar to previous results but under a more standard assumption

improved inapproximability of lattice and coding problems with preprocessing we show that the closest vector problem with preprocessing cvpp is np hard to approximate to within spl radic 3 spl epsi for any spl epsi 0 in addition we show that the nearest codeword problem with preprocessing ncpp is np hard to approximate to within 3 spl epsi these results improve previous results of feige and micciancio we also present the first inapproximability result for the relatively nearest codeword problem with preprocessing rncp finally we describe an n approximation algorithm to cvpp

on the hardness of approximating minimum vertex cover we prove the minimum vertex cover problem to be np hard to approximate to within a factor of 1 3606 extending on previous pcp and hardness of approximation technique to that end one needs to develop a new proof framework and to borrow and extend ideas from several fields

new construction for a qma complete three local hamiltonian we present a new way of encoding a quantum computation into a three local hamiltonian our construction is novel in that it does not include any terms that induce legal illegal clock transitions therefore the weights of the terms in the hamiltonian do not scale with the size of the problem as in previous constructions this improves the construction by kempe and regev quantum inf comput 3 258 264 2003 e print quant ph 0302079 who were the first to prove that three local hamiltonian is complete for the complexity class qma the quantum analog of np quantum k sat a restricted version of the local hamiltonian problem using only projector terms was introduced by bravyi e print quant ph 0602108 as an analog of the classical k sat problem bravyi proved that quantum 4 sat is complete for the class qma with one sided error qma1 and that quantum 2 sat is in p we give an encoding of a quantum circuit into a quantum 4 sat hamiltonian using only three local terms as an intermediate step to this th

product state approximations to quantum states we show that for any many body quantum state there exists an unentangled quantum state such that most of the two body reduced density matrices are close to those of the original state this is a statement about the monogamy of entanglement which cannot be shared without limit in the same way as classical correlation our main application is to hamiltonians that are sums of two body terms for such hamiltonians we show that there exist product states with energy that is close to the ground state energy whenever the interaction graph of the hamiltonian has high degree this proves the validity of mean field theory and gives an explicitly bounded approximation error if we allow states that are entangled within small clusters of systems but product across clusters then good approximations exist when the hamiltonian satisfies one or more of the following properties 1 high degree 2 small expansion or 3 a ground state where the blocks in the partition have sublinear entanglement previously this was known only in the case of small expansion or in the regime where the entanglement was close to zero our approximations allow an extensive error in energy which is the scale considered by the quantum pcp probabilistically checkable proof and nlts no low energy trivial state conjectures thus our results put restrictions on the possible hamiltonians that could be used for a possible proof of the qpcp or nlts conjectures by contrast the classical pcp constructions are often based on constraint graphs with high degree likewise we show that the parallel repetition that is possible with classical constraint satisfaction problems cannot also be possible for quantum hamiltonians unless qpcp is false the main technical tool behind our results is a collection of new classical and quantum de finetti theorems which do not make any symmetry assumptions on the underlying states

locally testable cyclic codes cyclic linear codes of block length n over a finite field f sub q are linear subspaces of f sub q sup n that are invariant under a cyclic shift of their coordinates a family of codes is good if all the codes in the family have constant rate and constant normalized distance distance divided by block length it is a long standing open problem whether there exists a good family of cyclic linear codes a code c is r testable if there exists a randomized algorithm which given a word x spl isin sub q sup n adaptively selects r positions checks the entries of x in the selected positions and makes a decision accept or reject x based on the positions selected and the numbers found such that 1 if x spl isin c then x is surely accepted ii if dist x c spl ges spl epsi n then x is probably rejected dist refers to hamming distance a family of codes is locally testable if all members of the family are r testable for some constant r this concept arose from holographic proofs pcp s recently it was asked whether there exist good locally testable families of codes in this paper the intersection of the two questions stated is addressed theorem there are no good locally testable families of cyclic codes over any fixed finite field in fact the result is stronger in that it replaces condition ii of local testability by the condition ii if dist x c spl ges spl epsi n then x has a positive chance of being rejected the proof involves methods from galois theory cyclotomy and diophantine approximation

gowers uniformity influence of variables and pcps we study the relation of query complexity and soundness in probabilistically checkable proofs pcps we present a pcp verifier for languages that are unique games hard and such that the verifier makes q queries has almost perfect completeness and has soundness error at most 2q 2 q varepsilon for arbitrarily small varepsilon 0 for values of q of the form 2 t 1 the soundness error is q 1 2 q varepsilon charikar makarychev and makarychev show that there is a constant beta such that every language that has a verifier of query complexity q and a ratio of soundness error to completeness smaller than beta q 2 q is decidable in polynomial time up to the value of the multiplicative constant and to the validity of the unique games conjecture our result is therefore tight as a corollary we show that approximating the maximum independent set problem in graphs of degree delta within a factor better than delta log delta alpha is unique games hard for a certain constant alpha 0 our main technical results are i a connection between the gowers uniformity of a boolean function and the influence of its variables and ii the proof that gowers uniform functions pass the hypergraph linearity test approximately with the same probability of a random function the connection between gowers uniformity and influence might have other applications

high rate codes with sublinear time decoding locally decodable codes are error correcting codes that admit efficient decoding algorithms any bit of the original message can be recovered by looking at only a small number of locations of a corrupted codeword the tradeoff between the rate of a code and the locality efficiency of its decoding algorithms has been well studied and it has widely been suspected that nontrivial locality must come at the price of low rate a particular setting of potential interest in practice is codes of constant rate for such codes decoding algorithms with locality o k were known only for codes of rate 1 where k is the length of the message furthermore for codes of rate 1 2 no nontrivial locality had been achieved in this article we construct a new family of locally decodable codes that have very efficient local decoding algorithms and at the same time have rate approaching 1 we show that for every 0 and 0 for infinitely many k there exists a code c which encodes messages of length k with rate 1 and is locally decodable from a constant fraction of errors using o k queries and time these codes which we call multiplicity codes are based on evaluating multivariate polynomials and their derivatives multiplicity codes extend traditional multivariate polynomial codes they inherit the local decodability of these codes and at the same time achieve better tradeoffs and flexibility in the rate and minimum distance

max np completeness made easy we introduce a simple technique to obtain reductions between optimization constraint satisfaction problems with respect to an approximation preserving reducibility more powerful than the commonly used l reducibility the technique applies the probabilistic method to reduce the size of disjunctions as a first application we prove the max np completeness of max 3sat without using the pcp theorem thus solving an open question posed in khanna et al 1994 successively we show that the planar restrictions of several optimization constraint satisfaction problems admit linear time approximation schemes thus improving the results of khanna and motwani 1996

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 maximizing welfare when utility functions are subadditive we consider the problem of maximizing welfare when allocating m items to n players with subadditive utility functions our main result is a way of rounding any fractional solution to a linear programming relaxation to this problem so as to give a feasible solution of welfare at least 1 2 that of the value of the fractional solution this approximation ratio of 1 2 is an improvement over an omega 1 log m ratio of dobzinski nisan and schapira proceedings of the 37th annual acm symposium on theory of computing baltimore md acm new york 2005 pp 610 618 we also show an approximation ratio of 1 1 e when utility functions are fractionally subadditive a result similar to this last result was previously obtained by dobzinski and schapira proceedings of the 17th annual acm siam symposium on discrete algorithms miami fl siam philadelphia 2006 pp 1064 1073 but via a different rounding technique that requires the use of a so called xos oracle the randomized rounding techniques that we use are oblivious in the sense that they only use the primal solution to the linear program relaxation but have no access to the actual utility functions of the players

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

polynomial time approximation schemes for euclidean traveling salesman and other geometric problems we present a polynomial time approximation scheme for euclidean tsp in fixed dimensions for every fixed c 1 and given any n nodes in r 2 a randomized version of the scheme finds a 1 1 c approximation to the optimum traveling salesman tour in o n log n o c time when the nodes are in r d the running time increases to o n log n o d c d 1 for every fixed c d the running time is n poly log n that is nearly linear in n the algorithmm can be derandomized but this increases the running time by a factor o n d the previous best approximation algorithm for the problem due to christofides achieves a 3 2 aproximation in polynomial time we also give similar approximation schemes for some other np hard euclidean problems minimum steiner tree k tsp and k mst the running times of the algorithm for k tsp and k mst involve an additional multiplicative factor k the previous best approximation algorithms for all these problems achieved a constant factor approximation we also give efficient approximation schemes for euclidean min cost matching a problem that can be solved exactly in polynomial time all our algorithms also work with almost no modification when distance is measured using any geometric norm such as e p for p 1 or other minkowski norms they also have simple parallel i e nc implementations

the pcp theorem by gap amplification the pcp theorem arora and safra 1998 arora et al 1998 says that every language in np has a witness format that can be checked probabilistically by reading only a constant number of bits from the proof the celebrated equivalence of this theorem and inapproximability of certain optimization problems due to feige et al 1996 has placed the pcp theorem at the heart of the area of inapproximability in this work we present a new proof of the pcp theorem that draws on this equivalence we give a combinatorial proof for the np hardness of approximating a certain constraint satisfaction problem which can then be reinterpreted to yield the pcp theorem our approach is to consider the unsat value of a constraint system which is the smallest fraction of unsatisfied constraints ranging over all possible assignments for the underlying variables we describe a new combinatorial amplification transformation that doubles the unsat value of a constraint system with only a linear blowup in the size of the system the amplification step causes an increase in alphabet size that is corrected by a standard pcp composition step iterative application of these two steps yields a proof for the pcp theorem the amplification lemma relies on a new notion of graph powering that can be applied to systems of binary constraints this powering amplifies the unsat value of a constraint system provided that the underlying graph structure is an expander we also extend our amplification lemma towards construction of assignment testers alternatively pcps of proximity which are slightly stronger objects than pcps we then construct pcps and locally testable codes whose length is linear up to a polylog factor and whose correctness can be probabilistically verified by making a constant number of queries namely we prove sat pcp 1 2 1 log2 n poly log n o 1

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

generalized xor non locality games with graph description on a square lattice we propose a family of non locality unique games for 2 parties based on a square lattice on an arbitrary surface we show that due to structural similarities with error correction codes of kitaev for fault tolerant quantum computation the games have classical values computable in polynomial time for d 2 measurement outcomes by representing games in their graph form for arbitrary d and underlying surface we provide their classification into equivalence classes with respect to relabeling of measurement outcomes for a selected set of permutations which define the winning conditions a case study of games with periodic boundary conditions is presented in order to verify their impact on classical and quantum values of the family of games it suggests that quantum values suffer independently from presence of different winning conditions that can be imposed due to periodicity as long as no local restrictions are in place

two query pcp with subconstant error we show that the np complete language 3sat has a pcp verifier that makes two queries to a proof of almost linear size and achieves subconstant probability of error eeo 1 the verifier performs only projection tests meaning that the answer to the first query determines at most one accepting answer to the second query the number of bits representing a symbol in the proof depends only on the error e previously by the parallel repetition theorem there were pcp theorems with two query projection tests but only arbitrarily small constant error and polynomial size there were also pcp theorems with subconstant error and almost linear size but a constant number of queries that is larger than 2 as a corollary we obtain a host of new results in particular our theorem improves many of the hardness of approximation results that are proved using the parallel repetition theorem a partial list includes the following 1 3sat cannot be efficiently approximated to within a factor of 7 8po 1 unless penp this holds even under almost linear reductions previously the best known np hardness factor was 7 8pe for any constant e 0 under polynomial reductions hastad 2 3lin cannot be efficiently approximated to within a factor of 1 2po 1 unless penp this holds even under almost linear reductions previously the best known np hardness factor was 1 2pe for any constant e 0 under polynomial reductions hastad 3 a pcp theorem with amortized query complexity 1 p o 1 and amortized free bit complexity o 1 previously the best known amortized query complexity and free bit complexity were 1pe and e respectively for any constant e 0 samorodnitsky and trevisan one of the new ideas that we use is a new technique for doing the composition step in the classical proof of the pcp theorem without increasing the number of queries to the proof we formalize this as a composition of new objects that we call locally decode reject codes ldrc the notion of ldrc was implicit in several previous works and we make it explicit in this work we believe that the formulation of ldrcs and their construction are of independent interest