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

algebraic methods for interactive proof systems a new algebraic technique for the construction of interactive proof systems is presented our technique is used to prove that every language in the polynomial time hierarchy has an interactive proof system this technique played a pivotal role in the recent proofs that ip pspace 28 and that mip nexp 4

differential approximation of min sat max sat and related problems we present differential approximation results both positive and negative for optimal satisfiability optimal constraint satisfaction and some of the most popular restrictive versions of them as an important corollary we exhibit an interesting structural difference between the landscapes of approximability classes in standard and differential paradigms

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

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

spherical and hyperbolic embeddings of data many computer vision and pattern recognition problems may be posed as the analysis of a set of dissimilarities between objects for many types of data these dissimilarities are not euclidean i e they do not represent the distances between points in a euclidean space and therefore cannot be isometrically embedded in a euclidean space examples include shape dissimilarities graph distances and mesh geodesic distances in this paper we provide a means of embedding such non euclidean data onto surfaces of constant curvature we aim to embed the data on a space whose radius of curvature is determined by the dissimilarity data the space can be either of positive curvature spherical or of negative curvature hyperbolic we give an efficient method for solving the spherical and hyperbolic embedding problems on symmetric dissimilarity data our approach gives the radius of curvature and a method for approximating the objects as points on a hyperspherical manifold without optimisation for objects which do not reside exactly on the manifold we develop a optimisation based procedure for approximate embedding on a hyperspherical manifold we use the exponential map between the manifold and its local tangent space to solve the optimisation problem locally in the euclidean tangent space this process is efficient enough to allow us to embed data sets of several thousand objects we apply our method to a variety of data including time warping functions shape similarities graph similarity and gesture similarity data in each case the embedding maintains the local structure of the data while placing the points in a metric space

a generalized kernel approach to dissimilarity based classification usually objects to be classified are represented by features in this paper we discuss an alternative object representation based on dissimilarity values if such distances separate the classes well the nearest neighbor method offers a good solution however dissimilarities used in practice are usually far from ideal and the performance of the nearest neighbor rule suffers from its sensitivity to noisy examples we show that other more global classification techniques are preferable to the nearest neighbor rule in such cases for classification purposes two different ways of using generalized dissimilarity kernels are considered in the first one distances are isometrically embedded in a pseudo euclidean space and the classification task is performed there in the second approach classifiers are built directly on distance kernels both approaches are described theoretically and then compared using experiments with different dissimilarity measures and datasets including degraded data simulating the problem of missing values

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

a vertex based finite volume method applied to non linear material problems in computational solid mechanics a vertex based finite volume fv method is presented for the computational solution of quasi static solid mechanics problems involving material non linearity and infinitesimal strains the problems are analysed numerically with fully unstructured meshes that consist of a variety of two and threedimensional element types a detailed comparison between the vertex based fv and the standard galerkin fe methods is provided with regard to discretization solution accuracy and computational efficiency for some problem classes a direct equivalence of the two methods is demonstrated both theoretically and numerically however for other problems some interesting advantages and disadvantages of the fv formulation over the galerkin fe method are highlighted

topographic mapping of large dissimilarity data sets topographic maps such as the self organizing map som or neural gas ng constitute powerful data mining techniques that allow simultaneously clustering data and inferring their topological structure such that additional features for example browsing become available both methods have been introduced for vectorial data sets they require a classical feature encoding of information often data are available in the form of pairwise distances only such as arise from a kernel matrix a graph or some general dissimilarity measure in such cases ng and som cannot be applied directly in this article we introduce relational topographic maps as an extension of relational clustering algorithms which offer prototype based representations of dissimilarity data to incorporate neighborhood structure these methods are equivalent to the standard vectorial techniques if a euclidean embedding exists while preventing the need to explicitly compute such an embedding extending these techniques for the general case of non euclidean dissimilarities makes possible an interpretation of relational clustering as clustering in pseudo euclidean space we compare the methods to well known clustering methods for proximity data based on deterministic annealing and discuss how far convergence can be guaranteed in the general case relational clustering is quadratic in the number of data points which makes the algorithms infeasible for huge data sets we propose an approximate patch version of relational clustering that runs in linear time the effectiveness of the methods is demonstrated in a number of examples

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

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

a new approach to classification of brainwaves abstract in past years statistical pattern recognition has been used in the classification of evoked potential waveforms the statistical approach has more recently been enhanced and often replaced by the structural approach in which patterns are depicted as being built out of subpatterns in various ways of composition this paper describes an application of a general approach to pattern recognition in which these principal approaches are combined the result is a two stage classification algorithm which places evoked potential waveforms into clinically significant classes fourier descriptors are used as feature representations of each waveform shape euclidean distance is used to determine an optimal number of these features for design and testing of the algorithm the development of a dissimilarity measure follows and is crucial to the overall success of the algorithm an interdistance matrix of the training sample which is all that is retained from the original input is projected into a pseudo euclidean vector space resulting in a minimal vector representation of the sample accuracies of 90 of correctly classified test waveforms have been obtained

numerical simulation of coupled fluid solid problems abstract a multi grid finite volume method is presented for the prediction of coupled fluid solid problems in complex geometries whereby a unified finite volume approach based on the concept of block structured grids is employed for the handling of the complexity of the geometry and the fluid solid coupling a high numerical efficiency is obtained with a non linear multi grid method using a pressure correction based scheme as a smoother a number of numerical experiments were used to verify the accuracy and numerical efficiency of the method additionally to illustrate the capabilities of the approach results for two practical applications are presented

a finite volume method for analysing the bending deformation of thick and thin plates the formulation of a novel finite volume based procedure for determining deformations and moment distributions within transversely loaded plates is described in detail the formulation employs the mindlin theory of plate deformation transverse displacement and bending moment distributions are determined for a number of test problems the effect of mesh refinement upon the accuracy of the displacement and moment fields is investigated the finite volume procedure exhibits a number of interesting features it is conservative in the sense that equilibrium is satisfied locally within each cell and across cell boundaries it has the capacity to represent the behaviour of thick and thin plates and it is geometrically versatile because the cell shapes can take the form of multifaceted polygons

on weighted vs unweighted versions of combinatorial optimization problems we investigate the approximability properties of several weighted problems by comparing them with the respective unweighted problems for an appropriate and very general definition of niceness we show that if a nice weighted problem is hard to approximate within r then its polynomially bounded weighted version is hard to approximate within r o 1 then we turn our attention to specific problems and we show that the unweighted versions of min vertex cover min sat max cut max dicut max 2sat and max exactksat are exactly as hard to approximate as their weighted versions we note in passing that min vertex cover is exactly as hard to approximate as min sat in order to prove the reductions for max 2sat max cut max dicut and max e3sat we introduce the new notion of mixing set and we give an explicit construction of such sets these reductions give new non approximability results for these problems

nondeterministic functions and the existence of optimal proof systems we provide new characterizations of two previously studied questions on nondeterministic function classes q1 do nondeterministic functions admit efficient deterministic refinements q2 do nondeterministic function classes contain complete functions we show that q1 for the class npmv t is equivalent to the question whether the standard proof system for sat is p optimal and to the assumption that every optimal proof system is p optimal assuming only the existence of a p optimal proof system for sat we show that every set with an optimal proof system has a p optimal proof system under the latter assumption we also obtain a positive answer to q2 for the class npmv t an alternative view on nondeterministic functions is provided by disjoint sets and tuples we pursue this approach for disjoint np pairs and its generalizations to tuples of sets from np and conp with disjointness conditions of varying strength in this way we obtain new characterizations of q2 for the class npsv question q1 for npsv is equivalent to the question of whether every disjoint np pair is easy to separate in addition we characterize this problem by the question of whether every propositional proof system has the effective interpolation property again these interpolation properties are intimately connected to disjoint np pairs and we show how different interpolation properties can be modeled by np pairs associated with the underlying proof system

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

completeness in approximation classes beyond apx we present a reduction that allows us to establish completeness results for several approximation classes mainly beyond apx using it we extend one of the basic results of s khanna r motwani m sudan and u vazirani on syntactic versus computational views of approximability siam j comput 28 1998 164 191 by proving sufficient conditions for getting complete problems for the whole log apx the class of problems approximable within ratios that are logarithms of the size of the instance as well as for any approximability class beyond apx we also introduce a new approximability class called poly apx dealing with graph problems approximable with ratios functions of the maximum degree of the input graph for this class also using the proposed reduction we establish complete problems

the boolean hierarchy and the polynomial hierarchy a closer connection we show that if the boolean hierarchy collapses to level k then the polynomial hierarchy collapses to bh 3 k where bh 3 k is the k th level of the boolean hierarchy over sigma p 2 this is an improvement over the known results which show that the polynomial hierarchy would collapse to p np np o log n this result is significant in two ways first the theorem says that a deeper collapse of the boolean hierarchy implies a deeper collapse of the polynomial hierarchy also this result points to some previously unexplored connections between the boolean and query hierarchies of delta p 2 and delta p 3 namely begin eqnarray bh k rm co bh k implies bh 3 k rm co bh 3 k jot p rm np k p rm np k 1 implies p rm np rm np k 1 p rm np rm np k 2 end eqnarray

completeness in approximation classes we introduce a formal framework for studying approximation properties of np optimization npo problems the classes we consider are those appearing in the literature namely the class of approximable problems within a constant e apx the class of problems having a polynomial time approximation scheme pas and the class of problems having a fully polynomial time approximation scheme fpas we define natural approximation preserving reductions and obtain completeness results for these classes a complete problem in a class can not have stronger approximation properties unless p np we also show that the degree structure of npo allows intermediate degrees that is if p np there are problems which are neither complete nor belong to a lower class

finite volume analysis of stress and deformation in hygro thermo elastic orthotropic body abstract a numerical method originally developed for the stress analysis in isotropic bodies of arbitrary shape subjected to thermo mechanical loads is extended to handle general anisotropic bodies subjected to hygro thermo mechanical loads starting from an integral form of the equations governing heat mass and momentum balance and using the constitutive equations which account for mechanical as well as thermal and hygral stresses an unconditionally stable second order in both time and space finite volume discretisation is performed resulting in a set of algebraic equations which are then solved by an efficient segregated iterative procedure in order to demonstrate the methods capabilities a transient analysis of stresses and deformation in an orthotropic body subjected to hygro thermal loads is performed and shown that results compare favourably with experimental data

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

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

indefinite proximity learning a review efficient learning of a data analysis task strongly depends on the data representation most methods rely on symmetric similarity or dissimilarity representations by means of metric inner products or distances providing easy access to powerful mathematical formalisms like kernel or branch and bound approaches similarities and dissimilarities are however often naturally obtained by nonmetric proximity measures that cannot easily be handled by classical learning algorithms major efforts have been undertaken to provide approaches that can either directly be used for such data or to make standard methods available for these types of data we provide a comprehensive survey for the field of learning with nonmetric proximities first we introduce the formalism used in nonmetric spaces and motivate specific treatments for nonmetric proximity data second we provide a systematization of the various approaches for each category of approaches we provide a comparative discussion of the individual algorithms and address complexity issues and generalization properties in a summarizing section we provide a larger experimental study for the majority of the algorithms on standard data sets we also address the problem of large scale proximity learning which is often overlooked in this context and of major importance to make the method relevant in practice the algorithms we discuss are in general applicable for proximity based clustering one class classification classification regression and embedding approaches in the experimental part we focus on classification tasks

a matrix free implicit unstructured multigrid finite volume method for simulating structural dynamics and fluid structure interaction a new three dimensional 3d matrix free implicit unstructured multigrid finite volume fv solver for structural dynamics is presented in this paper the solver is first validated using classical 2d and 3d cantilever problems it is shown that very accurate predictions of the fundamental natural frequencies of the problems can be obtained by the solver with fast convergence rates this method has been integrated into our existing fv compressible solver x lv y zhao et al an efficient parallel unstructured multigrid preconditioned implicit method for simulating 3d unsteady compressible flows with moving objects journal of computational physics 215 2 2006 661 690 based on the immersed membrane method imm x lv y zhao et al as mentioned above results for the interaction between the fluid and an immersed fixed free cantilever are also presented to demonstrate the potential of this integrated fluid structure interaction approach

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

reductions completeness and the hardness of approximability in computability and in complexity theory reductions are widely used for mapping sets into sets in order to prove undecidability or hardness results in the study of the approximate solvability of hard discrete optimization problems suitable kinds of reductions called approximation preserving reductions can also be used to transfer from one problem to another either positive results solution techniques or negative results non approximability results in this paper various kinds of approximation preserving reductions are surveyed and their properties discussed the role of completeness under approximation preserving reductions is also analyzed and its relationship with hardness of approximability is explained

structure in approximation classes the study of the approximability properties of np hard optimization problems has recently made great advances mainly due to the results obtained in the field of proof checking the last important breakthrough proves the apx completeness of several important optimization problems and thus reconciles two distinct views of approximation classes syntactic and computational s khanna et al in proc 35th ieee symp on foundations of computer science ieee computer society press los alamitos ca 1994 pp 819 830 in this paper we obtain new results on the structure of several computationally defined approximation classes in particular after defining a new approximation preserving reducibility to be used for as many approximation classes as possible we give the first examples of natural npo complete problems and the first examples of natural apx intermediate problems moreover we state new connections between the approximability properties and the query complexity of npo problems

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

cell centered finite volume method for elastic deformation of heterogeneous media with full tensor properties abstract we propose a cell centered finite volume method for the heterogeneous anisotropic linear elasticity problem the internal traction vector is represented as a discrete flux on the interface this elastic flux is decomposed into a discrete two point flux approximation and a semi discrete transversal part we introduce an interpolation method in the presence of discontinuity in the full tensor material properties the scheme yields an accurate reconstruction of the gradient of the displacement in each cell the formulation is based on the assumption of linearity of the displacement and we enforce continuity of the internal traction vector and the displacement at the interface the treatment of the boundary conditions can be complicated in finite volume methods here we describe a general treatment of boundary conditions that does not entail the introduction of additional degrees of freedom the finite volume method is tested for a series of challenging elasticity problems

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

structural properties of bounded relations with an application to np optimization problems we introduce the notion bounded relation which comprises most resource bounded reducibilities which can be found in the literature including non uniform bounded reducibilities such as tp poly we state conditions on bounded relations which are again satisfied for most bounded reducibilities and which imply that every countable partial ordering can be embedded into every proper interval of the recursive degrees as corollaries we obtain that every countable partial ordering can be embedded into every proper interval of rec tp poly as well as into every proper interval between either two maximization or two minimization problems in the structures npo e and npo l for the two latter structures we show further that the result about embeddings of partial orderings extends to embeddings of arbitrary countable distributive lattices where in addition the least or the greatest element of the lattice can be preserved among other corollaries we obtain that for both structures every non trivial np optimization problem bounds a minimal pair in connection with embeddings into npo we introduce a representation of maximization or minimization problems by polynomial time computable functions we consider decidability issues w r t this representation and show for example that there is no effective procedure which decides for a subrecursive index of a polynomial time computable function whether the corresponding maximization problem is approximable within a constant factor

a low order finite element method for three dimensional linear elasticity problems with general meshes abstract the paper is concerned with a low order finite element method namely the staggered cell centered finite element method which has been proposed and analyzed in ong et al 2015 for two dimensional compressible and nearly incompressible linear elasticity problems in this work we extend the results to the three dimensional case and focus on the creating of the meshes in particular from a general primal mesh m we construct a polygonal dual mesh m and its submesh m in a way such that each dual control volume of m corresponds to a primal vertex and is a union macro element of some fixed number of adjacent tetrahedral elements of m the displacement is approximated by piecewise trilinear functions on the subdual mesh m and the pressure by piecewise constant functions on the dual mesh m as for two dimensional case such construction of the meshes and approximation spaces satisfies the macroelement condition which implies stability and convergence of the scheme numerical experiments are carried out to investigate the performance of the proposed method on various mesh types

bounds for assembly line balancing heuristics we analyze the asymptotic worst case performance ratio of polynomial time heuristics for the assembly line balancing problem assuming that p np we show that no polynomial heuristic has worst case ratio less than 1 5 this ratio is known to be not worse than 2 for any reasonable heuristic tighter results are established for problems with only short tasks i e when all processing times do not exceed a given fraction of the cycle time these results apply even to planar series parallel precedence constraints and contrast with known results for situations with an empty precedence relation i e the bin packing problem in particular if p np there are no fully polynomial approximation schemes for assembly line balancing even with short tasks and planar series parallel precedence constraints

when hamming meets euclid the approximability of geometric tsp and steiner tree we prove that the traveling salesman problem sc min tsp is sf max snp hard and thus sf np hard to approximate within some constant r 1 even if all cities lie in a euclidean space of dimension log n n is the number of cities and distances are computed with respect to any lp norm the running time of recent approximation schemes for geometric sc min tsp is doubly exponential in the number of dimensions our result implies that this dependence is necessary unless np has subexponential algorithms as an intermediate step we also prove the hardness of approximating sc min tsp in hamming spaces finally we prove a similar but weaker inapproximability result for the steiner minimal tree problem sc min st the reduction for sc min tsp uses error correcting codes the reduction for sc min st uses the integrality property of sc min cut linear programming relaxations the only previous inapproximability results for metric sc min tsp involved metrics where all distances are 1 or 2

the superconvergent patch recovery and a posteriori error estimates part 1 the recovery technique this is the first of two papers concerning superconvergent recovery techniques and a posteriori error estimation in this paper a general recovery technique is developed for determining the derivatives stresses of the finite element solutions at nodes the implementation of the recovery technique is simple and cost effective the technique has been tested for a group of widely used linear quadratic and cubic elements for both one and two dimensional problems numerical experiments demonstrate that the recovered nodal values of the derivatives with linear and cubic elements are superconvergent one order higher accuracy is achieved by the procedure with linear and cubic elements but two order higher accuracy is achieved for the derivatives with quadratic elements in particular an o h4 convergence of the nodal values of the derivatives for a quadratic triangular element is reported for the first time the performance of the proposed technique is compared with the widely used smoothing procedure of global l2 projection and other methods it is found that the derivatives recovered at interelement nodes by using l2 projection are also superconvergent for linear elements but not for quadratic elements numerical experiments on the convergence of the recovered solutions in the energy norm are also presented higher rates of convergence are again observed the results presented in this part of the paper indicate clearly that a new powerful and economical process is now available which should supersede the currently used post processing procedures applied in most codes

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

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

polynomial terse sets abstract let a be a set and k n be such that we wish to know the answers to x1 a x2 a xk a for various k tuples x1 x2 xk if this problem requires k queries to a in order to be solved in polynomial time then a is called polynomial terse or pterse we show the existence of both arbitrarily complex pterse and non pterse sets and that p np iff every np complete set is pterse we also show connections with p immunity p selective p generic sets and the boolean hierarchy in our framework unique satisfiability and a variation of it called ksat is in some sense close to satisfiability

quantifiers and approximation we investigate the relationship between logical expressibility of np optimization problems and their approximation properties first such attempt was made by papadimitrou and yannakakis 1988 who defined the class of npo problems max np we show that many important optimization problems do not belong to max np and that in fact there are problems in p which are not in max np the problems that we consider fit naturally in a new complexity class that we call max 1 we prove that several natural optimization problems are complete for max 1 under approximation preserving reductions all these complete problems are not approximable unless p np this motivates the definition of subclasses of max 1 that only contain problems which are presumably eaiser with respect to approximation in particular the class that we call rmax 2 contains approximable problems and problems like max clique that are not known to be nonapproximable we prove the max clique and several other optimization problems are complete for rmax 2 all the complete problems in rmax 2 share the interesting property that they either are nonapproximable or are approximable to any degree of accuracy

what is distance and why do we need the metric model for pattern learning abstract the concept of distance its role in pattern recognition and some advantages of the new model for pattern learning proposed recently by the author are discussed the universality flexibility and the ability to connect intrinsically the low level process that selects the primitives for the pattern representation with the higher level recognition process make the model clearly superior to other models proposed so far the fundamentally new analytical feature of the model which allows the learning machine to reconfigure itself efficiently is the introduction of continuity in the classical discrete computational model

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

the high fidelity generalized method of cells with arbitrary cell geometry and its relationship to the parametric finite volume micromechanics abstract this contribution is motivated by the recent ijss article titled formulation of the high fidelity generalized method of cells with arbitrary cell geometry for refined micromechanics and damage in composites by haj ali and aboudi in the present discussion we critically compare these authors approach with the parametric finite volume direct averaging micromechanics fvdam theory developed in a sequence of papers dating to 2004 we show that the extension proposed by haj ali and aboudi follows the framework of the parametric fvdam theory and not the original hfgmc model as claimed just as importantly we demonstrate that the proposed extension is fundamentally flawed as it violates accepted principles which every mechanics theory must satisfy namely correct reducibility and coordinate frame indifference direct comparison of predictions generated by the parametric fvdam theory the proposed extension called hfgmc with arbitrary cell geometry and finite element method demonstrates no need for the extension carried out in the specific manner proposed by haj ali and aboudi which in fact produces inferior results

on the information and representation of non euclidean pairwise data two common data representations are mostly used in intelligent data analysis namely the vectorial and the pairwise representation pairwise data which satisfy the restrictive conditions of euclidean spaces can be faithfully translated into a euclidean vectorial representation by embedding non metric pairwise data with violations of symmetry reflexivity or triangle inequality pose a substantial conceptual problem for pattern recognition since the amount of predictive structural information beyond what can be measured by embeddings is unclear we show by systematic modeling of non euclidean pairwise data that there exists metric violations which can carry valuable problem specific information furthermore euclidean and non metric data can be unified on the level of structural information contained in the data stable component analysis selects linear subspaces which are particularly insensitive to data fluctuations experimental results from different domains support our pattern recognition strategy

the np completeness of edge coloring we show that it is np complete to determine the chromatic index of an arbitrary graph the problem remains np complete even for cubic graphs

on γ reducibility versus polynomial time many one reducibility abstract we prove that each element of a class of functions denoted by npc t p whose graphs can be accepted in nondeterministic polynomial time can be evaluated in deterministic polynomial time if and only if reducibility is equivalent to polynomial time many one reducibility we also modify the proof technique used to obtain part of this result to obtain the stronger result that if every reduction can be replaced by a polynomial time turing reduction then every function in npc t p can be evaluated in deterministic polynomial time

numerical method for coupled fluid flow heat transfer and stress analysis using unstructured moving meshes with cells of arbitrary topology abstract a numerical method is presented that can be used for both solid body stress analysis and fluid flow predictions independently as well as in a coupled manner the method uses an integral form of equations governing mass energy and momentum balance for an arbitrary control volume a detailed description is provided of a novel second order accurate spatial discretisation technique which can accommodate unstructured moving meshes with cells of arbitrary topology this is accompanied by a fully implicit temporal discretisation which makes the method stable for any time step size the resulting set of coupled non linear algebraic equations is solved by employing a segregated approach leading to a decoupled set of linear algebraic equations for each dependent variable with a sparse diagonally dominant coefficient matrix these equations are solved by an iterative conjugate gradient solver which retains the sparsity of the coefficient matrix thus achieving a very efficient use of computer resources the method has been tested on a number of fluid flow and stress analysis problems and numerical calculations show favourable agreement with analytical and or experimental results in order to demonstrate the full capabilities of the present method an example of numerically complex coupled fluid flow stress analysis and heat transfer calculations is presented

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

erratum the polynomial time hierarchy collapses if the boolean hierarchy collapses it is shown that if the boolean hierarchy bh collapses then there exists a sparse set s such that text co np subseteq text np s and therefore the polynomial time hierarchy ph col

hop constrained steiner trees with multiple root nodes we consider a network design problem that generalizes the hop and diameter constrained steiner tree problem as follows given an edge weighted undirected graph with two disjoint subsets representing roots and terminals find a minimum weight subtree that spans all the roots and terminals so that the number of hops between each relevant node and an arbitrary root does not exceed a given hop limit h the set of relevant nodes may be equal to the set of terminals or to the union of terminals and root nodes this article proposes integer linear programming models utilizing one layered graph for each root node different possibilities to relate solutions on each of the layered graphs as well as additional strengthening inequalities are then discussed furthermore theoretical comparisons between these models and to previously proposed flow and path based formulations are given to solve the problem to optimality we implement branch and cut algorithms for the layered graph formulations our computational study shows their clear advantages over previously existing approaches

a unified approach to fuzzy pattern recognition abstract goldfarb has proposed a unified approach to pattern recognition in goldfarb s approach the data from a pseudometric space are isometrically embedded in a pseudo euclidean space the data representation is built for every particular data set the resulting data space will be data dependent the aim of this paper is to extend goldfarb s approach for fuzzy clustering a fuzzy clustering procedure for a pseudometric data set is given a generalisation of this algorithm to obtain the cluster substructure of a fuzzy class is also proposed

locally testable codes and pcps of almost linear length we initiate a systematic study of locally testable codes that is error correcting codes that admit very efficient membership tests specifically these are codes accompanied with tests that make a constant number of random queries into any given word and reject non codewords with probability proportional to their distance from the code locally testable codes are believed to be the combinatorial core of pcps however the relation is less immediate than commonly believed nevertheless we show that certain pcp systems can be modified to yield locally testable codes on the other hand we adapt techniques that we develop for the construction of the latter to yield new pcps our main results are locally testable codes and pcps of almost linear length specifically we prove the existence of the following constructs locally testable binary linear codes in which k information bits are encoded by a codeword of length k exp o log k this improves over previous results that either yield codewords of exponential length or obtained almost quadratic length codewords for sufficiently large nonbinary alphabet pcp systems of almost linear length for sat the length of the proof is n exp o log n and verification in performed by a constant number i e 19 of queries as opposed to previous results that used proof length n 1 p o 1 q for verification by q queries the novel techniques in use include a random projection of certain codewords and pcp oracles that preserves local testability an adaptation of pcp constructions to obtain linear pcp oracles for proving conjunctions of linear conditions and design of pcps with some new soundness properties a direct construction of locally testable linear codes of subexponential length

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

a finite volume method for incompressible linear elasticity abstract this paper describes development of a finite volume based method for modelling of incompressible linear elastic body problems the method is based on the solution of the integral form of conservation equations governing momentum balance and the introduction of pressure as an additional variable the incompressibility is enforced by employing the volumetric constraint equation a collocated variable arrangement is used and the spatial domain is discretised using finite volumes of arbitrary polyhedral shapes a segregated approach is employed to solve resulting set of coupled linear algebraic equations embedding a simple based algorithm for displacement pressure coupling numerical experiments showed that the developed method appears to be locking free

a unified formulation for continuum mechanics applied to fluid structure interaction in flexible tubes this paper outlines the development of a new procedure for analysing continuum mechanics problems with a particular focus on fluid structure interaction in flexible tubes a review of current methods of fluid structure coupling highlights common limitations of high computational cost and solution instability it is proposed that these limitations can be overcome by an alternative approach in which both fluid and solid components are solved within a single discretized continuum domain a single system of momentum and continuity equations is therefore derived that governs both fluids and solids and which are solved with a single mesh using finite volume discretization schemes the method is validated first by simulating dynamic oscillation of a clamped elastic beam it is then applied to study the case of interest wave propagation in highly flexible tubes in which a predicted wave speed of 8 58m s falls within 2 of an approximate analytical solution the method shows further good agreement with analytical solutions for tubes of increasing rigidity covering a range of wave speeds from those found in arteries to that in the undisturbed fluid

novel multiclass classifiers based on the minimization of the within class variance in this paper a novel class of multiclass classifiers inspired by the optimization of fisher discriminant ratio and the support vector machine svm formulation is introduced the optimization problem of the so called minimum within class variance multiclass classifiers mwcvmc is formulated and solved in arbitrary hilbert spaces defined by mercer s kernels in order to find multiclass decision hyperplanes surfaces afterwards mwcvmcs are solved using indefinite kernels and dissimilarity measures via pseudo euclidean embedding the power of the proposed approach is first demonstrated in the facial expression recognition of the seven basic facial expressions i e anger disgust fear happiness sadness and surprise plus the neutral state problem in the presence of partial facial occlusion by using a pseudo euclidean embedding of hausdorff distances and the mwcvmc the experiments indicated a recognition accuracy rate achieved up to 99 the mwcvmc classifiers are also applied to face recognition and other classification problems using mercer s kernels

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

finite volume method for thermo elasto plastic stress analysis abstract in this paper a new numerical method for prediction of stresses and displacements in thermo elasto plastic material is presented the method is based on the direct solution of the governing equations for thermal energy and momentum balance expressed in terms of temperature and displacements as dependent variables via corresponding constitutive relations and discretized by the finite volume technique as a result the unsteady fields of temperature displacements and stresses as well as residual stresses in a solid body subjected to a non uniform temperature and or external loads are obtained the method has been applied to a number of test cases and the results are compared with the analytical solution calculations obtained by another numerical method or experimental data results show that the method is accurate stable and efficient and that it can be used as a useful tool for predicting thermo mechanical deformation processes in a solid body

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

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

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

bounded queries to sat and the boolean hierarchy we study the complexity of decision problems that can be solved by a polynomial time turing machine that makes a bounded number of queries to an np oracle depending on whether we allow some queries to depend on the results of other queries we obtain two probably different hierarchies we present several results relating the bounded np query hierarchies to each other and to the boolean hierarchy we also consider the similarly defined hierarchies of functions that can be computed by a polynomial time turing machine that makes a bounded number of queries to an np oracle we present relations among these two hierarchies and the boolean hierarchy in particular we show for all k that there are functions computable with 2k parallel queries to an np set that are not computable in polynomial time with k serial queries to any oracle unless p np as a corollary k 1 parallel queries to an np set allow us to compute more functions than are computable with only k parallel queries to an np set unless p np the same is true of serial queries similar results hold for all tt self reducible sets using a mind change technique we show that 2k 1 parallel queries to an np set allow us to accept in polynomial time exactly the same sets as can be accepted in polynomial time with k serial queries to an np set in fact the same is true for any class in place of np that is closed under polynomial time positive bounded truth table reductions this contrasts with the expected result for function computations with an np oracle beigel 1988 in addition we show that the boolean hierarchy and the bounded query hierarchies of languages either stand or collapse together finally we show that if the boolean hierarchy collapses to any level but the zeroth deterministic polynomial time then for all k there are functions computable in polynomial time with k parallel queries to an np set that are not computable in polynomial time with k 1 serial queries to any set np complete sets are p superterse

a unified approach to pattern recognition abstract the paper is an outline of a new approach to pattern recognition developed by the author a fuller introduction to the approach will appear soon 1 within the proposed framework the two principal approaches to pattern recognition vector and syntactic are unified