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

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

interaction equations for short and long dispersive waves abstract we show the time local well posedness for a system of nonlinear dispersive equations for the water wave interaction formula it is shown that for any initial data u0 v0 hs r hs 1 2 r s 0 the solution for the above equation uniquely exists in a subset ofc t t hs c t t hs 1 2 and depends continuously on the data by virtue of a special structure of the nonlinear coupling the solution is stable under a singular limiting process

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

a new self stabilizing maximal matching algorithm the maximal matching problem has received considerable attention in the self stabilizing community previous work has given several self stabilizing algorithms that solve the problem for both the adversarial and the fair distributed daemon the sequential adversarial daemon as well as the synchronous daemon in the following we present a single self stabilizing algorithm for this problem that unites all of these algorithms in that it has the same time complexity as the previous best algorithms for the sequential adversarial the distributed fair and the synchronous daemon in addition the algorithm improves the previous best time complexities for the distributed adversarial daemon from o n 2 and o dm to o m where n is the number of processes m is the number of edges and d is the maximum degree in the graph

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

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

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

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

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

periodic and localized solutions of the long wave short wave resonance interaction equation in this paper we investigate the 2 1 dimensional long wave short wave resonance interaction lsri equation and show that it possess the painleve property we then solve the lsri equation using painleve truncation approach through which we are able to construct solution in terms of three arbitrary functions utilizing the arbitrary functions present in the solution we have generated a wide class of elliptic function periodic wave solutions and exponentially localized solutions such as dromions multidromions instantons multi instantons and bounded solitary wave solutions

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

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

the existence of a global attractor for the 2 1 dimensional long wave short wave resonance interaction equation abstract in this paper we prove the existence of a global attractor and an asymptotic smoothing effect of the solution for the 2 1 dimensional long wave short wave resonance interaction equation

homogenization in perforated domains and interior lipschitz estimates abstract we establish interior lipschitz estimates at the macroscopic scale for solutions to systems of linear elasticity with rapidly oscillating periodic coefficients and mixed boundary conditions in domains periodically perforated at a microscopic scale e by establishing h 1 convergence rates for such solutions the interior estimates are derived directly without the use of compactness via an argument presented in 3 that was adapted for elliptic equations in 2 and 11 as a consequence we derive a liouville type estimate for solutions to the systems of linear elasticity in unbounded periodically perforated domains

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

lipschitz estimates in almost periodic homogenization we establish uniform lipschitz estimates for second order elliptic systems in divergence form with rapidly oscillating almost periodic coefficients we give interior estimates as well as estimates up to the boundary in bounded c1 domains with either dirichlet or neumann data the main results extend those in the periodic setting due to avellaneda and lin for interior and dirichlet boundary estimates and later kenig lin and shen for the neumann boundary conditions in contrast to these papers our arguments are constructive and thus the constants are in principle computable and the results for the neumann conditions are new even in the periodic setting since we can treat nonsymmetric coefficients we also obtain uniform w1 p estimates 2015 wiley periodicals inc

the global solution of the 2 1 dimensional long wave short wave resonance interaction equation in this paper we prove the existence and uniqueness of the global smooth solution to the 2 1 dimensional long wave short wave resonance interaction equation

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

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

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

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

convergence for elliptic equations in periodic perforated domains abstract convergence for the solutions of elliptic equations in periodic perforated domains is concerned let denote the size ratio of the holes of a periodic perforated domain to the whole domain it is known that by energy method the gradient of the solutions of elliptic equations is bounded uniformly in in l 2 space also when approaches 0 the elliptic solutions converge to a solution of some simple homogenized elliptic equation in this work above results are extended to general w 1 p space for p 1 more precisely a uniform w 1 p estimate in for p 1 and a w 1 p convergence result for p n n 2 for the elliptic solutions in periodic perforated domains are derived here n is the dimension of the space domain one also notes that the l p norm of the second order derivatives of the elliptic solutions in general cannot be bounded uniformly in

free bits pcps and nonapproximability towards tight results this paper continues the investigation of the connection between probabilistically checkable proofs pcps and the approximability of np optimization problems the emphasis is on proving tight nonapproximability results via consideration of measures such as the free bit complexity and the amortized free bit complexity of proof systems the first part of the paper presents a collection of new proof systems based on a new error correcting code called the long code we provide a proof system that has amortized free bit complexity of 2 epsilon implying that approximating maxclique within n frac13 e and approximating the chromatic number within n frac15 e are hard assuming np neq corp for any e 0 we also derive the first explicit and reasonable constant hardness factors for min vertex cover msat 2 and max cut and we improve the hardness factor for msat 3 we note that our nonapproximability factors for maxsnp problems are appreciably close to the values known to be achievable by polynomial time algorithms finally we note a general approach to the derivation of strong nonapproximability results under which the problem reduces to the construction of certain gadgets the increasing strength of nonapproximability results obtained via the pcp connection motivates us to ask how far this can go and whether pcps are inherent in any way the second part of the paper addresses this the main result is a reversal of the connection due to feige et al fglss connection j acm 43 1996 pp 268 292 where the latter had shown how to translate proof systems for np into np hardness of approximation results for maxclique we show how any np hardness of approximation result for maxclique yields a proof system for np roughly our result says that for any constant f if maxclique is np hard to approximate within n 1 1 f then np subseteq overline fpcp log f the latter being the class of languages possessing proofs of logarithmic randomness and amortized free bit complexity f this suggests that pcps are inherent to obtaining nonapproximability results furthermore the tight relation suggests that reducing the amortized free bit complexity is necessary for improving the nonapproximability results for maxclique the third part of our paper initiates a systematic investigation of the properties of pcp and fpcp free pcp as a function of the following various parameters randomness query complexity free bit complexity amortized free bit complexity proof size etc we are particularly interested in triviality results which indicate which classes are not powerful enough to capture np we also distill the role of randomized reductions in this area and provide a variety of useful transformations between proof checking complexity classes

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

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

long time behavior for the weakly damped driven long wave short wave resonance equations abstract in this paper we consider the cauchy problem of the long wave short wave resonance equations by making use of a strichartz type inequality for the solutions decomposing suitably the solution semigroup into a decay parts and a more regular parts and ruling out the vanishing and dichotomy of the solutions we prove the existence of the global attractor and the asymptotic smoothing effect of the solutions