subset algebra lift operators for 0 1 integer programming we extend the sherali adams lovasz schrijver balas ceria cornuejols and lasserre lift and project methods for 0 1 optimization by considering liftings to subset algebras our methods yield polynomial time algorithms for solving a relaxation of a set covering problem at least as strong as that given by the set of all valid inequalities with small coefficients and more generally all valid inequalities where the right hand side is not very large relative to the positive coefficients in the left hand side applied to generalizations of vertex packing problems our methods yield in polynomial time relaxations that have unbounded rank using for example the n operator

lifts of convex sets and cone factorizations in this paper we address the basic geometric question of when a given convex set is the image under a linear map of an affine slice of a given closed convex cone such a representation or lift of the convex set is especially useful if the cone admits an efficient algorithm for linear optimization over its affine slices we show that the existence of a lift of a convex set to a cone is equivalent to the existence of a factorization of an operator associated to the set and its polar via elements in the cone and its dual this generalizes a theorem of yannakakis that established a connection between polyhedral lifts of a polytope and nonnegative factorizations of its slack matrix symmetric lifts of convex sets can also be characterized similarly when the cones live in a family our results lead to the definition of the rank of a convex set with respect to this family we present results about this rank in the context of cones of positive semidefinite matrices our methods provide new tools for understanding cone lifts of convex sets

the norm convergence of the trotter kato product formula with error bound the norm convergence of the trotter kato product formula with error bound is shown for the semigroup generated by that operator sum of two nonnegative selfadjoint operators a and b which is selfadjoint

solving lift and project relaxations of binary integer programs we propose a method for optimizing the lift and project relaxations of binary integer programs introduced by lovasz and schrijver in particular we study both linear and semidefinite relaxations the key idea is a restructuring of the relaxations which isolates the complicating constraints and allows for a lagrangian approach we detail an enhanced subgradient method and discuss its efficient implementation computational results illustrate that our algorithm produces tight bounds more quickly than state of the art linear and semidefinite solvers

an improved semidefinite programming relaxation for the satisfiability problem the satisfiability sat problem is a central problem in mathematical logic computing theory and artificial intelligence an instance of sat is specified by a set of boolean variables and a propositional formula in conjunctive normal form given such an instance the sat problem asks whether there is a truth assignment to the variables such that the formula is satisfied it is well known that sat is in general np complete although several important special cases can be solved in polynomial time semidefinite programming sdp refers to the class of optimization problems where a linear function of a matrix variable x is maximized or minimized subject to linear constraints on the elements of x and the additional constraint that x be positive semidefinite we are interested in the application of sdp to satisfiability problems and in particular in how sdp can be used to detect unsatisfiability in this paper we introduce a new sdp relaxation for the satisfiability problem this sdp relaxation arises from the recently introduced paradigm of higher liftings for constructing semidefinite programming relaxations of discrete optimization problems to derive the sdp relaxation we first formulate sat as an optimization problem involving matrices relaxing this formulation yields an sdp which significantly improves on the previous relaxations in the literature the important characteristics of the sdp relaxation are its ability to prove that a given sat formula is unsatisfiable independently of the lengths of the clauses in the formula its potential to yield truth assignments satisfying the sat instance if a feasible matrix of sufficiently low rank is computed and the fact that it is more amenable to practical computation than previous sdps arising from higher liftings we present theoretical and computational results that support these claims

some remarks on operator norm convergence of the trotter product formula on banach spaces abstract we extend the trotter kato chernoff theory of strong approximation of c 0 semigroups on banach spaces to operator norm approximation of analytic semigroups with error estimate as application we obtain a criterion for the operator norm convergence of the trotter product formula on banach spaces with error estimate n 1 log n provided one of the generators has a bounded h functional calculus for both results we present versions for approximation in operator ideal norms such as the trace norm or the hilbert schmidt norm finally we give some remarks on the operator norm convergence of the trotter product formula for semigroups acting on a scale of l p spaces

least squares adaptive lattice and transversal filters a unified geometric theory a unified theory is presented to characterize least squares adaptive filters in either lattice or transversal filter form for nonstationary processes the derivations are based upon a geometric formulation of least squares estimation and on the concept of displacement rank a few basic geometric relations are shown to underlie the various algorithms insights into the fundamental concepts that unify lattice and transversal filter approaches to least squares adaptive filters are also given the general results are illustrated by applications to the so called pre windowed and growing memory covariance formulations of the deterministic least squares problem

on the set covering polytope i all the facets with coefficients in 0 1 2 while the set packing polytope through its connection with vertex packing has lent itself to fruitful investigations little is known about the set covering polytope we characterize the class of valid inequalities for the set covering polytope with coefficients equal to 0 1 or 2 and give necessary and sufficient conditions for such an inequality to be minimal and to be facet defining we show that all inequalities in the above class are contained in the elementary closure of the constraint set and that 2 is the largest value ofk such that all valid inequalities for the set covering polytope with integer coefficients no greater thank are contained in the elementary closure we point out a connection between minimal inequalities in the class investigated and certain circulant submatrices of the coefficient matrix finally we discuss conditions for an inequality to cut off a fractional solution to the linear programming relaxation of the set covering problem and to improve the lower bound given by a feasible solution to the dual of the linear programming relaxation

sum of squares hierarchy lower bounds for symmetric formulations we introduce a method for proving sum of squares sos lasserre hierarchy lower bounds when the initial problem formulation exhibits a high degree of symmetry our main technical theorem allows us to reduce the study of the positive semidefiniteness to the analysis of well behaved univariate polynomial inequalities we illustrate the technique on two problems one unconstrained and the other with constraints more precisely we give a short elementary proof of grigoriev laurent lower bound for finding the integer cut polytope of the complete graph we also show that the sos hierarchy requires a non constant number of rounds to improve the initial integrality gap of 2 for the min knapsack linear program strengthened with cover inequalities

solving a polynomial equation some history and recent progress the classical problem of solving an nth degree polynomial equation has substantially influenced the development of mathematics throughout the centuries and still has several important applications to the theory and practice of present day computing we briefly recall the history of the algorithmic approach to this problem and then review some successful solution algorithms we end by outlining some algorithms of 1995 that solve this problem at a surprisingly low computational cost

a look ahead levinson algorithm for general toeplitz systems the authors present an extension of levinson s algorithm that is guaranteed to be weakly stable for a large class of general toeplitz matrices namely those that do not have many consecutive ill conditioned leading principal submatrices the new algorithm adapts itself to the given toeplitz matrix by skipping over all the ill conditioned leading principal submatrices encountered during the solution process this is done by a look ahead strategy that monitors the condition of the leading principal submatrices and if necessary switches to a block step of suitable size the overhead of the look ahead algorithm is typically small compared to the classical levinson algorithm and in addition a reliable condition number estimate is produced

a new algorithm for solving toeplitz systems of equations abstract we present some recurrences that are the basis for an algorithm to invert an n n toeplitz system of linear equations with computational complexity o n log 2 n the recurrences used are closely related to the zohar trench and bareiss algorithms but do not have any obvious connection with other asymptotically fast algorithms for the inversion of toeplitz systems

valid inequalities for mixed integer linear programs this tutorial presents a theory of valid inequalities for mixed integer linear sets it introduces the necessary tools from polyhedral theory and gives a geometric understanding of several classical families of valid inequalities such as lift and project cuts gomory mixed integer cuts mixed integer rounding cuts split cuts and intersection cuts and it reveals the relationships between these families the tutorial also discusses computational aspects of generating the cuts and their strength

lower bound for the number of iterations in semidefinite hierarchies for the cut polytope hierarchies of semidefinite relaxations for 0 1 polytopes have been constructed by lasserre 2001a and by lovaisz and schrijver 1991 the cut polytope of a graph onn nodes can be expressed as a projection of such a semidefinite relaxation after at mostn steps we show that n 2 iterations are needed for finding the cut polytope of the complete graphk n

a comparison of the sherali adams lov 341sz schrijver and lasserre relaxations for 0 1 programming sherali and adams sa90 lov asz and schrijver ls91 and recently lasserre las01b have proposed lift and project methods for constructing hierarchies of successive linear or semidefinite relaxations of a 0 1 polytope p subseteq or n converging to p in n steps lasserre s approach uses results about representations of positive polynomials as sums of squares and the dual theory of moments we present the three methods in a common elementary framework and show that the lasserre construction provides the tightest relaxations of p as an application this gives a direct simple proof for the convergence of the lasserre s hierarchy we describe applications to the stable set polytope and to the cut polytope

trotter kato product formula and operator norm convergence we find necessary and sufficient conditions for the operator norm convergence of the trotter kato product formula using them we prove that this convergence takes place i if the resolvent of one of the involved operators is compact either ii if one operator is relatively compact with respect to another one or iii if the product of resolvents of the involved operators is compact

recursive partial realization from the combined sequence of markov parameters and moments abstract in this paper we develop the recursive structure of partial realization from a two sided sequence of markov parameters and moments the specific feature of this recursive structure is that it accommodates sequences growing in two directions like in the two point pade approximation we present a recursive algorithm for the minimal partial realization from a sequence of markov parameters and moments that amounts to recursive lu factorization of a toeplitz behavior matrix leaping over nongeneric sections it turns out that this algorithm reveals a new continued fraction expansion which extends the classical cauer third form as a corollary various results related to toeplitz matrices two point pade approximations continued fractions etc are unified within this partial realization setting

a stabilized superfast solver for indefinite hankel systems abstract we present a stabilized superfast solver for indefinite hankel systems whose size is a power of 2 the hankel system is transformed into a loewner system which is solved by using an inversion formula for loewner matrices this explicit formula for the inverse of a loewner matrix contains certain parameters that are computed by solving two linearized rational interpolation problems on the unit circle the heart of our hankel solver is a superfast algorithm to solve these interpolation problems this algorithm is stabilized via pivoting iterative improvement and by giving the so called difficult interpolation points an adequate treatment we have implemented our algorithm in fortran 90 numerical examples illustrate the effectiveness of our approach

operator splitting for non autonomous evolution equations abstract we establish general product formulas for the solutions of non autonomous abstract cauchy problems the main technical tools are evolution semigroups allowing the direct application of existing results on autonomous problems the results obtained are illustrated by the example of an autonomous diffusion equation perturbed with time dependent potential we also prove convergence rates for the sequential splitting applied to this problem

parallel and superfast algorithms for hankel systems of equations utilizing kernel structure properties a unified construction of hankel matrix inversion algorithms is presented three types of algorithms are obtained 1 o n 2 complexity levinson type 2 o n parallel complexity schur type and 3 o n log2 n complexity asymptotically fast ones all algorithms work without additional assumption like strong nonsingularity

strengthening convex relaxations of 0 1 sets using boolean formulas in convex integer programming various procedures have been developed to strengthen convex relaxations of sets of integer points on the one hand there exist several general purpose methods that strengthen relaxations without specific knowledge of the set s of feasible integer points such as popular linear programming or semi definite programming hierarchies on the other hand various methods have been designed for obtaining strengthened relaxations for very specific sets s that arise in combinatorial optimization we propose a new efficient method that interpolates between these two approaches our procedure strengthens any convex set containing a set s subseteq 0 1 n by exploiting certain additional information about s namely the required extra information will be in the form of a boolean formula phi defining the target set s the new relaxation is obtained by feeding the convex set into the formula phi we analyze various aspects regarding the strength of our procedure as one application interpreting an iterated application of our procedure as a hierarchy our findings simplify improve and extend previous results by bienstock and zuckerberg on covering problems

work preserving speed up of parallel matrix computations brent s scheduling principle provides a general simulation scheme when fewer processors are available than specified by the fastest parallel algorithm such a scheme preserves under slow down the actual number of executed operations also called work in this paper we take the complementary viewpoint and rather than consider the work preserving slow down of some fast parallel algorithm we investigate the problem of the achievable speed ups of computation while preserving the work of the best known sequential algorithm for the same problem the proposed technique eminently applicable to problems of matrix computational flavor achieves its result through the interplay of two algorithms with significantly different features analogous but structurally different interplays have been used previously to improve the algorithmic efficiency of graph computations selection and list ranking we demonstrate the efficacy of our technique for the computation of path algebras in graphs and digraphs and various fundamental computations in linear algebra some of the fundamental new algorithms may have practical value for instance we substantially improve the algorithmic performance of the parallel solution of triangular and toeplitz linear systems of equations and the computation of the transitive closure of digraphs

trotter kato product formula and symmetrically normed ideals abstract it is proven that the trotter product formula converges in the norm of a symmetrically normed ideal of compact operators away from t0 0 if the kac operator the transfer matrix f t e tb 2e tae tb 2 belongs to this ideal for t t0 the result is generalized to the trotter kato product formula moreover if the perturbation b is small relative to a then error bounds for convergence are obtained the results apply to the dixmier trace

notes on the chernoff product formula abstract we revise the strong convergent chernoff product formula and extend it in a hilbert space to convergence in the operator norm topology main results deal with the self adjoint chernoff product formula the non self adjoint case concerns the quasi sectorial contractions

a lift and project cutting plane algorithm for mixed 0 1 programs we propose a cutting plane algorithm for mixed 0 1 programs based on a family of polyhedra which strengthen the usual lp relaxation we show how to generate a facet of a polyhedron in this family which is most violated by the current fractional point this cut is found through the solution of a linear program that has about twice the size of the usual lp relaxation a lifting step is used to reduce the size of the lp s needed to generate the cuts an additional strengthening step suggested by balas and jeroslow is then applied we report our computational experience with a preliminary version of the algorithm this approach is related to the work of balas on disjunctive programming the matrix cone relaxations of lovasz and schrijver and the hierarchy of relaxations of sherali and adams

when does the positive semidefiniteness constraint help in lifting procedures we study the lift and project procedures of lovaisz and schrijver for 0 1 integer programming problems we prove that the procedure using the positive semidefiniteness constraint is not better than the one without it in the worst case various examples are considered we also provide geometric conditions characterizing when the positive semidefiniteness constraint does not help

strong formulations for mixed integer programs valid inequalities and extended formulations we examine progress over the last fifteen years in finding strong valid inequalities and tight extended formulations for simple mixed integer sets lying both on the easy and hard sides of the complexity frontier most progress has been made in studying sets arising from knapsack and single node flow sets and a variety of sets motivated by different lot sizing models we conclude by citing briefly some of the more intriguing new avenues of research

parallel complexity of computations with general and toeplitz like matrices filled with integers and extensions computations with toeplitz and toeplitz like matrices are fundamental for many areas of algebraic and numerical computing the list of computational problems reducible to toeplitz and toeplitz like computations includes in particular the evaluation of the greatest common divisor gcd the least common multiple lcm and the resultant of two polynomials computing pade approximation and the berlekamp massey recurrence coefficients as well as numerous problems reducible to these transition to toeplitz and toeplitz like computations is currently the basis for the design of the parallel randomized nc rnc algorithms for these computational problems our main result is in constructing nearly optimal randomized parallel algorithms for toeplitz and toeplitz like computations and consequently for numerous related computational problems including the computational problems listed above where all the input values are integers and all the output values are computed exactly this includes randomized parallel algorithms for computing the rank the determinant and a basis for the null space of an n n toeplitz or toeplitz like matrix a filled with integers as well as a solution x to a linear system a x f if the system is consistent our algorithms use o log n log n log a parallel time and o n log n processors each capable of performing in unit time an arithmetic operation a comparision or a rounding of a rational number to a closest integer the cost bounds cover the cost of the verification of the correctness of the output the computations by these algorithms can be performed with the precision of o n log a bits which matches the precision required in order to represent the output except for the rank computation where the precision of the computation decreases the algorithms involve either a single random parameter or at most 2n 1 parameters the cited processor bounds are less by roughly factor n than ones supported by the known algorithms that run in polylogarithmic arithmetic time and do not use rounding to the closest integers technically we first devise new algorithms supporting our old nearly optimal complexity estimates for parallel computations with general matrices filled with integers then we decrease dramatically by roughly factor n1 376 the processor bounds required in these algorithms in the case where the input matrix is toeplitz like our algorithms exploit and combine some new techniques which may be of independent interest e g in the study of parallel and sequential computation of recursive factorization of integer matrices as well as our earlier techniques of variable diagonal relating to each other several known algebraic and numerical methods stream contraction and the truncation of displacement generators in toeplitz like computations our development and application of these techniques may be of independent

optimal and nearly optimal algorithms for approximating polynomial zeros abstract we substantially improve the known algorithms for approximating all the complex zeros of an n th degree polynomial p x our new algorithms save both boolean and arithmetic sequential time versus the previous best algorithms of schonhage 1 pan 2 and neff and reif 3 in parallel nc implementation we dramatically decrease the number of processors versus the parallel algorithm of neff 4 which was the only nc algorithm known for this problem so far specifically under the simple normalization assumption that the variable x has been scaled so as to confine the zeros of p x to the unit disc x x 1 our algorithms which promise to be practically effective approximate all the zeros of p x within the absolute error bound 2 b by using order of n arithmetic operations and order of b n n 2 boolean bitwise operations in both cases up to within polylogarithmic factors the algorithms allow their optimal work preserving nc parallelization so that they can be implemented by using polylogarithmic time and the orders of n arithmetic processors or b n n 2 boolean processors all the cited bounds on the computational complexity are within polylogarithmic factors from the optimum in terms of n and b under both arithmetic and boolean models of computation in the boolean case under the additional realistic assumption that n o b

complexity of computations with matrices and polynomials this paper reviews the complexity of polynomial and matrix computations as well as their various correlations to each other and some major techniques for the design of algebraic and numerical algorithms

cones of matrices and set functions and 0 1 optimization it has been recognized recently that to represent a polyhedron as the projection of a higher dimensional but simpler polyhedron is a powerful tool in polyhedral combinatorics a general method is developed to construct higher dimensional polyhedra or in some cases convex sets whose projection approximates the convex hull of 0 1 valued solutions of a system of linear inequalities an important feature of these approximations is that one can optimize any linear objective function over them in polynomial time in the special case of the vertex packing polytope a sequence of systems of inequalities is obtained such that the first system already includes clique odd hole odd antihole wheel and orthogonality constraints in particular for perfect and many other graphs this first system gives the vertex packing polytope for various classes of graphs including t perfect graphs it follows that the stable set polytope is the projection of a polytope with a polynomial number of facets an extension o

the mixing mir set with divisible capacities we study the set s x y in re times z n x b j y j geq b j j 1 ldots n where b j b j in re 0 j 1 n and b 1 b n the set s generalizes the mixed integer rounding mir set of nemhauser and wolsey and the mixing mir set of gunluk and pochet in addition it arises as a substructure in general mixed integer programming mip such as in lot sizing despite its importance a number of basic questions about s remain unanswered including the tractability of optimization over s and how to efficiently find a most violated cutting plane valid for p conv s we address these questions by analyzing the extreme points and extreme rays of p we give all extreme points and extreme rays of p in the worst case the number of extreme points grows exponentially with n however we show that in some interesting cases it is bounded by a polynomial of n in such cases it is possible to derive strong cutting planes for p efficiently finally we use our results on the extreme points of p to give a polynomial time algorithm for solving optimization over s

the weak and strong stability of algorithms in numerical linear algebra the stability of algorithms in numerical linear algebra is discussed the concept of stability is extended to notions of weak stability and strong stability justifications are given for these extensions and the implications of error analyses in terms of these definitions are discussed the concept of weak stability helps to clarify some of the controversy which has arisen concerning the stability of algorithms for toeplitz systems

parallel computation of polynomial gcd and some related parallel computations over abstract fields several fundamental problems of computations with polynomials and structured matrices are well known for their resistance to effective parallel solution this includes computing the gcd the 1cm and the resultant of two polynomials as well as any selected entry of either the extended euclidean scheme for these polynomials or the pade approximation table associated to a fixed power series the solution of the berlekamp massey problem of recovering the n coefficients of a linear recurrence sequence from its first 2n terms the solution of a toeplitz linear system of equations computing the ranks and the null spaces of toeplitz matrices and several other computations with toeplitz hankel vandermonde and cauchy generalized hilbert matrices and with matrices having similar structures our algorithms enable us to reach new record estimates for randomized parallel arithmetic complexity of all these computations over any field of constants that is o log n h time and o n2 log log n log n h g arithmetic processors where n is the input size g varies from 1 to 2 and h varies from 2 to 4 depending on the computational problem and on the fixed field of constants the estimates include the cost of correctness tests for the solution the results have further applications to many related computations over abstract fields and rely on some new techniques in particular on recursive pade approximation and regularization of pade approximation via randomization which might be of some independent technical interest in particular an auxiliary algorithm computes over any field the coefficients of a polynomial from the power sums of its zeros which is an important problem of algebraic coding

complexity of parallel matrix computations abstract we estimate parallel complexity of several matrix computations under both boolean and arithmetic machine models using deterministic and probabilistic approaches those computations include the evaluation of the inverse the determinant and the characteristic polynomial of a matrix recently processor efficiency of the previous parallel algorithms for numerical matrix inversion has been substantially improved in pan and reif 1987 reaching optimum estimates up to within a logarithmic factor that work however applies neither to the evaluation of the determinant and the characteristic polynomial nor to exact matrix inversion nor to the numerical inversion of ill conditioned matrices we present four new approaches to the solution of those latter problems having several applications to combinatorial computations in order to extend the suboptimum time and processor bounds of pan and reif 1987 to the case of computing the inverse determinant and characteristic polynomial of an arbitrary integer input matrix in addition processor efficient algorithms using polylogarithmic parallel time are devised for some other matrix computations such as triangular and qr factorizations of a matrix and its reduction to hessenberg form

fast gaussian elimination with partial pivoting for matrices with displacement structure fast o n 2 implementation of gaussian elimination with partial pivoting is designed for matrices possessing cauchy like displacement structure we show how toeplitz like toeplitz plus hankel like and vandermonde like matrices can be transformed into cauchy like matrices by using discrete fourier cosine or sine transform matrices in particular this allows us to propose a new fast o n 2 toeplitz solver gko which incorporates partial pivoting a large set of numerical examples showed that gko demonstrated stable numerical behavior and can be recommended for solving linear systems especially with nonsymmetric indefinite and ill conditioned positive definite toeplitz matrices it is also useful for block toeplitz and mosaic toeplitz toeplitz block matrices the algorithms proposed in this paper suggest an attractive alternative to look ahead approaches where one has to jump over ill conditioned leading submatrices which in the worst case requires o n 3 operations

an exact recursion for the composite nearest neighbor degeneracy for a 2 n lattice space a set theoretic argument is utilized to develop a recursion relation that yields exactly the composite nearest neighbor degeneracy for simple indistinguishable particles distributed on a 2 n lattice space the associated generating functions as well as the expectation of the resulting statistics are also treated

the trotter kato product formula for gibbs semigroups the trace norm convergence of the trotter lie product formula has recently been proved for particular classes of gibbs semigroups in the present paper we prove it for the whole generality including generalization of the product formula proposed by kato

periodic quasi orthogonal spline bases and applications to least squares curve fitting of digital images presents a new covariant basis dubbed the quasi orthogonal q spline basis for the space of n degree periodic uniform splines with k knots this basis is obtained analogously to the b spline basis by scaling and periodically translating a single spline function of bounded support the construction hinges on an important theorem involving the asymptotic behavior in the dimension of the inverse of banded toeplitz matrices the authors show that the gram matrix for this basis is nearly diagonal hence the name quasi orthogonal the new basis is applied to the problem of approximating closed digital curves in 2d images by least squares fitting since the new spline basis is almost orthogonal the least squares solution can be approximated by decimating a convolution between a resolution dependent kernel and the given data the approximating curve is expressed as a linear combination of the new spline functions and new control points another convolution maps these control points to the classical b spline control points a generalization of the result has relevance to the solution of regularized fitting problems

mixed 0 1 programming by lift and project in a branch and cut framework we investigate the computational issues that need to be addressed when incorporating general cutting planes for mixed 0 1 programs into a branch and cut framework the cuts we use are of the lift and project variety some of the issues addressed have a theoretical answer but others are of an experimental nature and are settled by comparing alternatives on a set of test problems the resulting code is a robust solver for mixed 0 1 programs we compare it with several existing codes on a wide range of test problems it performs as well as or better than some of the best currently available mixed integer programming codes

improved parallel computations with toeplitz like and hankel like matrices abstract the known fast algorithms for computations with general toeplitz hankel toeplitz like and hankel like matrices are inherently sequential we develop some new techniques in order to devise fast parallel algorithms for such computations including the evaluation of krylov sequences for such matrices traces of their power sums characteristic polynomials and generalized inverses this has further extensions to computing the solution or a least squares solution to a linear system of equations with such a matrix and to several polynomial computations such as computing the gcd lcm pade approximation and extended euclidean scheme for two polynomials as well as to computing the minimum span of a linear recurrence sequence the algorithms can be applied over any field of constants with the resulting advantages of using modular arithmetic the algorithms consist of simple computational blocks mostly reduced to fast fourier transforms and have potential practical value we also develop the techniques for extending all our results to the case of matrices representable as the sums of toeplitz like and hankel like matrices

a new approach to fast polynomial interpolation and multipoint evaluation abstract the fastest known algorithms for the problems of polynomial evaluation and multipoint interpolation are devastatingly unstable numerically because of their recursive use of polynomial divisions we apply a completely distinct approach to compute approximate solutions to both problems equally fast but with improved numerical stability our approach relies on new techniques so far not used in this area we reduce the problems to vandermonde matrix computations and then exploit some recent methods for improving computations with structured matrices

the trotter lie product formula for gibbs semigroups the trace norm and hilbert schmidt norm convergence of the trotter lie formula is proved for some classes of gibbs semigroups

characterizing polytopes in the 0 1 cube with bounded chvatal gomory rank let s 0 1 n and r be any polytope contained in 0 1 n with r 0 1 n s we prove that r has bounded chvatal gomory rank cg rank provided that s has bounded notch and bounded gap where the notch is the minimum integer p such that all p dimensional faces of the 0 1 cube have a nonempty intersection with s and the gap is a measure of the size of the facet coefficients of conv s let h s denote the subgraph of the n cube induced by the vertices not in s we prove that if h s does not contain a subdivision of a large complete graph then the notch and the gap are bounded by our main result this implies that the cg rank of r is bounded as a function of the treewidth of h s we also prove that if s has notch 3 then the cg rank of r is always bounded both results generalize a recent theorem of cornuejols and lee cornuejols g lee d 2016 on some polytopes contained in the 0 1 hypercube that have a small chvatal rank louveaux q skutella m eds proc 18th internat conf integer programm

parallel algorithms for algebraic problems fast parallel algorithms are presented for the following problems in symbolic manipulation of univariate polynomials computing all entries of the extended euclidean scheme of two polynomials over an arbitrary field gcd and 1cm of many polynomials factoring polynomials over finite fields and the squarefree decomposition of polynomials over fields of characteristic zero and over finite fields for the following estimates assume that the input polynomials have degree at most n and the finite field has p d elements the euclidean algorithm is deterministic and runs in parallel time o log 2 n all the other algorithms are probabilistic las vegas in the general case but when applicable to bf q or bf r they can be implemented deterministically over these fields the algorithms for gcd and lcm use parallel time o log 2 n the factoring algorithm runs in parallel time o log 2 n log 2 d 1 log p the algorithm for squarefree decomposition runs in parallel time o log 2 n

parametrization of newton s iteration for computations with structured matrices and applications abstract we apply a new parametrized version of newton s iteration in order to compute over any field f of constants the solution or at least squares solution to linear system b x v with an n n toeplitz or toeplitz like matrix b as well as the determinant of b and the coefficients of its characteristic polynomial det i b dramatically improving the processor efficiency of the known fast parallel algorithms our algorithms together with some previously known and some recent results of 1 5 as well as with our new techniques for computing polynomial god s and lcm s imply respective improvement of the known estimates for parallel arithmetic complexity of several fundamental computations with polynomials and with both structured and general matrices

an approximation to wiener measure and quantization of the hamiltonian on manifolds with non positive sectional curvature this paper gives a rigorous interpretation of a feynman path integral on a riemannian manifold m with non positive sectional curvature an l2 riemannian metric gp is given on the space of piecewise geodesic paths hp m adapted to the partition p of 0 1 whence a finite dimensional approximation of wiener measure is developed it is proved that as mesh p 0 the approximate wiener measure converges in an l1 sense to the measure exp 2 3203 01scal s ds d on the wiener space w m with wiener measure this gives a possible prescription for the path integral representation of the quantized hamiltonian as well as yielding such a result for the natural geometric approximation schemes originating in andersson and driver 1999 3 and followed by lim 2007 34

analysis of coppersmith s block wiedemann algorithm for the parallel solution of sparse linear systems by using projections by a block of vectors in place of a single vector it is possible to parallelize the outer loop of iterative methods for solving sparse linear systems we analyze such a scheme proposed by coppersmith for wiedemann s coordinate recurrence algorithm which is based in part on the krylov subspace approach we prove that by use of certain randomizations on the input system the parallel speed up is roughly by the number of vectors in the blocks when using as many processors our analysis is valid for fields of entries that have sufficiently large cardinality our analysis also deals with an arising subproblem of solving a singular block toeplitz system by use of the theory of toeplitz like matrices

on cutting plane proofs in combinatorial optimization abstract gomory s cutting plane technique can be viewed as a recursive procedure for proving the validity of linear inequalities over the set of all integer vectors in a prescribed polyhedron the number of rounds of cutting planes needed to obtain all valid linear inequalities is known as the rank of the polyhedron we prove that polyhedra featured in popular formulations of the stable set problem the set covering problem the set partitioning problem the knapsack problem the bipartite subgraph problem the maximum cut problem the acyclic subdigraph problem the asymmetric traveling salesman problem and the traveling salesman problem have arbitrarily high rank in particular we prove conjectures of barahona grotschel and mahjoub chvatal grotschel and pulleyblank and junger

optimal size integer division circuits division is a fundamental problem for arithmetic and algebraic computation this paper describes boolean circuits of bounded fan in for integer division finding reciprocals that have size o m n and depth o log n log log n where m n is the size complexity of o log n depth integer multiplication circuits currently m n is known to be o log n log log n but any improvement in this bound that preserves circuit depth will be reflected by a similar improvement in the size complexity of our division algorithm previously no one has been able to derive a division circuit with size o n log c n for any c and simultaneous depth less than omega log 2 n the circuit families described in this paper are logspace uniform that is they can be constructed by a deterministic turing machine in space o log n the results match the best known depth bounds for logspace uniform circuits and are optimal in size the general method of high order iterative formulas is of independent

efficient numerical methods in non uniform sampling theory summary we present a new second generation reconstruction algorithm for irregular sampling i e for the problem of recovering a band limited function from its non uniformly sampled values the efficient new method is a combination of the adaptive weights method which was developed by the two first named authors and the method of conjugate gradients for the solution of positive definite linear systems the choice of adaptive weights can be seen as a simple but very efficient method of preconditioning further substantial acceleration is achieved by utilizing the toeplitztype structure of the system matrix this new algorithm can handle problems of much larger dimension and condition number than have been accessible so far furthermore if some gaps between samples are large then the algorithm can still be used as a very efficient extrapolation method across the gaps

on the matrix cut rank of polyhedra lovasz and schrijver 1991 described a semidefinite operator for generating strong valid inequalities for the 0 1 vectors in a prescribed polyhedron among their results they showed that n iterations of the operator are sufficient to generate the convex hull of 0 1 vectors contained in a polyhedron in n space we give a simple example having chvatal rank 1 that meets this worst case bound of n we describe another example requiring n iterations even when combining the semidefinite and gomory chvatal operators this second example is used to show that the standard linear programming relaxation of a k city traveling salesman problem requires at least k 8 iterations of the combined operator this bound is best possible up to a constant factor as k 1 iterations suffice

approximate fixed rank closures of covering problems consider a 0 1 integer program min ctx ax b x 0 1 n where a is nonnegative we show that if the number of minimal covers of ax b is polynomially bounded then for any e 0 and any fixed q there is a polynomially large lift and project relaxation whose value is at least 1 e times the value of the rank q relaxation a special case of this result is that given by set covering problems or generally problems where the coefficients in a and b are bounded

computation of the lasserre ranks of some polytopes over the years various lift and project methods have been proposed to construct hierarchies of successive linear or semidefinite relaxations of a 0 1 polytope p rn that converge to p in n steps many such methods have been shown to require n steps in the worst case in this paper we show that the method of lasserre also requires n steps in the worst case

approximation of the semigroup generated by the hamiltonian of reggeon field theory in bargmann space abstract the reggeon field theory is governed by a non self adjoint operator constructed as a polynomial in a a the standard bose annihilation and creation operators in zero transverse dimension this hamiltonian acting in bargmann space is defined by h a 2 a 2 a a i a a a a where i 2 1 and are real numbers and the operators a a satisfy the commutation relation a a i as the quantum mechanical system described by h has a velocity dependent potential containing powers of momentum up to the fourth the problem of existence of hamiltonian path integral for the evolution operator e t h of this theory is of interest on its own in particular can we express e t h as a limit of integral operators in this article one considerably reduces the difficulty by studying the trotter product formula of h to reach two objectives the first objective is to prove a very specific error estimate for the error in a trotter product formula in trace norm for h viewed as the sum of the operators a 2 a 2 and a a i a a a a the second objective of this work is to give a approximation of the semigroup generated by h when h is split in the sum of a 2 a 2 a a and i a a a a we note that this case is entirely different in fact the usual trotter product formula is not defined because the interaction operator a a a a is not the infinitesimal generator of a semigroup on bargmann space for 0 and 0 we choose an approximation operator i i a a a a e a 2 a 2 a a and we give a connection between and e h this choice allows us to give in a intissar note on the path integral formulation of reggeon field theory preprint a generalized trotter product formula for t a a i a a a a i e for limit case as 0 and answers to the above question

fast and efficient parallel solution of sparse linear systems this paper presents a parallel algorithm for the solution of a linear system a bf x bf b with a sparse n times n symmetric positive definite matrix a associated with the graph g a that has n vertices and has an edge for each nonzero entry of a if g a has an s n separator family and a known s n separator tree then the algorithm requires only o log 3 n time and e m s n log n processors for the evaluation of the solution vector bf x a 1 bf b where e is the number of edges in g a and m n is the number of processors sufficient for multiplying two n times n rational matrices in time o log n furthermore for this computational cost the algorithm computes a recursive factorization of a such that the solution of any other linear system a bf x bf b with the same matrix a requires only o log 2 n time and e log n s n 2 processors

reducing matching to polynomial size linear programming the question of whether the maximum weight matching problem can be reduced to a linear program of polynomial size is studied a partial answer to it is given i e it is shown that the chinese postman problem and optimum matching reduces to a sequence of o m 2 log n minimum mean cycle problems it is shown that this last problem can be formulated as a linear program of polynomial size this gives a polynomial algorithm for matching based on any polynomial method for linear programming a combinatorial algorithm for finding minimum mean cycles in undirected graphs is also given

fast and efficient parallel solution of dense linear systems the most efficient previously known parallel algorithms for the inversion of a nonsingular n n matrix a or solving a linear system ax b over the rational numbers require o log2n time and m n n processors provided that m n processors suffice in order to multiply two n n rational matrices in time o logn furthermore the known polylog arithmetic time algorithms for those problems are numerically unstable in this paper we apply newton s iteration and initially choose an approximate inverse matrix by following ben israel this quadratically convergent and numerically stable iterative method takes o log2n parallel time using m n processors to compute the inverse within the relative precision 2 nc for a positive constant c of an n n rational matrix a with the condition number at most nd for a constant d this is the optimum processor bound and by a factor of n improvement of the previously known processor bounds for polylogarithmic time matrix inversion the algorithm does not require to precompute the condition number of the input matrix but it just converges slower for ill conditioned input matrices

tightening simple mixed integer sets with guaranteed bounds in this paper we study how to reformulate knapsack sets and simple mixed integer sets in order to obtain provably tight polynomially large formulations

fast parallel computation of polynomials using few processors it is shown that any multivariate polynomial of degree d that can be computed sequentially in c steps can be computed in parallel in o log d log c log d steps using only cd o 1 processors

theoretical challenges towards cutting plane selection while many classes of cutting planes are at the disposal of integer programming solvers our scientific understanding is far from complete with regards to cutting plane selection i e the task of selecting a portfolio of cutting planes to be added to the lp relaxation at a given node of the branch and bound tree in this paper we review the different classes of cutting planes available known theoretical results about their relative strength important issues pertaining to cut selection and discuss some possible new directions to be pursued in order to accomplish cutting plane selection in a more principled manner finally we review some lines of work that we undertook to provide a preliminary theoretical underpinning for some of the issues related to cut selection

on the rank of mixed 0 1 polyhedra for a polytope in the 0 1 n cube eisenbrand and schulz showed recently that the maximum chvatal rank is bounded above by o n 2logn and bounded below by 1 e n for some e 0 chvatal cuts are equivalent to gomory fractional cuts which are themselves dominated by gomory mixed integer cuts what do these upper and lower bounds become when the rank is defined relative to gomory mixed integer cuts an upper bound of n follows from existing results in the literature in this note we show that the lower bound is also equal to n this result still holds for mixed 0 1 polyhedra with n binary variables