decision lists and related boolean functions we consider boolean functions represented by decision lists and study their relationships to other classes of boolean functions it turns out that the elementary class of 1 decision lists has interesting relationships to independently defined classes such as disguised horn functions read once functions nested differences of concepts threshold functions and 2 monotonic functions in particular 1 decision lists coincide with fragments of the mentioned classes we further investigate the recognition problem for this class as well as the extension problem in the context of partially defined boolean functions pdbfs we show that finding an extension of a given pdbf in the class of 1 decision lists is possible in linear time this improves on previous results moreover we present an algorithm for enumerating all such extensions with polynomial delay

budgeted matching and budgeted matroid intersection via the gasoline puzzle many polynomial time solvable combinatorial optimization problems become np hard if an additional complicating constraint is added to restrict the set of feasible solutions in this paper we consider two such problems namely maximum weight matching and maximum weight matroid intersection with one additional budget constraint we present the first polynomial time approximation schemes for these problems similarly to other approaches for related problems our schemes compute two solutions to the lagrangian relaxation of the problem and patch them together to obtain a near optimal solution however due to the richer combinatorial structure of the problems considered here standard patching techniques do not apply to circumvent this problem we crucially exploit the adjacency relations on the solution polytope and surprisingly the solution to an old combinatorial puzzle

approximation of pareto optima in multiple objective shortest path problems we study methods for approximating the set of pareto optimal paths in multiple objective shortest path problems known generalizations of standard shortest path methods will compute this set but can suffer from rapidly increasing computational and storage demands as problem size increases in an effort to avoid such difficulties we develop approximation methods that can estimate the pareto optima to any required degree of accuracy the approximation methods are fully polynomial that is they operate in time and space bounded by a polynomial in problem size and accuracy of approximation the greater the accuracy the more time required to reach a solution we show how approximation methods may be applied to yield fully polynomial approximation schemes for a variety of np complete single objective problems

min max min robust combinatorial optimization the idea of k adaptability in two stage robust optimization is to calculate a fixed number k of second stage policies here and now after the actual scenario is revealed the best of these policies is selected this idea leads to a min max min problem in this paper we consider the case where no first stage variables exist and propose to use this approach to solve combinatorial optimization problems with uncertainty in the objective function we investigate the complexity of this special case for convex uncertainty sets we first show that the min max min problem is as easy as the underlying certain problem if k is greater than the number of variables and if we can optimize a linear function over the uncertainty set in polynomial time we also provide an exact and practical oracle based algorithm to solve the latter problem for any underlying combinatorial problem on the other hand we prove that the min max min problem is np hard for every fixed number k even when the uncertainty set is a polyhedron given by an inner description for the case that k is smaller or equal to the number of variables we finally propose a fast heuristic algorithm and evaluate its performance

a note on convergence of the multigrid v cycle several recent papers have discussed the convergence of the multigrid v cycle in particular there are several results for the symmetric case where the numbers of smoothings before the fine to coarse transfer and after the coarse to fine transfer are the same in most instances the smoother h i e 1a has been limited to the case where e is positive definite and the eigenvalues h of h satisfy 0

a p cycle analysis of multigrid method we suggest a new strategy for multigrid iterations based on our analysis for p cycle multigrid method first we improve a recent result on multigrid v cycle analysis given by bramble and pasciak to p cycle multigrid method we show that the convergence rate improves slowly with p eventually bounded by a constant which is in agreement with many previous experiments based on the this observation we use p cycle only in the first few multigrid iterations which is more effective than the standard p cycle

min max and min max regret versions of combinatorial optimization problems a survey abstract min max and min max regret criteria are commonly used to define robust solutions after motivating the use of these criteria we present general results then we survey complexity results for the min max and min max regret versions of some combinatorial optimization problems shortest path spanning tree assignment min cut min s t cut knapsack since most of these problems are np hard we also investigate the approximability of these problems furthermore we present algorithms to solve these problems to optimality

on the use of inexact subdomain solvers for bddc algorithms the standard bddc balancing domain decomposition by constraints preconditioner is shown to be equivalent to a preconditioner built from a partially subassembled finite element model this results in a system of linear algebraic equations which is much easier to solve in parallel than the fully assembled model the cost is then often dominated by that of the problems on the subdomains an important role is also played both in theory and practice by an averaging operator and in addition exact dirichlet solvers are used on the subdomains in order to eliminate the residual in the interior of the subdomains the use of inexact solvers for these problems and even the replacement of the dirichlet solvers by a trivial extension are considered it is established that one of the resulting algorithms has the same eigenvalues as the standard bddc algorithm and the connection of another with the feti dp algorithm with a lumped preconditioner is also considered multigrid methods are used in the experimental work and under certain assumptions it is established that the iteration count essentially remains the same as when exact solvers are used while considerable gains in the speed of the algorithm can be realized since the cost of the exact solvers grows superlinearly with the size of the subdomain problems while the multigrid methods are linear

the analysis of multigrid algorithms for pseudodifferential operators of order minus one multigrid algorithms are developed to solve the discrete systems approximating the solutions of operator equations involving pseudodifferential operators of order minus one classical multigrid theory deals with the case of differential operators of positive order the pseudodifferential operator gives rise to a coercive form on h sup minus 1 2 omega effective multigrid algorithms are developed for this problem these algorithms are novel in that they use the inner product on h sup minus 1 omega as a base inner product for the multigrid development the authors show that the resulting rate of iterative convergence can at worst depend linearly on the number of levels in these novel multigrid algorithms in addition it is shown that the convergence rate is independent of the number of levels and unknowns in the case of a pseudodifferential operator defined by a single layer potential finally the results of numerical experiments illustrating the theory are presented 19 refs 1 fig 2 tabs

bulk robust combinatorial optimization we commence an algorithmic study of bulk robustness a new model of robustness in combinatorial optimization unlike most existing models bulk robust combinatorial optimization features a highly nonuniform failure model instead of an interdiction budget bulk robust counterparts provide an explicit list of interdiction sets comprising the admissible set of scenarios thus allowing to model correlations between failures of different components in the system interdiction sets of variable cardinality and more the resulting model is suitable for capturing failures of complex structures in the system we provide complexity results and approximation algorithms for bulk robust counterparts of the minimum matroid basis problems and the shortest path problem our results rely on various techniques and outline the rich and heterogeneous combinatorial structure of bulk robust optimization

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

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

two level hierarchical fem method for modeling passive microwave devices in recent years multigrid methods have been proven to be very efficient for solving large systems of linear equations resulting from the discretization of positive definite differential equations by either the finite difference method or theh version of the finite element method in this paper an iterative method of the multiple level type is proposed for solving systems of algebraic equations which arise from thep version of the finite element analysis applied to indefinite problems a two levelv cycle algorithm has been implemented and studied with a gauss seidel iterative scheme used as a smoother the convergence of the method has been investigated and numerical results for a number of numerical examples are presented

a multi grid method with a priori and a posteriori level choice for the regularization of nonlinear ill posed problems in this paper we study a multi grid method for the numerical solution of nonlinear systems of equations arising from the discretization of ill posed problems where the special eigensystem structure of the underlying operator equation makes it necessary to use special smoothers we provide uniform contraction factor estimates and show that a nested multigrid iteration together with an a priori or a posteriori chosen stopping index defines a regularization method for the ill posed problem i e a stable solution method that converges to an exact solution of the underlying infinite dimensional problem as the data noise level goes to zero with optimal rates under additional regularity conditions

pattern based modeling and solution of probabilistically constrained optimization problems we propose a new modeling and solution method for probabilistically constrained optimization problems the methodology is based on the integration of the stochastic programming and combinatorial pattern recognition fields it permits the fast solution of stochastic optimization problems in which the random variables are represented by an extremely large number of scenarios the method involves the binarization of the probability distribution and the generation of a consistent partially defined boolean function pdbf representing the combination f p of the binarized probability distribution f and the enforced probability level p we show that the pdbf representing f p can be compactly extended as a disjunctive normal form dnf the dnf is a collection of combinatorial p patterns each defining sufficient conditions for a probabilistic constraint to hold we propose two linear programming formulations for the generation of p patterns that can be subsequently used to derive a linear programming inner approximation of the original stochastic problem a formulation allowing for the concurrent generation of a p pattern and the solution of the deterministic equivalent of the stochastic problem is also proposed the number of binary variables included in the deterministic equivalent formulation is not an increasing function of the number of scenarios used to represent uncertainty results show that large scale stochastic problems in which up to 50 000 scenarios are used to describe the stochastic variables can be consistently solved to optimality within a few seconds

convergence estimates for multigrid algorithms without regularity assumptions a new technique for proving rate of convergence estimates of multi grid algorithms for symmetric positive definite problems will be given in this paper the standard multigrid theory requires a regularity and approxima tion assumption in contrast the new theory requires only an easily verified approximation assumption this leads to convergence results for multigrid re finement applications problems with irregular coefficients and problems whose coefficients have large jumps in addition the new theory shows why it suffices to smooth only in the regions where new nodes are being added in multigrid refinement applications

optimal order multigrid methods for solving exterior boundary value problems the coupling of boundary elements and finite elements combines the advantage of boundary elements for treating domains extended to infinity and that of finite elements in treating the nonhomogeneity of equations and the complexity of domains in the case of the laplacian by taking a circle or a sphere as the artificial coupling boundary it is shown that the corresponding boundary integral equation can be solved without any cost and the coupled system is reduced to a simple finite element system two multigrid methods are proposed to solve this finite element linear system both methods are of optimal order and can be used to solve such finite element equations as efficiently as to solve those arising from interior boundary value problems numerical experiments are included to show the efficiency and advantages of the methods an apparent significance of the methods is that the boundary elements appear neither in the discretization nor in the coding

a posteriori boundary element error estimation an a posteriori error estimator is presented for the boundary element method in a general framework it is obtained by solving local residual problems for which a local concept is introduced to accommodate the fact that integral operators are nonlocal operators the estimator is shown to have an upper and a lower bound by the constant multiples of the exact error in the energy norm for symm s and hypersingular integral equations numerical results are also given to demonstrate the eectiveness of the estimator for these equations it can be used for adaptive h p and hp methods c 1999 elsevier science b v all rights reserved msc 65n35 65n15 65r20 65d07 45l10

on online learning of decision lists a fundamental open problem in computational learning theory is whether there is an attribute efficient learning algorithm for the concept class of decision lists rivest 1987 blum 1996 we consider a weaker problem where the concept class is restricted to decision lists with d alternations for this class we present a novel online algorithm that achieves a mistake bound of o rdlog n where r is the number of relevant variables and n is the total number of variables the algorithm can be viewed as a strict generalization of the famous winnow algorithm by littlestone 1988 and improves the o r2dlog n mistake bound of balanced winnow our bound is stronger than a similar pac learning result of dhagat and hellerstein 1994 a combination of our algorithm with the algorithm suggested by rivest 1987 might achieve even better bounds

remarks on multigrid convergence theorems multigrid has become an important iterative method for the solution of discrete elliptic equations however there is much to be done in the theory of convergence proofs at the present time there are two general two level methods for general convergence proofs an algebraic approach and a duality approach while these theories do not give sharp estimates they provide good general rigorous convergence theorems in this note we study the relationship between these theories while the approach and thought process leading to these theories are different the results are essentially the same indeed the basic estimates required by these theories are the same

weak solvability of interior transmission problems via mixed finite elements and dirichlet to neumann mappings we study the weak solvability of an interior linear nonlinear transmission problem arising in steady heat transfer and potential theory for the variational formulation we use a dirichlet to neumann mapping on the interface which is obtained from the application of the boundary integral method to the linear domain and we utilize a mixed finite element method in the nonlinear region existence and uniqueness of solution for the continuous formulation are provided and general approximation results for a fully discrete galerkin method are derived in particular a compatibility condition between the mesh sizes involved is deduced in order to conclude the solvability and stability of this galerkin scheme

error free and best fit extensions of partially defined boolean functions abstract in this paper we address a fundamental problem related to the induction of boolean logic given a set of data represented as a set of binary truen vectors or positive examples and a set of falsen vectors or negative examples we establish a boolean function or an extension f so thatfis true resp false in every given true resp false vector we shall further require that such an extension belongs to a certain specified class of functions e g class of positive functions class of horn functions and so on the class of functions represents our a priori knowledge or hypothesis about the extensionf which may be obtained from experience or from the analysis of mechanisms that may or may not cause the phenomena under consideration the real world data may contain errors e g measurement and classification errors might come in when obtaining data or there may be some other influential factors not represented as variables in the vectors in such situations we have to give up the goal of establishing an extension that is perfectly consistent with the given data and we are satisfied with an extensionfhaving the minimum number of misclassifications both problems i e the problem of finding an extension within a specified class of boolean functions and the problem of finding a minimum error extension in that class will be extensively studied in this paper for certain classes we shall provide polynomial algorithms and for other cases we prove their np hardness

functions comuted by monotone boolean formulas with no repeated variables abstract if g is a monotone boolean function depending on all its variables the property that each prime implicant of g intersects each prime clause of g in a singleton is a well known necessary condition for g to be computable by a formula with no repeated variable and only using the connectives and or we prove that the condition is also sufficient our proof uses matroid theory

multigrid convergence for nonsymmetric indefinite variational problems and one smoothing step we prove that the multigrid method for linear systems arising from nonsymmetric and indefinite elliptic variational problems converges for all sufficiently small h with one smoothing step only previous proofs required that the number of smoothing steps be within certain bounds which were quite difficult to compute our theory is based on sharp energy norm convergence estimates for the symmetric definite case and lower order terms are treated as perturbations the resulting estimates are so sharp that we obtain even the convergence of the v cycle

nonconforming multigrid method for nonsymmetric and indefinite problems abstract the convergence of w cycle multigrid algorithms applied to nonsymmetric and indefinite elliptic boundary value problems using p 1 nonconforming finite elements is proved

preconditioning in h div and applications we consider the solution of the system of linear algebraic equations which arises from the finite element discretization of boundary value problems associated to the differential operator i grad div the natural setting for such problems is in the hilbert space h div and the variational formulation is based on the inner product in h div we show how to construct preconditioners for these equations using both domain decomposition and multigrid techniques these preconditioners are shown to be spectrally equivalent to the inverse of the operator as a consequence they may be used to precondition iterative methods so that any given error reduction may be achieved in a finite number of iterations with the number independent of the mesh discretization we describe applications of these results to the efficient solution of mixed and least squares finite element approximations of elliptic boundary value problems

convergence of the multigrid v cycle algorithm for second order boundary value problems without full elliptic regularity the multigrid v cycle algorithm using the richardson relaxation scheme as the smoother is studied in this paper for second order elliptic boundary value problems the contraction number of the v cycle algorithm is shown to improve uniformly with the increase of the number of smoothing steps without assuming full elliptic regularity as a consequence the v cycle convergence result of braess and hackbusch is generalized to problems without full elliptic regularity

double horn functions abstract in this paper we define double horn functions which are the boolean functions f such that both f and its complement i e negation f are horn and investigate their semantical and computational properties double horn functions embody a balanced treatment of positive and negative information in the course of the extension problem of partially defined boolean functions pdbfs where a pdbf is a pair t f of disjoint sets t f 0 1 n of true and false vectors respectively and an extension of t f is a boolean function f that is compatible with t and f we derive syntactic and semantic characterizations of double horn functions and determine the number of such functions the characterizations are then exploited to give polynomial time algorithms i that recognize double horn functions from horn dnfs disjunctive normal forms and ii that compute the prime dnf from an arbitrary formula as well as its complement and its dual furthermore we consider the problem of determining a double horn extension of a given pdbf we describe a polynomial time algorithm for this problem and moreover an algorithm that enumerates all double horn extensions of a pdbf with polynomial delay however finding a shortest double horn extension in terms of the size of a formula representing it is shown to be intractable

on the convergence of the multigrid method for a hypersingular integral equation of the first kind we present a multigrid method to solve linear systems arising from galerkin schemes for a hypersingular boundary integral equation governing three dimensional neumann problems for the laplacian our algorithm uses damped jacobi iteration gauss seidel iteration or sor as preand post smoothers if the integral equation holds on a closed lipschitz surface we prove convergence ofv andw cycles with any number of smoothing steps if the integral equation holds on an open lipschitz surface covering crack problems we show convergence of thew cycle numerical experiments are given which underline the theoretical results

multigrid convergence for a singular perturbation problem abstract so far there has been no analysis of multigrid methods applied to singularly perturbed dirichlet boundary value problems only for periodic boundary conditions does the fourier transformation mode analysis apply and it is not obvious that the convergence results carry over to the dirichlet case since the eigenfunctions are quite different in the two cases in this paper we prove a close relationship between multigrid convergence for the easily analysable case of periodic conditions and the convergence for the dirichlet case

robust discrete optimization and network flows we propose an approach to address data uncertainty for discrete optimization and network flow problems that allows controlling the degree of conservatism of the solution and is computationally tractable both practically and theoretically in particular when both the cost coefficients and the data in the constraints of an integer programming problem are subject to uncertainty we propose a robust integer programming problem of moderately larger size that allows controlling the degree of conservatism of the solution in terms of probabilistic bounds on constraint violation when only the cost coefficients are subject to uncertainty and the problem is a 0 1 discrete optimization problem on n variables then we solve the robust counterpart by solving at most n 1 instances of the original problem thus the robust counterpart of a polynomially solvable 0 1 discrete optimization problem remains polynomially solvable in particular robust matching spanning tree shortest path matroid intersection etc are polynomially solvable we also show that the robust counterpart of an np hard approximable 0 1 discrete optimization problem remains approximable finally we propose an algorithm for robust network flows that solves the robust counterpart by solving a polynomial number of nominal minimum cost flow problems in a modified network

the convergence rate of a multigrid method with gauss seidel relaxation for the poisson equation the numerical solution of the poisson equation is treated by a multigrid method for a uniform grid the convergence rate can be estimated even for the iteration with a v cycle independently of the shape of the domain as long as it is convex and polygonal the smoothing effect of the gaus seidel relaxation is described by a discrete seminorm which is weaker than the energy norm

a dual dual mixed formulation for nonlinear exterior transmission problems we combine a dual mixed finite element method with a dirichletto neumann mapping derived by the boundary integral equation method to study the solvability and galerkin approximations of a class of exterior nonlinear transmission problems in the plane as a model problem we consider a nonlinear elliptic equation in divergence form coupled with the laplace equation in an unbounded region of the plane our combined approach leads to what we call a dual dual mixed variational formulation since the main operator involved has itself a dual type structure we establish existence and uniqueness of solution for the continuous and discrete formulations and provide the corresponding error analysis by using raviart thomas elements the main tool of our analysis is given by a generalization of the usual babuska brezzi theory to a class of nonlinear variational problems with constraints

optimal order nonnested multigrid methods for solving finite element equations i on quasi uniform meshes we prove that the multigrid method works with optimal computational order even when the multiple meshes are not nested when a coarse mesh is not a submesh of the finer one the coarse level correction usually does not have the a projection property and does amplify the iterative error in some components nevertheless the low frequency components of the error can still be caught by the coarse level correction since the amplified highfrequency errors will be damped out by the fine level smoothing efficiently the optimal work order of the standard multigrid method can still be maintained however unlike the case of nested meshes a nonnested multigrid method with one smoothing does not converge in general no matter whether it is a v cycle or a w cycle method it is shown numerically that the convergence rates of nonnested multigrid methods are not necessarily worse than those of nested ones since nonnested multigrid methods accept quite arbitrarily related meshes we may then combine the efficiencies of adaptive refinements and of multigrid algorithms

an optimal iterative process for the johnson nedelec method of coupling boundary and finite elements we reformulate the discretization of the johnson nedelec method 11 of coupling boundary elements and finite elements for an exterior bidimensional laplacian this new formulation leads to optimal error estimates and allows the use of simple quadrature formulas for calculation of the boundary element matrix we show that if the parameter of discretization is sufficiently small the fully discrete scheme is well posed and the error estimates remain unaltered the rest of the paper is devoted to the study of an efficient algorithm for solving the resulting discrete linear systems

a conjugate gradient method and a multigrid algorithm for morley s finite element approximation of the biharmonic equation the numerical solution of the linear equations arising from morley s nonconforming displacement method is studied a suitable preconditioning for the conjugate gradient method is described furthermore the nonconformity of the discretization necessitates some non standard ingredients of multigrid algorithms

fem bem coupling methods for tokamak plasma axisymmetric free boundary equilibrium computations in unbounded domains abstract incorporating boundary conditions at infinity into simulations on bounded computational domains is a repeatedly occurring problem in scientific computing the combination of finite element methods fem and boundary element methods bem is the obvious instrument and we adapt here for the first time the two standard fem bem coupling approaches to the free boundary equilibrium problem the johnson nedelec coupling and the bielak maccamy coupling we recall also the classical approach for fusion applications dubbed according to its first appearance von hagenow lackner coupling and present the less used alternative introduced by albanese blum and de barbieri in 2 we show that the von hagenow lackner coupling suffers from undesirable non optimal convergence properties that suggest that other coupling schemes in particular johnson nedelec or albanese blum de barbieri are more appropriate for non linear equilibrium problems moreover we show that any of such coupling methods requires newton like iteration schemes for solving the corresponding non linear discrete algebraic systems

measuring teachability using variants of the teaching dimension in a typical algorithmic learning model a learner has to identify a target object from partial information conversely in a teaching model a teacher has to give information that allows the learners to identify a target object we devise two variants of the classical teaching model for boolean concept classes based on the teaching dimension and describe them by teaching dimension like combinatorial parameters in the first model the learners choose consistent hypotheses with least complexity we show that 1 decision lists are the harder to teach the longer they are and that 2 term dnfs are the harder to teach the more terms they have this contrasts with the teachability results for these classes in the teaching dimension model in our second model the learners choose consistent hypotheses based on the assumption that the teacher is optimal we show that monomials can be taught with a linear number of examples whereas some 1 decision lists need exponentially many

a multilevel algorithm for the biharmonic problem a finite element discretization of the mixed variable formulation of the biharmonic problem is considered a multilevel algorithm for the numerical solution of the discrete equations is described convergence is proved under the assumption ofh 3 regularity

split level iteration many techniques are used for solving large sparse linear systems some of the most popular methods for solving the more difficult problems combine direct and iterative algorithms to yield efficient multistage procedures although each stage is standard the manner in which stages are combined is often novel both multigrid and preconditioned conjugate gradients fall into this class of numerical methods in lo two promising algorithms were described with supportive numerical studies the first was cyclic application of model problem adi iteration followed by a conjugate gradient iteration on the deviation of the model problem from the actual problem some new developments relating to this procedure are described in a companion paper in this journal 121 the second was a two level iteration in which a variational correction with a coarse grid basis was applied cyclically to the result of conventional iteration over a fine grid this was the first published implementation of a multigrid method based on the variational techniques that provide the foundations for finite element computation in these initial two level variational studies certain advantages of additive over multiplicative coarse mesh correction were not recognized the coefficient matrix for additive correction does not depend on the fine mesh iterate and need thus be computed only once moreover it need only be factored once for direct solution the right hand side of the correction equations varies each cycle brandt 4 and others have exploited this in multigrid and related methods some numerical comparisons of additive and multiplicative correction for two level iteration are given in 111 also it was not recognized that under suitable conditions one may use the two level iteration as a preconditioner for conjugate gradients thus combining the two initially distinct methods as suggested subsequently in 1 7 9 some insight into the nature of such an iteration may be gained by a qualitative analysis of error behavior the fine mesh iteration acts primarily on high mcde error components and the coarse mesh correction reduces low mode components in some implementations the two iteration matrices commute but in most they do not in the commuting case careful choice of iteration parameters precludes benefits from the conjugate gradient third stage in the noncommuting case the conjugate gradient stage should act on error persisting because of component mixing between the other two stages the automatic extrapolation in the conjugate gradient method greatly reduces convergence sensitivity to choice of iteration parameters for the fine mesh iteration

the complexity of very simple boolean formulas with applications the concepts of textbf sat hardness and textbf sat completeness modulo npolylogn time and linear size reducibility denoted by textbf sat hard npolylogn n and textbf sat complete npolylogn n respectively are introduced regardless of whether textbf p textbf np or textbf p neq textbf np it is shown that intuitivelyeach textbf sat hard npolylogn n problem requires essentially at least as much deterministic time as andeach textbf sat complete npolylogn n problem requires essentially the same deterministic time as the satisfiability problem for 3cnf formulas it is proved that the leqq satisfiability tautology unique satisfiability equivalence and minimization problems are already textbf sat complete npolylogn n for very simple boolean formulas and for very simple systems of boolean equations these completeness results are used to characterize the deterministic time complexities of a number of problems for lattices propositional calculi

multigrid in h div and h curl summary we consider the solution of systems of linear algebraic equations which arise from the finite element discretization of variational problems posed in the hilbert spaces bf h div and bf h curl in three dimensions we show that if appropriate finite element spaces and appropriate additive or multiplicative schwarz smoothers are used then the multigrid v cycle is an efficient solver and preconditioner for the discrete operator all results are uniform with respect to the mesh size the number of mesh levels and weights on the two terms in the inner products

horn extensions of a partially defined boolean function given a partially defined boolean function pdbf t f we investigate in this paper how to find a horn extension f 0 1 n mapsto 0 1 which is consistent with t f where t subseteq 0 1 n denotes a set of true boolean vectors or positive examples and f subseteq 0 1 n denotes a set of false boolean vectors or negative examples given a pdbf t f it is known that the existence of a horn extension can be checked in polynomial time as there are many horn extensions however we consider those extensions f which have maximal and minimal sets t f of the true vectors of f respectively for a pdbf t f there always exists the unique maximal i e maximum horn extension but there are in general many minimal horn extensions we first show that a polynomial time membership oracle can be constructed for the maximum extension even if its disjunctive normal form dnf can be very long our main contribution is to show that checking if a given horn dnf represents a minimal extension and generating a horn dnf of a minimal horn extension can both be done in polynomial time we also can check in polynomial time if a pdbf t f has the unique minimal horn extension however the problems of finding a horn extension f with the smallest t f and of obtaining a horn dnf whose number of literals is smallest are both np hard

a domain decomposition method based on bem and fem for linear exterior boundary value problems abstract we develop the finite dimensional analysis of a new domain decomposition method for linear exterior boundary value problems arising in potential theory and heat conductivity our approach uses a dirichlet to neumann mapping to transform the exterior problem into an equivalent boundary value problem on a bounded domain then the domain is decomposed into a finite number of annular subregions and the local steklov poincare operators are expressed conveniently either by bem or fem in order to obtain a symmetric interface problem the global steklov poincare problem is solved by using both a richardson type scheme and the preconditioned conjugate gradient method which yield iteration by subdomain algorithms well suited for parallel processing finally contractivity results and finite dimensional approximations are provided

variational formulations of transmission problems via fem bem and dtn mappings the purpose of this work is to review some of the main results on the weak solvability and galerkin approximations of interior and exterior transmission problems arising in potential theory and elastostatics both linear and nonlinear model problems are included in our presentation we consider variational formulations based on the combination of mixed finite elements with either boundary integral equations or dirichlet to neumann dtn mappings we provide existence and uniqueness of solution for the continuous formulations and give general approximation estimates for the associated galerkin schemes some open questions are also mentioned

version spaces and the consistency problem a version space is a collection of concepts consistent with a given set of positive and negative examples mitchell artificial intelligence 18 1982 203 226 proposed representing a version space by its boundary sets the maximally general g and maximally specific consistent concepts s for many simple concept classes the size of g and s is known to grow exponentially in the number of positive and negative examples this paper argues that previous work on alternative representations of version spaces has disguised the real question underlying version space reasoning we instead show that tractable reasoning with version spaces turns out to depend on the consistency problem i e determining if there is any concept consistent with a set of positive and negative examples indeed we show that tractable version space reasoning is possible if and only if there is an efficient algorithm for the consistency problem our observations give rise to new concept classes for which tractable version space reasoning is now possible e g 1 decision lists monotone depth two formulas and halfspaces

a new convergence proof for the multigrid method including the v cycle for a positive definite finite element equation we describe a multigrid iteration and prove convergence under natural assumptions on the discretization and the elliptic problem hitherto existing convergence proofs require a sufficiently large number of smoothing iterations and exclude the v cycle the presented proof applies to procedures with any number of smoothing iterations and to the v cycle

new approaches to multi objective optimization a natural way to deal with multiple partially conflicting objectives is turning all the objectives but one into budget constraints many classical optimization problems such as maximum spanning tree and forest shortest path maximum weight perfect matching maximum weight independent set basis in a matroid or in the intersection of two matroids become np hard even with one budget constraint still for most of these problems efficient deterministic and randomized approximation schemes are known not much is known however about the case of two or more budgets filling this gap at least partially is the main goal of this paper in more detail we obtain the following main results using iterative rounding for the first time in multi objective optimization we obtain multi criteria ptass which slightly violate the budget constraints for spanning tree matroid basis and bipartite matching with k o 1 k o 1 budget constraints we present a simple mechanism to transform multi criteria approximation schemes into pure approximation schemes for problems whose feasible solutions define an independence system this gives improved algorithms for several problems in particular this mechanism can be applied to the above bipartite matching algorithm hence obtaining a pure ptas we show that points in low dimensional faces of any matroid polytope are almost integral an interesting result on its own this gives a deterministic approximation scheme for k k budgeted matroid independent set we present a deterministic approximation scheme for k k budgeted matching in general graphs where k o 1 k o 1 interestingly to show that our procedure works we rely on a non constructive result by stromquist and woodall which is based on the ham sandwich theorem

an optimal order process for solving finite element equations a k level iterative procedure for solving the algebraic equations which arise from the finite element approximation of elliptic boundary value problems is presented and analyzed the work estimate for this procedure is proportional to the number of unknowns an optimal order result general geometry is permitted for the underlying domain but the shape plays a role in the rate of convergence through elliptic regularity finally a short discussion of the use of this method for parabolic problems is presented

approximating minimum size k connected spanning subgraphs via matching an efficient heuristic is presented for the problem of finding a minimum size k connected spanning subgraph of an undirected or directed simple graph g v e there are four versions of the problem and the approximation guarantees are as follows minimum size k node connected spanning subgraph of an undirected graph 1 1 k minimum size k node connected spanning subgraph of a directed graph 1 1 k minimum size k edge connected spanning subgraph of an undirected graph 1 2 k 1 minimum size k edge connected spanning subgraph of a directed graph 1 4 sqrt k the heuristic is based on a subroutine for the degree constrained subgraph b matching problem it is simple and deterministic and runs in time o k e 2 the following result on simple undirected graphs is used in the analysis the number of edges required for augmenting a graph of minimum degree k to be k edge connected is at most k v k 1 for undirected graphs and k 2 a deterministic parallel nc version of the heuristic finds a 2 node connected or 2 edge connected spanning subgraph whose size is within a factor of 1 5 epsilon of minimum where epsilon 0 is a constant

iterative solution methods for plate bending problems multigrid and preconditioned cg algorithm the numerical solution of the linear equations arising from a discretization of the plate bending problem based on the zienkiewicz triangle is studied the finite element scheme proceeds from the mindlin reissner formulation with modified shear energy however the kirchhoff condition is imposed on discrete points a multigrid algorithm is analyzed and numerical examples are provided furthermore a suitable preconditioning is established for the use of conjugate gradients

overcoming corner singularities using multigrid methods we consider the poisson equation delta u f with homogeneous dirichlet boundary condition on a two dimensional polygonal domain o we develop a finite element multigrid method on quasi uniform grids that obtains cal o h m 1 epsilon convergence in the h 1 o norm for any positive epsilon when f in h m o the cost of the method is proportional to the number of elements in the triangulation the results of this paper can be generalized to other equations and other boundary conditions

multigrid computation of vector potentials in this paper we analyze the use of edge finite elements and the multigrid method to approximate the problem of computing a static magnetic field in a cavity we show how the mixed finite element discrete problem may be reformulated as a symmetric positive definite system this involves the proof of a discrete analogue of the second kind friedrichs inequality then we show how the multigrid method may be applied to compute the magnetic vector potential the difficulty here is that the bilinear forms on different meshes are not related in the standard way two different intergrid transfer operators are defined in the paper for the multigrid method the convergence theory and the numerical tests include both algorithms