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

orthogonalized generalized iso geometric analysis ogiga and its applications to problems of fracture mechanics abstract ill conditioning of system matrices is one of the drawbacks of generalized extended isogeometric analysis giga xiga which results in increase in computational cost in the case of iterative solvers or erroneous results in the case of direct solvers linear dependency between the basis functions which can be measured in terms of smallest angle among all the bases is one of the attributes of ill conditioning in xiga in the present study a simplified method to find the smallest angle among all the bases is introduced also to reduce the problem of ill conditioning orthogonalized generalized isogeometric analysis ogiga is proposed in which the standard and the corresponding enriched basis functions are orthogonalized in order to study the performance of present method various fracture mechanics problems like 2 d homogenous cracks bi material interface cracks inclined cracks and 3 d crack problems are chosen the effect of crack position and the radius of enrichment zone on accuracy scaled condition number and smallest angle among all the bases is studied the results obtained from the proposed ogiga method are compared with results obtained from standard xiga and xiga with enrichment function orthogonalization oe xiga agathos et al 2019 in all the cases considered the conditioning of system matrices is significantly improved by employing present methodology and it can be easily extended to wide variety of problems

moderately exponential approximation for makespan minimization on related machines we consider in this paper the classical q c m a x scheduling problem the objective is to minimize the maximum completion time called makespan while scheduling independent jobs in parallel on machines that have different speeds while several approximation schemes has been proposed and in particular a recent eptas jansen 2010 12 the current best direct algorithm i e an algorithm specifically designed for reaching a given approximation ratio is still due to chen 1991 4 with a 1 382 ratio our objective in this work is not to provide yet another improvement of the asymptotic dependencies in email protected ensuring a email protected ratio but to design faster direct algorithms by targeting respectively 43 and 54 ratios indeed instantiating any of the existing approximation scheme for e 13 respectively 14 leads to polynomial complexities but not to practical algorithms because of the hidden large constants in the computational complexity thus our approach focuses on a moderately exponential algorithm and provides a email protected dual approximation algorithm running in o m 1 1 3 e l o g 3 b 1 m n where m is the number of machines n the number of jobs b an integer lower than m depending on the instance this result is obtained through an oracle framework where the algorithm guesses possible answers from an oracle the terseness of the answers points out the critical information needed while solving any instance such an approach leads to a better comprehension of the problem similarly we obtain the same kind of results for a email protected ratio moreover the proposed techniques seem promising for tackling classical specific cases like scheduling on identical machines as the complexity becomes a low degree polynomial when the speeds are non arbitrary

the price of envy freeness in machine scheduling we consider k envy free assignments for scheduling problems in which the completion time of each machine is not k times larger than the one she could achieve by getting the jobs of another machine for a given factor k 1 we introduce and investigate the notion of price of k envy freeness defined as the ratio between the makespan of the best k envy free assignment and that of an optimal allocation achievable without envy freeness constraints we provide exact or asymptotically tight bounds on the price of k envy freeness for all the basic scheduling models that is unrelated related and identical machines moreover we show how to efficiently compute such allocations with a worsening multiplicative factor being at most the best approximation ratio for the minimum makespan problem guaranteed by a polynomial time algorithm for each specific model finally we extend our results to the case of restricted assignments and to the objective of minimizing the sum of the completion times of all the machines

a polynomial time approximation scheme for the multiple knapsack problem the multiple knapsack problem mkp is a natural and well known generalization of the single knapsack problem and is defined as follows we are given a set of n items and m bins knapsacks such that each item i has a profit p i and a size s i and each bin j has a capacity c j the goal is to find a subset of items of maximum profit such that they have a feasible packing in the bins mkp is a special case of the generalized assignment problem gap where the profit and the size of an item can vary based on the specific bin that it is assigned to gap is apx hard and a 2 approximation for it is implicit in the work of shmoys and tardos math program a 62 1993 pp 461 474 and thus far this was also the best known approximation for mkp the main result of this paper is a polynomial time approximation scheme ptas for mkp apart from its inherent theoretical interest as a common generalization of the well studied knapsack and bin packing problems it appears to be the strongest special case of gap that is not apx hard we substantiate this by showing that slight generalizations of mkp are apx hard thus our results help demarcate the boundary at which instances of gap become apx hard an interesting aspect of our approach is a ptas preserving reduction from an arbitrary instance of mkp to an instance with o log n distinct sizes and profits

inexact schwarz algebraic multigrid preconditioners for crack problems modeled by extended finite element methods summary traditional algebraic multigrid amg preconditioners are not well suited for crack problems modeled by extended finite element methods xfem this is mainly because of the unique xfem formulations which embed discontinuous fields in the linear system by addition of special degrees of freedom these degrees of freedom are not properly handled by the amg coarsening process and lead to slow convergence in this paper we proposed a simple domain decomposition approach that retains the amg advantages on well behaved domains by avoiding the coarsening of enriched degrees of freedom the idea was to employ a multiplicative schwarz preconditioner where the physical domain was partitioned into healthy or unfractured and cracked subdomains first the healthy subdomain containing only standard degrees of freedom was solved approximately by one amg v cycle followed by concurrent direct solves of cracked subdomains this strategy alleviated the need to redesign special amg coarsening strategies that can handle xfem discretizations numerical examples on various crack problems clearly illustrated the superior performance of this approach over a brute force amg preconditioner applied to the linear system copyright 2011 john wiley sons ltd

algebraic multigrid amg experiences and comparisons algebraic multigrid methods are designed for the solution of sparse linear systems of equations using multigrid principles in contrast to standard multigrid methods amg does not take advantage of the origin of a particular system of equations at hand nor does it exploit any underlying geometrical situation fully automatically and based solely on algebraic information contained in the given matrix amg constructs a sequence of grids and corresponding operators a special amg algorithm will be presented for a wide range of problems including certain problems which do not have a continuous background this algorithm yields an iterative method which exhibits a convergence behavior typical for multigrid methods

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

stable gfem sgfem improved conditioning and accuracy of gfem xfem for three dimensional fracture mechanics abstract in this paper we present an extension of the stable generalized fem sgfem for 3 d fracture mechanics problems numerical experiments show that the use of available singular enrichment bases derived from the 2 d asymptotic crack tip solution leads to a severe loss of accuracy in the sgfem an enrichment scheme based on singular bases and linear polynomials is shown to recover the optimal convergence of the sgfem the proposed sgfem for 3 d fractures delivers the same rate of growth in condition number as the standard fem for proper choice of singular enrichment functions the accuracy and conditioning obtained with the sgfem is compared with the generalized fem gfem for different types of singular enrichment bases a fully 3 d fracture mechanics problem is considered to highlight issues that arise only in 3 d problems the convergence of stress intensity factors sifs extracted from gfem and sgfem solutions and the effect of the size of enrichment sub domains on the accuracy of extracted sifs are also studied it is shown that the accuracy of sifs may deteriorate with mesh refinement when the topological enrichment strategy is adopted the proposed sgfem delivers much better accuracy than the gfem for both geometrical and topological enrichment strategies

a method of finite element tearing and interconnecting and its parallel solution algorithm a novel domain decomposition approach for the parallel finite element solution of equilibrium equations is presented the spatial domain is partitioned into a set of totally disconnected subdomains each assigned to an individual processor lagrange multipliers are introduced to enforce compatibility at the interface nodes in the static case each floating subdomain induces a local singularity that is resolved in two phases first the rigid body modes are eliminated in parallel from each local problem and a direct scheme is applied concurrently to all subdomains in order to recover each partial local solution next the contributions of these modes are related to the lagrange multipliers through an orthogonality condition a parallel conjugate projected gradient algorithm is developed for the solution of the coupled system of local rigid modes components and lagrange multipliers which completes the solution of the problem when implemented on local memory multiprocessors this proposed method of tearing and interconnecting requires less interprocessor communications than the classical method of substructuring it is also suitable for parallel vector computers with shared memory moreover unlike parallel direct solvers it exhibits a degree of parallelism that is not limited by the bandwidth of the finite element system of equations

adaptive modeling of damage growth using a coupled fem bem approach summary we propose a coupled boundary element method bem and a finite element method fem for modelling localized damage growth in structures bem offers the flexibility of modelling large domains efficiently while the non linear damage growth is accurately accounted by a local fem mesh an integral type nonlocal continuum damage mechanics with adapting fem mesh is used to model multiple damage zones and follow their propagation in the structure strong form coupling bem hosted is achieved using lagrange multipliers because the non linearity is isolated in the fem part of the system of equations the system size is reduced using schur complement approach then the solution is obtained by a monolithic newton method that is used to solve both domains simultaneously the coupled bem fem approach is verified by a set of convergence studies where the reference solution is obtained by a fine fem in addition the method is applied to multiple fractures growth benchmark problems and shows good agreement with the literature copyright 2015 john wiley sons ltd

a truthful constant approximation for maximizing the minimum load on related machines abstract designing truthful mechanisms for scheduling on related machines is a very important problem in single parameter mechanism design we consider the covering objective that is we are interested in maximizing the minimum completion time of a machine this problem falls into the class of problems where the optimal allocation can be truthfully implemented a major open issue for this class is whether truthfulness affects the polynomial time implementation we provide the first constant factor approximation for deterministic truthful mechanisms in particular we come up with an approximation guarantee of 2 e significantly improving on the previous upper bound of min m 2 e s m s 1

approximation algorithms for scheduling unrelated parallel machines we consider the following scheduling problem there arem parallel machines andn independent jobs each job is to be assigned to one of the machines the processing of jobj on machinei requires timep ij the objective is to find a schedule that minimizes the makespan our main result is a polynomial algorithm which constructs a schedule that is guaranteed to be no longer than twice the optimum we also present a polynomial approximation scheme for the case that the number of machines is fixed both approximation results are corollaries of a theorem about the relationship of a class of integer programming problems and their linear programming relaxations in particular we give a polynomial method to round the fractional extreme points of the linear program to integral points that nearly satisfy the constraints in contrast to our main result we prove that no polynomial algorithm can achieve a worst case ratio less than 3 2 unlessp np we finally obtain a complexity classification for all special cases with a fixed number of processing times

the multigrid method in solid mechanics part i algorithm description and behaviour a multigrid algorithm is described that can be used to obtain the finite element solution of linear elastic solid mechanics problems the method is applied to some two dimensional problems to evaluate its strengths and weaknesses extensive studies are made to determine the convergence behaviour of the method in general this depends on many factors the number of degrees of freedom in the discretization characteristics of the algorithm poisson s ratio when it is close to 0 5 the amount of bending deformation in the problem under consideration and the degree of non uniformity in the mesh only certain values of the multigrid parameters allow a converged solution to be obtained with a computational effort proportional to the number of degrees of freedom these values include the optimum ones i e those that lead to convergence with the least computational effort the constant of proportionality is only independent of the number of degrees of freedom and still depends on the other factors listed above

approximation schemes for the restricted shortest path problem this note contains two fully polynomial approximation schemes for the shortest path problem with an additional constraint the main difficulty in constructing such algorithms arises since no trivial lower and upper bounds on the solution value whose ratio is polynomially bounded are known in spite of this difficulty one of the algorithms presented here is strongly polynomial applications to other problems are also discussed

numerical experiments using hierarchical finite element method for nonlinear heat conduction in plates in this paper we consider a nonlinear hierarchical finite element method for heat conduction problems over two or three dimensional plates problems considered are nonlinear because the heat conductivity parameter depends upon the temperature itself this paper explores a new technique recently proposed by the first author which transforms a nonlinear parabolic problem to a linear problem at the discrete level we present several numerical examples which demonstrate the efficiency of the current technique

superconvergence of the iterated galerkin methods for hammerstein equations in this paper the well known iterated galerkin method and iterated galerkin kantorovich regularization method for approximating the solution of fredholm integral equations of the second kind are generalized to hammerstein equations with smooth and weakly singular kernels the order of convergence of the galerkin method and those of superconvergence of the iterated methods are analyzed numerical examples are presented to illustrate the superconvergence of the iterated galerkin approximation for hammerstein equations with weakly singular kernels

property testing and its connection to learning and approximation in this paper we consider the question of determining whether a function f has property p or is e far from any function with property p a property testing algorithm is given a sample of the value of f on instances drawn according to some distribution in some cases it is also allowed to query f on instances of its choice we study this question for different properties and establish some connections to problems in learning theory and approximation in particular we focus our attention on testing graph properties given access to a graph g in the form of being able to query whether an edge exists or not between a pair of vertices we devise algorithms to test whether the underlying graph has properties such as being bipartite k colorable or having a p clique clique of density p with respect to the vertex set our graph property testing algorithms are probabilistic and make assertions that are correct with high probability while making a number of queries that is independent of the size of the graph moreover the property testing algorithms can be used to efficiently i e in time linear in the number of vertices construct partitions of the graph that correspond to the property being tested if it holds for the input graph

performance guarantees of local search for minsum scheduling problems we study the worst case performance guarantee of locally optimal solutions for the problem of minimizing the total weighted and unweighted completion time on parallel machine environments our method makes use of a mapping that maps a schedule into an inner product space so that the norm of the mapping is closely related to the cost of the schedule we apply the method to study the most basic local search heuristics for scheduling namely jump and swap and establish their worst case performance in the case of unrelated restricted related and restricted identical machines

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

the rs method for material failure simulations a new method for propagating arbitrary failure modes is presented arbitrary failure modes are resolved on a refined local patch of elements and then embedded into the coarse grid using partition of unity method strong discontinuities are propagated by means of element erosion in the superimposed patch of elements only the method coined as the rs version of the finite element method or reduced order s method has been integrated in abaqus and verified on several test problems copyright 2007 john wiley sons ltd

optimal convergence properties of the feti domain decomposition method abstract the finite element tearing and interconnecting feti method is a practical and efficient domain decomposition dd algorithm for the solution of self adjoint elliptic partial differential equations for large scale structural problems discretized with shell and beam elements this method was found to outperform popular iterative algorithms and direct solvers on both serial and parallel computers and to compare favorably with leading dd methods in this paper we discuss some numerical properties of the feti method that were not addressed before in particular we show that the mathematical treatment of the floating subdomains and the specific conjugate projected gradient algorithm that characterize the feti method are equivalent to the construction and solution of a coarse problem that propagates the error globally accelerates convergence and ensures a performance that is independent of the number of subdomains we also show that when the interface problem is optimally preconditioned and the mesh is partitioned into well structured subdomains with good aspect ratios the performance of the feti method is also independent of the mesh size however we also argue that the feti and other leading dd methods for unstructured problems lose in practice these scalability properties when the mesh contains junctures with rotational degrees of freedom or the decomposition is irregular and characterized by arbitrary subdomain aspect ratios finally we report that for realistic problems optimal preconditioners are not necessarily computationally efficient and can be outperformed by non optimal ones

error estimates and automatic time step control for nonlinear parabolic problems i in this note we extend results by one of the authors on time discretization error estimates and related automatic time step control for stiff ordinary differential equations to the case of a nonlinear parabolic problem the method for time discretization is the so called discontinuous galerkin method based on using piecewise polynomials of degree q geqq 0 we consider in this note the case q 0 corresponding to a variant of the backward euler method we prove a new almost optimal error estimate and present a related new algorithm for automatic time step control this algorithm is very simple but yet is efficient and gives control of the global error

numerical solution of parabolic integro differential equations by the discontinuous galerkin method the numerical solution of a parabolic equation with memory is considered the equation is first discretized in time by means of the discontinuous galerkin method with piecewise constant or piecewise linear approximating functions the analysis presented allows variable time steps which as will be shown call then efficiently be selected to match singularities in the solution induced by singularities in the kernel of the memory term or by nonsmooth initial data the combination with finite element discretizat in space is also studied

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

a discontinuous galerkin method for parabolic problems with modified hp finite element approximation technique abstract a recent paper hideaki kaneko kim s bey gene j w hou discontinuous galerkin finite element method for parabolic problems preprint november 2000 nasa is generalized to a case where the spatial region is taken in r 3 the region is assumed to be a thin body such as a panel on the wing or fuselage of an aerospace vehicle the traditional h as well as hp finite element methods are applied to the surface defined in the x y variables while through the thickness the technique of the p element is employed time and spatial discretization scheme developed in kaneko et al 2000 based upon an assumption of certain weak singularity of u t 2 is used to derive an optimal a priori error estimate for the current method

smoothed performance guarantees for local search we study popular local search and greedy algorithms for standard machine scheduling problems the performance guarantee of these algorithms is well understood but the worst case lower bounds seem somewhat contrived and it is questionable whether they arise in practical applications to find out how robust these bounds are we study the algorithms in the framework of smoothed analysis in which instances are subject to some degree of random noise while the lower bounds for all scheduling variants with restricted machines are rather robust we find out that the bounds are fragile for unrestricted machines in particular we show that the smoothed performance guarantee of the jump and the lex jump algorithm are in contrast to the worst case independent of the number of machines they are theta phi and theta log phi respectively where 1 phi is a parameter measuring the magnitude of the perturbation the latter immediately implies that also the smoothed price of anarchy is theta log phi for routing games on parallel links additionally we show that for unrestricted machines also the greedy list scheduling algorithm has an approximation guarantee of theta log phi

truthful mechanism design via correlated tree rounding a powerful algorithmic technique for truthful mechanism design is the maximal in distributional range midr paradigm unfortunately many such algorithms use heavy algorithmic machinery e g the ellipsoid method and approximate solution of convex programs in this paper we present a correlated rounding technique for designing mechanisms that are truthful in expectation it is elementary and can be implemented quickly the main property we rely on is that the domain offers fractional optimum solutions with a tree structure in auctions based on the generalized assignment problem each bidder has a publicly known knapsack constraint that captures the subsets of items that are of value to him he has a private valuation for each item and strives to maximize the value of assigned items minus payment for this domain we design a truthful 2 approximate midr mechanism for social welfare maximization it avoids using the ellipsoid method or convex programming in contrast to some previous work our mechanism achieves exact truthfulness in restricted related scheduling with selfish machines each job comes with a public weight and it must be assigned to a machine from a public job specific subset each machine has a private speed and strives to maximize payments minus workload of jobs assigned to it here we design a mechanism for makespan minimization this is a single parameter domain but the approximation status of the optimization problem is similar to unrelated machine scheduling the best known algorithm obtains a non truthful 2 approximation for unrelated machines and there is 1 5 hardness our mechanism matches this bound with a truthful 2 approximation

strongly stable generalized finite element method application to interface problems abstract in this paper we present the theoretical justification for the stable generalized finite element method sgfem when applied to smooth interface problems we prove that a generalized finite element method gfem is stable i e its conditioning is a not worse than that of the finite element method and b robust with respect to the mesh if the enrichment space of the gfem satisfies two axioms we provide element based sufficient conditions on the enrichment space that will guarantee that the axioms are satisfied we show that the enrichment space of the gfem used to address the interface problems in 2d satisfies the sufficient conditions and thus the two axioms yielding an sgfem the idea of a strongly stable gfem associated with one of the two axioms has been introduced strong stability of the gfem is important for designing efficient iterative methods to solve the underlying linear system a proof of the optimal convergence of the gfem for the interface problem has also been derived in this paper the work in this paper is the continuation of kergrene et al 2016 where the stability of the gfem when applied to interface problems was established through numerical experiments the numerical results in kergrene et al 2016 indicated that the gfem is indeed strongly stable

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

on the stability of the discontinuous galerkin method for the heat equation this paper analyzes stability properties of a class of discontinuous galerkin methods for the heat equation it is shown that the finite element projection associated with these methods is stable with respect to a mesh dependent norm a discrete analogue of the space time l 2 norm optimal order regularity error bounds in l 2 0 t l 2 omega are derived

robustness in stable generalized finite element methods sgfem applied to poisson problems with crack singularities abstract in this paper we study the performance of the generalized finite element method gfem applied to the poisson problem with crack singularities recently a gfem with modified heaviside enrichments was proposed in gupta et al 2013 to approximate the solution of a 2d elasticity problem with a crack it was shown that the gfem is indeed a stable gfem sgfem i e it yields the optimal order of convergence and its conditioning is not worse than that of the standard finite element method fem however the robustness of the conditioning of the gfem with respect to the position of the mesh relative to the crack was not addressed in gupta et al 2013 in this paper we observed that the conditioning of the gfem with the enrichments used in gupta et al 2013 is not robust when applied to approximate the solution of the poisson problem moreover the order of convergence may not be optimal when the position of the mesh is changed with respect to the crack interfaces we proposed using additional singular enrichments at the nodes close to the crack near the crack tip we proved that the gfem with enrichments proposed in this paper yields optimal order of convergence irrespective of the position of the mesh moreover with a local orthogonalization procedure we have shown through numerical experiments that the conditioning of this gfem is not worse than that of the standard fem and the conditioning is robust with respect to the position of the mesh thus the gfem with the enrichments suggested in this paper is indeed an sgfem when applied to a poisson problem with the crack singularities

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

a unified approach to truthful scheduling on related machines we present a unified framework for designing deterministic monotone polynomial time approximation schemes ptass for a wide class of scheduling problems on uniformly related machines this class includes among others minimizing the makespan maximizing the minimum load and minimizing the p norm of the machine loads vector previously this kind of result was only known for the makespan objective monotone algorithms have the property that an increase in the speed of a machine cannot decrease the amount of work assigned to it our results imply the existence of a truthful mechanism that can be implemented in polynomial time where the social goal is approximated within arbitrary precision

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

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

adaptive finite element methods for parabolic problems i a linear model problem this paper is the first part in a series of papers on adaptive finite element methods for parabolic problems in this paper an adaptive algorithm is presented and analyzed for choosing the space a

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

a robust preconditioning technique for the extended finite element method the extended finite element method enhances the approximation properties of the finite element space by using additional enrichment functions but the resulting stiffness matrices can become ill conditioned in that case iterative solvers need a large number of iterations to obtain an acceptable solution in this paper a procedure is described to obtain stiffness matrices whose condition number is close to the one of the finite element matrices without any enrichments a domain decomposition is employed and the algorithm is very well suited for parallel computations the method was tested in numerical experiments to show its effectiveness the experiments have been conducted for structures containing cracks and material interfaces we show that the corresponding enrichments can result in arbitrarily ill conditioned matrices the method proposed here however provides well conditioned matrices and can be applied to any sort of enrichment the complexity of this approach and its relation to the domain decomposition is discussed computation times have been measured for a structure containing multiple cracks for this structure the computation times could be decreased by a factor of 2

on line bin stretching we are given a sequence of items that can be packed into m unit size bins in the classical bin packing problem we fix the size of the bins and try to pack the items in the minimum number of such bins in contrast in the bin stretching problem we fix the number of bins and try to pack the items while stretching the size of the bins as least as possible we present two on line algorithms for the bin stretching problem that guarantee a stretching factor of 5 3 for any number m of bins we then combine the two algorithms and design an algorithm whose stretching factor is 1 625 for any m the analysis for the performance of this algorithm is tight the best lower bound for any algorithm is 4 3 for any m 2 we note that the bin stretching problem is also equivalent to the classical scheduling load balancing problem in which the value of the makespan maximum load is known in advance copyright 2001 elsevier science b v

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

the design and analysis of the generalized finite element method abstract in this paper we introduce the generalized finite element method gfem as a combination of the classical finite element method fem and the partition of unity method pum the standard finite element spaces are augmented by adding special functions which reflect the known information about the boundary value problem and the input data e g the singular functions obtained from the local asymptotic expansion of the exact solution in the neighborhood of a corner point etc the special functions are multiplied with the partition of unity corresponding to the standard linear vertex shape functions and pasted together with the existing finite element basis to construct an augmented conforming finite element space in this way the local approximability afforded by the special functions is included in the approximation while maintaining the existing infrastructure of finite element codes the major features of the gfem are 1 the essential boundary conditions can be imposed exactly as in the standard fem unlike other partition of unity based methods where this is a major issue 2 the accuracy of the numerical integration of the entries of the stiffness matrix and load vector is controlled adaptively so that the errors in integration of the special functions do not affect the accuracy of the constructed approximation this issue also has not been sufficiently addressed in other implementations of partition of unity or meshless methods and 3 linear dependencies in the system of equations are resolved by employing an easy modification of the direct linear solver the power of the gfem for solving problems in domains with complex geometry with less error and less computer resources than the standard fem is illustrated by numerical examples

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

the multigrid method in solid mechanics part ii practical applications no proportionality is observed between the total number of operations to convergence and the problem size despite such behaviour the multigrid algorithm proves to be an effective matrix equation solver for solid mechanics problems it is orders of magnitude faster than a direct factorization method and yields converged solutions several times faster than the jacobi preconditioned conjugate gradient method

error estimates with smooth and nonsmooth data for a finite element method for the cahn hilliard equation a finite element method for the cahn hilliard equation a semilinear parabolic equation of fourth order is analyzed both in a spatially semidiscrete case and in a completely discrete case based on the backward euler method error bounds of optimal order over a finite time interval are obtained for solutions with smooth and nonsmooth initial data a detailed study of the regularity of the exact solution is included the analysis is based on local lipschitz conditions for the nonlinearity with respect to sobolev norms and the existence of a ljapunov functional for the exact and the discretized equations is essential a result concerning the convergence of the attractor of the corresponding approximate nonlinear semigroup upper semicontinuity with respect to the discretization parameters is obtained as a simple application of the nonsmooth data error estimate

improved approximation algorithms for scheduling parallel jobs on identical clusters the multiple cluster scheduling problem corresponds to minimizing the maximum completion time makespan of a set of n parallel rigid and non preemptive jobs submitted to n identical clusters it cannot be approximated with a ratio better than 2 unless p np we present in this paper the methodology that encompasses several existing results 1 2 we detail first how to apply it for obtaining a 5 2 approximation then we use it to provide a new 7 3 approximation running in o log n h max n n log n where h max is the processing time of the longest job finally we apply it to a restriction of the problem to jobs of limited size leading to a 2 approximation which is the best possible ratio since the restriction remains 2 inapproximable

a simple first order well conditioned and optimally convergent generalized extended fem for two and three dimensional linear elastic fracture mechanics abstract this paper proposes a first order generalized extended finite element method g xfem for 2 d and 3 d linear elastic fracture mechanics problems the conditioning of the method is of the same order as the standard fem and it is robust with respect to the position of the mesh relative to 2 d or 3 d fractures the method achieves an optimal rate of convergence in energy norm even when the solution of the problem in the neighborhood of the fracture front is not contained in the spaces spanned by the adopted singular enrichments control of conditioning associated with heaviside enrichment functions is achieved by shifting them by their nodal values combined with a diagonal pre conditioner in the case of singular enrichment functions a simple extension of the enrichment shifting concept denoted as discontinuous shifting is adopted these enrichment modifications are space preserving the solution space spanned by the modified enrichments is the same as the space spanned by the unmodified ones this is often not the case for conditioning control strategies proposed in the literature the computational cost of these enrichment modifications is negligible and their implementation in existing g xfem software is straightforward the optimal convergence well conditioning and robustness of the method are numerically illustrated with the aid of representative 2 d and 3 d problems including the case of a non planar fracture with a curved fracture front

an algebraic multigrid approach to solve extended finite element method based fracture problems summary this article proposes an algebraic multigrid amg approach to solve linear systems arising from applications where strong discontinuities are modeled by the extended finite element method the application of amg methods promises optimal scalability for solving large linear systems however the straightforward or black box use of existing amg techniques for extended finite element method problems is often problematic in this paper we highlight the reasons for this behavior and propose a relatively simple adaptation that allows one to leverage existing amg software mostly unchanged numerical tests demonstrate that optimal iterative convergence rates can be attained that are comparable with amg convergence rates associated with linear systems for standard finite element approximations without discontinuities published 2012 this article is a us government work and is in the public domain in the usa

minimizing average completion time in the presence of release dates a natural and basic problem in scheduling theory is to provide good average quality of service to a stream of jobs that arrive over time in this paper we consider the problem of schedulingn jobs that are released over time in order to minimize the average completion time of the set of jobs in contrast to the problem of minimizing average completion time when all jobs are available at time 0 all the problems that we consider are np hard and essentially nothing was known about constructing good approximations in polynomial time we give the first constant factor approximation algorithms for several variants of the single and parallel machine models many of the algorithms are based on interesting algorithmic and structural relationships between preemptive and nonpreemptive schedules and linear programming relaxations of both many of the algorithms generalize to the minimization of averageweighted completion time as well 1998 the mathematical programming society inc published by elsevier science b v

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

direct estimation of generalized stress intensity factors using a three scale concurrent multigrid x fem a concurrent multigrid method is devised for the direct estimation of stress intensity factors and higher order coefficients of the elastic crack tip asymptotic field the proposed method bridges three characteristic length scales that can be present in fracture mechanics the structure the crack and the singularity at the crack tip for each of them a relevant model is proposed first a truncated analytical reduced order model based on williams expansion is used to describe the singularity at the tip then it is coupled with a standard extended finite element fe method model which is known to be suitable for the scale of the crack a multigrid solver finally bridges the scale of the crack to that of the structure for which a standard fe model is often accurate enough dedicated coupling algorithms are presented and the effects of their parameters are discussed the efficiency and accuracy of this new approach are exemplified using three benchmarks

time discretization of parabolic problems by the hp version of the discontinuous galerkin finite element method the discontinuous galerkin finite element method dgfem for the time discretization of parabolic problems is analyzed in the context of the hp version of the galerkin method error bounds which are explicit in the time steps as well as in the approximation orders are derived and it is shown that the hp dgfem gives spectral convergence in problems with smooth time dependence in conjunction with geometric time partitions it is proved that the hp dgfem results in exponential rates of convergence for piecewise analytic solutions exhibiting singularities induced by incompatible initial data or piecewise analytic forcing terms for the h version dgfem algebraically graded time partitions are determined that give the optimal algebraic convergence rates a fully discrete hp scheme is discussed exemplarily for the heat equation the use of certain mesh design principles for the spatial discretizations yields exponential rates of convergence in time and space numerical examples confirm the theoretical results

discontinuous galerkin finite element method for parabolic problems in this paper we develop a time and its corresponding spatial discretization scheme based upon the assumption of a certain weak singularity of u t t l 2 w email protected u email protected 2 for the discontinuous galerkin finite element method for one dimensional parabolic problems optimal convergence rates in both time and spatial variables are obtained a discussion of automatic time step control method is also included

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

adaptive finite element methods for parabolic problems iv nonlinear problems we extend our program on adaptive finite element methods for parabolic problems to a class of nonlinear scalar problems we prove a posteriori error estimates design corresponding adaptive algorithms and present some numerical results

robust and scalable domain decomposition solvers for unfitted finite element methods abstract unfitted finite element methods e g extended finite element techniques or the so called finite cell method have a great potential for large scale simulations since they avoid the generation of body fitted meshes and the use of graph partitioning techniques two main bottlenecks for problems with non trivial geometries however the linear systems that arise from these discretizations can be much more ill conditioned due to the so called small cut cell problem the state of the art approach is to rely on sparse direct methods which have quadratic complexity and are thus not well suited for large scale simulations in order to solve this situation in this work we investigate the use of domain decomposition preconditioners balancing domain decomposition by constraints for unfitted methods we observe that a straightforward application of these preconditioners to the unfitted case has a very poor behavior as a result we propose a customization of the classical bddc methods based on the stiffness weighting operator and an improved definition of the coarse degrees of freedom in the definition of the preconditioner these changes lead to a robust and algorithmically scalable solver able to deal with unfitted grids a complete set of complex 3d numerical experiments shows the good performance of the proposed preconditioners

convergence of algebraic multigrid based on smoothed aggregation we prove aconvergence estimate for the algebraic multigrid method with prolongations defined by aggregation using zero energy modes followed by a smoothing the method input is the problem matrix and a matrix of the zero energy modes the estimate depends only polylogarithmically on the mesh size and requires only a weak approximation property for the aggregates which can be a priori verified computationally construction of the prolongator in the case of a general second order system is described and the assumptions of the theorem are verified for a scalar problem discretized by linear conforming finite elements

an efficient polynomial time approximation scheme for load balancing on uniformly related machines we consider basic problems of non preemptive scheduling on uniformly related machines for a given schedule defined by a partition of the jobs into m subsets corresponding to the m machines c i c i denotes the completion time of machine i our goal is to find a schedule that minimizes or maximizes sum i 1 m c i p i 1 m c i p for a fixed value of p such that 0 1 p 1 the minimization problem is equivalent to the well known problem of minimizing the ell p l p norm of the vector of the completion times of the machines and for 0 p 1 0 p 1 the maximization problem is of interest our main result is an efficient polynomial time approximation scheme eptas for each one of these problems our schemes use a non standard application of the so called shifting technique we focus on the work total size of jobs assigned to each machine and introduce intervals of work that are forbidden these intervals are defined so that the resulting effect on the goal function is sufficiently small this allows the partition of the problem into sub problems with subsets of machines and jobs whose solutions are combined into the final solution using dynamic programming our results are the first eptas s for this natural class of load balancing problems

adaptive finite element methods for parabolic problems ii optimal error estimates in l l 2 and l l optimal error estimates are derived for a complete discretization of linear parabolic problems using space time finite elements the discretization is done first in time using the discontinuous galerkin method and then in space using the standard galerkin method the underlying partitions in time and space need not be quasi uniform and the partition in space may be changed from time step to time step the error bounds show in particular that the error may be controlled globally in time on a given tolerance level by controlling the discretization error on each individual time step on the same given level i e without error accumulation effects the derivation of the estimates is based on the orthogonality of the galerkin procedure and the use of strong stability estimates the particular and precise form of these error estimates makes it possible to design efficient adaptive methods with reliable automatic error control for parabolic problems in the norms under consideration

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

substructuring fe xfe approaches applied to three dimensional crack propagation two substructuring methods are investigated in order to allow for the use of the extended finite element method x fem within commercial finite element fe codes without need for modifying their kernel the global fe problem is decomposed into two subdomains the safe domain and the cracked domain based on the value of the level sets representing the crack the safe domain is treated by the host fe software while the cracked domain is treated by an independent xfe code the first substructuring method consists of calculating the schur matrix of a cracked super element with the xfe code the second technique introduces the finite element tearing and interconnecting method feti which ensures the compatibility of the displacements at the interface between the cracked and safe subdomains the stiffness matrices and nodal forces are provided by the xfe and fe codes for the cracked and safe subdomains respectively the solutions obtained with these two techniques are rigorously equivalent to those computed with the stand alone xfe code first the computational efficiency of the two approaches is demonstrated second a validation is proposed towards comparison with reference values of the stress intensity factors in simple 3d cracked geometries finally this contribution presents an application of the fe xfe feti method to the computation of the stress intensity factor induced by a crack inside a hydraulic cylinder under internal pressure

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

a polynomial approximation scheme for scheduling on uniform processors using the dual approximation approach in this paper we present a polynomial approximation scheme for the minimum makespan problem on uniform parallel processors more specifically the problem is to find a schedule for a set of independent jobs on a collection of machines of different speeds so that the last job to finish is completed as quickly as possible we give a family of polynomial time algorithms a such that a delivers a solution that is within a relative error of e of the optimum the technique employed is the dual approximation approach where infeasible but superoptimal solutions for a related dual problem are converted to the desired feasible but possibly suboptimal solution

adaptive s method for linear elastostatics abstract a methodology has been developed to simultaneously treat phenomena at several scales the key features of the method are i a detailed error diagnosis which identifies critical regions and ii a feedback procedure which overlays portions of the finite element mesh where unacceptable errors have been identified by local finite element mesh es and then subsequently improves the quality of the solution in the global and local response until the desired level of accuracy is achieved the method can be routinely used as a global local method and or as an adaptive technique which provides a measure of reliability of global and local computations numerical experiments for problems in homogeneous and heterogeneous laminated composites media are presented to validate the present formulation

an overlapping domain decomposition preconditioning method for monolithic solution of shear bands metals subjected to high strain rate impact often exhibit a sudden and profound drop in the material s load bearing capability a ductile failure phenomenon known as shear banding shear bands characterized as material instabilities are driven by shear heating and described as narrow regions that have sustained intense plastic deformation and high temperature rise this coupled thermo mechanical localization problem can be formulated as a nonlinear system with two balance equations momentum and energy and two constitutive laws for elasticity and plasticity in our formulation mixed finite elements are used to discretize the equations in space and an implicit finite difference scheme is used to advance the system in time where at every time step a newton type method is used to solve the system monolithically to that end a block jacobian matrix is formed analytically using g teaux derivatives the resulting jacobian is sparse nonsymmetric and its sparsity pattern and eigenvalue content vary with the different stages of shear bands formation consequently posing a significant challenge to iterative solvers to address this issue an overlapping domain decomposition preconditioner that takes into account the physics of shear bands and the domain in which it forms is proposed the key idea is to resolve the shear band domain accurately while only solving the remaining domain approximately with selective updates when necessary the preconditioner is formed on the basis of an additive schwartz method that is applied to a schur complement partition of the system the proposed preconditioner is implemented in parallel and tested on three different benchmark problems as compared against off the shelf solvers we study the effect of hh and kk refinement that are obtained from isogeometric discretizations of the system the overlapping strategy and its parallel performance excellent performance is demonstrated in serial and parallel on all benchmark examples

an aptas for generalized cost variable sized bin packing bin packing is a well known problem which has a large number of applications classical bin packing is a simple model in which all bins are identical in the bin packing problem with variable sized bins we are given a supply of a variety of sizes this latter model assumes however that the cost of a bin is always defined to be its exact size in this paper we study the more general problem where an available bin size is associated with a fixed cost which may be smaller or larger than its size the costs of different bin sizes are unrelated this generalized problem has various applications in storage and scheduling in order to generalize previous work we design new rounding and allocation methods our main result is an asymptotic polynomial time approximation scheme for the generalized problem

acceleration of the generalized global basis ggb method for nonlinear problems two heuristic strategies intended to enhance the performance of the generalized global basis ggb method h waisman j fish r s tuminaro j shadid the generalized global basis ggb method international journal for numerical methods in engineering 61 8 1243 1269 applied to nonlinear systems are presented the standard ggb accelerates a multigrid scheme by an additional coarse grid correction that filters out slowly converging modes this correction requires a potentially costly eigen calculation this paper considers reusing previously computed eigenspace information the ggb a scheme enriches the prolongation operator with new eigenvectors while the modified method mggb selectively reuses the same prolongation both methods use the criteria of principal angles between subspaces spanned between the previous and current prolongation operators numerical examples clearly indicate significant time savings in particular for the mggb scheme

error estimates and adaptive time step control for a class of one step methods for stiff ordinary differential equations we prove new optimal a priori error estimates for a class of implicit one step methods for stiff ordinary differential equations obtained by using the discontinuous galerkin method with piecewise polynomials of degree zero and one starting from these estimates we propose a new algorithm for automatic time step control and we discuss the relation between this algorithm and earlier algorithms implemented in packages for the numerical solution of stiff ordinary differential equations

afptas results for common variants of bin packing a new method for handling the small items we consider two well known natural variants of bin packing and show that these packing problems admit asymptotic fully polynomial time approximation schemes afptass in bin packing problems a set of one dimensional items of size at most 1 is to be assigned packed to subsets of sum at most 1 bins it has been known for a while that the most basic problem admits an afptas in this paper we develop new methods that allow us to extend this result to other variants of bin packing consisting of a family of two parameter bin packing problems we demonstrate the use of our methods by designing afptass for the following problems the first problem is bin packing with cardinality constraints where a parameter k is given such that a bin may contain up to k items the goal is to minimize the number of bins used the second problem is bin packing with rejection where every item has a rejection penalty associated with it an item needs to be either packed to a bin or rejected and the goal is to minimize the number of bins used and the total rejection penalty of unpacked items this resolves the complexity of two important variants of the bin packing problem our approximation schemes use a novel method for packing the small items this new method is the core of the improved running times of our schemes over the running times of the previous results which are only asymptotic polynomial time approximationschemes

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