l induced norm analysis of sampled data systems via piecewise constant and linear approximations this paper deals with the l 1 analysis of linear sampled data systems by which we mean the computation of the l induced norm of linear sampled data systems two computation methods based on piecewise constant and piecewise linear approximations are provided through fast lifting by which the sampling interval 0 h is divided into m subintervals with an equal width even though the central part of the method with the former approximation essentially coincides with a conventional method via fast sample fast hold fsfh approximation after all we show that both methods successfully lead to the upper and lower bounds of the l induced norm whose gap converges to 0 at the rate of 1 m in the former approximation and 1 m 2 in the latter extended approximation such achievements are in sharp contrast with an existing result on the former i e fsfh approximation which only shows the convergence rate of the error in the resulting estimate of the l induced norm without providing any readily computable upper and lower bounds a numerical example is given to illustrate the effectiveness of these methods

l2 gain analysis for a class of hybrid systems with applications to reset and event triggered control a lifting approach in this paper we study the stability and l 2 gain properties of a class of hybrid systems that exhibit linear flow dynamics periodic time triggered jumps and arbitrary nonlinear jump maps this class of hybrid systems is relevant for a broad range of applications including periodic event triggered control sampled data reset control sampled data saturated control and certain networked control systems with scheduling protocols for this class of continuous time hybrid systems we provide new stability and l 2 gain analysis methods inspired by ideas from lifting we show that the stability and the contractivity in l 2 sense meaning that the l 2 gain is smaller than 1 of the continuous time hybrid system is equivalent to the stability and the contractivity in l 2 sense meaning that the l 2 gain is smaller than 1 of an appropriate discrete time nonlinear system these new characterizations generalize earlier more conservative conditions provided in the literature we show via a reset control example and an event triggered control application for which stability and contractivity in l 2 sense is the same as stability and contractivity in l 2 sense of a discrete time piecewise linear system that the new conditions are significantly less conservative than the existing ones in the literature moreover we show that the existing conditions can be reinterpreted as a conservative l 2 gain analysis of a discretetime piecewise linear system based on common quadratic storage lyapunov functions these new insights are obtained by the adopted lifting based perspective on this problem which leads to computable l 2 gain and thus l 2 gain conditions despite the fact that the linearity assumption which is usually needed in the lifting literature is not satisfied

the generalized basis reduction algorithm let fx be a convex function defined in rn which is symmetric about the origin and homogeneous of degree 1 and let l be the lattice of integers zn a definition of a reduced basis b1 bn of the lattice with respect to the distance function f is presented and we describe an algorithm which yields a reduced basis in polynomial time for fixed n in the special case in which the bodies x fx t are ellipsoids the definition of a reduced basis is identical with that given by lenstra lenstra and lovasz 1982 and the algorithm is the well known basis reduction algorithm we show that the basis vector b1 in a reduced basis is an approximation to a shortest nonzero lattice point with respect to f and relate the basis vectors bi to minkowski s successive minima the results lead to an algorithm for integer programming which executes in polynomial time for fixed n but which avoids the ellipsoidal approximations required by lenstra s algorithm we also discuss the properties of a korkine zolotarev basis for the lattice

a polynomial time approximation scheme for the two stage multiprocessor flow shop problem in this paper we investigate the two stage multiprocessor flow shop scheduling problem f2 p cmax where the numbers m1 and m2 of machines available in the two stages are part of the input we demonstrate the existence of a polynomial time approximation scheme for this problem this result solves the simplest case of an open problem that has been posed by leslie hall in a recent paper hall 1995 an extension of our algorithm yields an approximation scheme for the closely related two stage multiprocessor job shop problem

computing the l 0 h induced norm of a compression operator via fast lifting abstract this paper studies computing the induced norm of a compression operator defined on the banach space l 0 h which is a difficult problem since it is an infinite rank operator two methods are provided for this problem each of which can compute an upper bound and a lower bound of the induced norm by using an idea of staircase or piecewise linear approximation staircase approximation and piecewise linear approximation are applied through fast lifting by which the interval 0 h is divided into m subintervals with equal width and the approximation errors in these methods are ensured to be reciprocally proportional to m or m 2 the effectiveness of the proposed methods is demonstrated through numerical examples

lattice basis reduction improved practical algorithms and solving subset sum problems we report on improved practical algorithms for lattice basis reduction we present a variant of the l3 algorithm with deep insertions and a practical algorithm for blockwise korkine zolotarev reduction a concept extending l3 reduction that has been introduced by schnorr 1987 empirical tests show that the strongest of these algorithms solves almost all subset sum problems with up to 58 random weights of arbitrary bit length within at most a few hours on a unisys 6000 70 or within a couple of minutes on a sparc 2 computer

injective stability for k1 of the orthogonal group abstract the injective stabilization bound of l n vaserstein for k1 of the orthogonal group over an affine algebra over a perfect c 1 field is discussed

the optimal lll algorithm is still polynomial in fixed dimension in this paper we consider the open problem of the complexity of the lll algorithm in the case when the approximation parameter of the algorithm has as its extreme value 1 this case is of interest because the output is then the strongest lovasz reduced basis experiments reported by lagarias and odlyzko j acm 32 1 1985 229 seem to show that the algorithm remains polynomial in average however no bound better than a naive exponential order one is established for the worst case complexity of the optimal lll algorithm even for fixed small dimension higher than 2 here we prove that for any fixed dimension n the number of iterations of the lll algorithm is linear with respect to the size of the input it is easy to deduce from vallee j algorithms 12 1991 556 that the linear order is optimal moreover in 3 dimensions we give a tight bound for the maximum number of iterations and we characterize precisely the output basis our bound also improves the known one for the usual non optimal lll algorithm

computing the structure of a finite abelian group we present an algorithm that computes the structure of a finite abelian group g from a generating system m the algorithm executes o m g group operations and stores o g group elements

on the rational polytopes with chvatal rank 1 abstract in this paper we study the following problem given a polytope p with chvatal rank 1 does p contain an integer point boyd and pulleyblank observed that this problem is in thecomplexity class np co np an indication that it is probably not np complete we solvethis problem in polynomial time for polytopes arising from the satisfiability problem of a givenformula with at least three literals per clause for simplices whose integer hull can be obtainedby adding at most a constant number of chvatal inequalities and for rounded polytopes weprove that any closed convex set whose chvatal closure is empty has an integer width of atmost n and we give an example showing that this bound is tight within an additive constantof 1 the promise that a polytope has chvatal rank 1 seems hard to verify though we provethat deciding emptiness of the chvatal closure of a given rational polytope p is np complete even when p is contained in the unit hypercube or is a rational simplex and even when p doesnot contain any integer point this has two implications i it is np hard to decide whethera given rational polytope p has chvatal rank 1 even when p is contained in the unit cube oris a rational simplex ii the optimization and separation problems over the chvatal closureof a given rational polytope contained in the unit hypercube or of a given rational simplex arenp hard these results improve earlier complexity results of cornuejols and li and eisenbrand finally we prove that for any positive integer k it is np hard to decide whether adding atmost k chvatal inequalities is sufficient to describe the integer hull of a given rational polytope

solving a system of linear diophantine equations with lower and upper bounds on the variables we develop an algorithm for solving a system of diophantine equations with lower and upper bounds on the variables the algorithm is based on lattice basis reduction it first finds a short vector satisfying the system of diophantine equations and a set of vectors belonging to the nullspace of the constraint matrix due to basis reduction all these vectors are relatively short the next step is to branch on linear combinations of the null space vectors which either yields a vector that satisfies the bound constraints or provides a proof that no such vector exists the research was motivated by the need for solving constrained diophantine equations as subproblems when designing integrated circuits for video signal processing our algorithm is tested with good results on real life data and on instances from the literature

computing bf l _ 2 gain of finite horizon systems with boundary conditions a bisection algorithm is developed for computing the l2 gain of a finite horizon system with boundary conditions upper and lower bounds of the gain are also derived for the initial step of the algorithm

optimal routing in double loop networks in this paper we study the problem of finding the shortest path in circulant graphs with an arbitrary number of jumps we provide algorithms specifically tailored for weighted undirected and directed circulant graphs with two jumps which compute the shortest path our method only requires o logn arithmetic operations and the total bit complexity is o log 2nloglognlogloglogn where n is the number of the graph s vertices this elementary and efficient shortest path algorithm has been derived from the closest vector problem cvp of lattices in dimension two and with an 1 norm

spectrum of monodromy operator for a time delay system with application to stability analysis this note studies the spectral properties of monodromy operators which play an important role in stability analysis of linear time invariant time delay feedback systems the note is motivated by the fact that this operator can actually be defined naturally on four spaces where the difference stems from different choices for the function space on which the infinite dimensional state of such a time delay system is assumed to take its value it is first shown that the spectrum of the monodromy operator is independent of the spaces on which it is defined this implies that stability of time delay systems is independent of the underlying function spaces it is further shown that the operator spectrum is continuous at monodromy operators which justifies the spectrum computation of the monodromy operator through its approximation by any sort of tractable operators a numerical study relevant to the theoretical development is provided and a practical implication of our theoretical study is suggested

computing the l2 induced norm of a compression operator abstract a new computationally viable approach is derived for computing the induced norm of a state space compression operator that is an integral operator defined on the finite length space l2 0 h determining this norm is a crucial component in the control analysis of both delay systems and sampled data systems the technique developed may have application to a wider variety of integral operators

generalized chvatal gomory closures for integer programs with bounds on variables integer programming problems that arise in practice often involve decision variables with one or two sided bounds in this paper we consider a generalization of chvatal gomory inequalities obtained by strengthening chvatal gomory inequalities using the bounds on the variables we prove that the closure of a rational polyhedron obtained after applying the generalized chvatal gomory inequalities is also a rational polyhedron this generalizes a result of dunkel and schulz on 0 1 problems to the case when some of the variables have upper or lower bounds or both while the rest of them are unbounded

on the separation of maximally violated mod k cuts separation is of fundamental importance in cutting plane based techniques for integer linear programming ilp in recent decades a considerable research effort has been devoted to the definition of effective separation procedures for families of well structured cuts in this paper we address the separation of chvatal rank 1 inequalities in the context of general ilp s of the form min c t x ax b x integer where a is an m n integer matrix and b an m dimensional integer vector in particular for any given integer k we study mod k cuts of the form t ax t b for any 0 1 k k 1 k m such that t a is integer following the line of research recently proposed for mod 2 cuts by applegate bixby chvatal and cook 1 and fleischer and tardos 19 we restrict to maximally violated cuts i e to inequalities which are violated by k 1 k by the given fractional point we show that for any given k such a separation requires o mn min m n time applications to both the symmetric and asymmetric tsp are discussed in particular for any given k we propose an o v 2 e time exact separation algorithm for mod k cuts which are maximally violated by a given fractional symmetric or asymmetric tsp solution with support graph g v e this implies that we can identify a maximally violated cut for the symmetric tsp whenever a maximally violated extended comb inequality exists finally facet defining mod k cuts for the symmetric and asymmetric tsp are studied

a subdivision based algorithm for the sparse resultant multivariate resultants generalize the sylvester resultant of two polynomials and characterize the solvability of a polynomial system they also reduce the computation of all common roots to a problem in linear algebra we propose a determinantal formula for the sparse resultant of an arbitrary system of n 1 polynomials in n variables this resultant generalizes the classical one and has significantly lower degree for polynomials that are sparse in the sense that their mixed volume is lower than their bezout number our algorithm uses a mixed polyhedral subdivision of the minkowski sum of the newton polytopes in order to construct a newton matrix its determinant is a nonzero multiple of the sparse resultant and the latter equals the gcd of at most n 1 such determinants this construction implies a restricted version of an effective sparse nullstellensatz for an arbitrary specialization of the coefficients there are two methods that use one extra variable and yield the sparse resultant this is the first algorithm to handle the general case with complexity polynomial in the resultant degree and simply exponential in n we conjecture its extension to producing an exact rational expression for the sparse resultant

injective stability for k 1 of classical modules in rao 1994 14 the second author and w van der kallen showed that the injective stabilization bound for k1 of the general linear group is d 1 over a regular affine algebra over a perfect c1 field where d is the krull dimension of the base ring which is finite and at least 2 in this article we prove that the injective stabilization bound for k1 of the symplectic group is d 1 over a geometrically regular ring containing a field where d is the dimension of the base ring which is finite and at least 2 using the local global principle for the transvection subgroup of the automorphism group of projective and symplectic modules we show that the injective stabilization bound is d 1 for k1 of projective and symplectic modules of global rank at least 1 and local rank at least 3 respectively in each of the two cases above

weighted sensitivity minimization for delay systems in this note we discuss the h infty sensitivity minimization problem for linear time invariant delay systems while the unweighted case reduces to simple nevanlinna pick interpolation the weighted case turns out to be much more complicated and demands certain functional analytic techniques for its solution

design and verify a new scheme for generating cutting planes a cutting plane procedure for integer programming ip problems usually involves invoking a black box procedure such as the gomory chvatal procedure to compute a cutting plane in this paper we describe an alternative paradigm of using the same cutting plane black box this involves two steps in the first step we design an inequality cx le d where c and d are integral independent of the cutting plane black box in the second step we verify that the designed inequality is a valid inequality by verifying that the set p cap x in mathbb r n mid cx ge d 1 cap mathbb z n is empty using cutting planes from the black box here p is the feasible region of the linear programming relaxation of the ip we refer to the closure of all cutting planes that can be verified to be valid using a specific cutting plane black box as the verification closure of the considered cutting plane black box this paper undertakes a systematic study of properties of verification closures of various cutting plane black box procedures

cutting plane proofs in polynomial space following chvatal cutting planes may be viewed as a proof system for establishing that a given system of linear inequalities has no integral solution we show that such proofs may be carried out in polynomial workspace

a function space approach to sampled data control systems and tracking problems this paper presents a new framework for hybrid sampled data control systems instead of considering the state only at sampling instants this paper introduces a function piece during the sampling period as the state and gives an infinite dimensional model with such a state space this gives the advantage that sampled data systems with built in intersample behavior can be regarded as linear time invariant discrete time systems as a result the approach makes it possible to introduce such time invariant concepts as transfer functions poles and zeros to the sampled data systems even with the presence of the intersample behavior in particular tracking problems can be studied in this setting in a simple and unified way and ripples are completely characterized as a mismatch between the intersample reference signal and transmission zero directions this leads to the internal model principle for sampled data systems

sampled data control of continuous time systems with an h optimality criterion abstract the synthesis of sampled data controllers for continuous time systems is studied in the context of h control theory the sampled data h optimal control problem consists of finding a sampled data controller which minimizes the l2 induced norm from the continuous time disturbance to the continuous time output the problem is here solved via a discrete system representation this is achieved by describing the closed loop system as a discrete system with finite state dimension and with a representation of the disturbance and the output signals of the discrete system as sequences of elements of certain infinite dimensional spaces via finite rank approximations of a compact integral operator associated with the system the sampled data h control problem can be solved to any degree of accuracy from the structure of the problem it follows that irrespective of the order of the approximation a feasible sampled data controller can be found which has the same order as the system

algorithms in algebraic number theory in this paper we discuss the basic problems of algorithmic algebraic number theory the emphasis is on aspects that are of interest from a purely mathematical point of view and practical issues are largely disregarded we describe what has been done and more importantly what remains to be done in the urea we hope to show that the study of algorithms not only increases our understanding of algebraic number fields but also stimulates our curiosity about them

a strongly polynomial algorithm for a concave production transportation problem with a fixed number of nonlinear variables we show that the production transportation problem involving an arbitrary fixed number of factories with concave production cost is solvable in strongly polynomial time the algorithm is based on a parametric approach which takes full advantage of the specific structure of the problem monotonicity of the objective function along certain directions small proportion of nonlinear variables and combinatorial properties implied by transportation constraints

a linear algorithm for integer programming in the plane we show that a 2 variable integer program defined by m constraints involving coefficients with at most bits can be solved with o m arithmetic operations on rational numbers of size o

polynomiality for bin packing with a constant number of item types we consider the bin packing problem with d different item sizes si and item multiplicities ai where all numbers are given in binary encoding this problem formulation is also known as the one dime

hermite and smith normal form algorithms over dedekind domains we show how the usual algorithms valid over euclidean domains such as the hermite normal form the modular hermite normal form and the smith normal form can be extended to dedekind rings in a sequel to this paper we will explain the use of these algorithms for computing in relative extensions of number fields

l_ 1 discretization for sampled data controller synthesis via piecewise linear approximation this paper develops a new discretization method with piecewise linear approximation for the l 1 optimal controller synthesis problem of sampled data systems which is the problem of minimizing the l infty induced norm of sampled data systems we apply fast lifting on the top of the lifting technique by which the sampling interval 0 h is divided into m subintervals with an equal width the signals on each subinterval are then approximated by linear functions by introducing two types of linearizing operators for input and output which leads to piecewise linear approximation of sampled data systems by using the arguments of preadjoint operators we provide an important inequality that forms a theoretical basis for tackling the l 1 optimal controller synthesis problem of sampled data systems more efficiently than the conventional method more precisely a mathematical basis for the piecewise linear approximation method associated with the convergence rate is shown through this inequality and this suggests that the piecewise linear approximation method may drastically outperform the conventional method in the l 1 optimal controller synthesis problem of sampled data systems we then provide a discretization procedure of sampled data systems by which the l 1 optimal controller synthesis problem is converted to the discrete time l 1 optimal controller synthesis problem finally effectiveness of the proposed method is demonstrated through a numerical example

on the structure of reduced kernel lattice bases lattice based reformulation techniques have been used successfully both theoretically and computationally one such reformulation is obtained from the kernel lattice associated with an input matrix some of the hard instances in the literature that have been successfully tackled by lattice based techniques have randomly generated input since the considered instances are very hard even in low dimension less experience is available for larger instances recently we have studied larger instances and observed that the lll reduced basis of the kernel lattice has a specific sparse structure in particular this translates into a map in which some of the original variables get a rich translation into a new variable space whereas some variables are only substituted in the new space if an original variable is important in the sense of branching or cutting planes this variable should be translated in a nontrivial way in this paper we partially explain through a probabilistic analysis the obtained struct

recognizing units in number fields we present a deterministic polynomial time algorithm that decides whether a power product ilki yn is a unit in the ring of integers of k where k is a number field yi are nonzero elements of k and ni are rational integers the main algorithm is based on the factor refinement method for ideals which might be of independent interest

factoring polynomials with rational coefficients in this paper we present a polynomial time algorithm to solve the following problem given a non zero polynomial fe q x in one variable with rational coefficients find the decomposition of f into irreducible factors in q x it is well known that this is equivalent to factoring primitive polynomials fez x into irreducible factors in z x here we call f z x primitive if the greatest common divisor of its coefficients the content of f is 1 our algorithm performs well in practice cf 8 its running time measured in bit operations is o nl2 n9 log fd3

cutting planes and the elementary closure in fixed dimension the elementary closurep of a polyhedronp is the intersection ofp with all its gomory chvatal cutting planes p is a rational polyhedron provided thatp is rational the known bounds for the number of inequalities definingp are exponential even in fixed dimension we show that the number of inequalities needed to describe the elementary closure of a rational polyhedron is polynomially bounded in fixed dimension ifp is a simplicial cone we construct a polytopeq whose integral elements correspond to cutting planes ofp the vertices of the integer hullq iinclude the facets ofp a polynomial upper bound on their number can be obtained by applying a result of cook et al 1992 finally we present a polynomial algorithm in varying dimension which computes cutting planes for a simplicial cone that correspond to vertices ofq i

valid linear inequalities for fixed charge problems many problems in the operations research management science literature can be formulated with both zero one and continuous variables however the exact optimization of such mixed zero one models remains a computational challenge in this paper we propose to study mixed problems from a mathematical point of view that is similar in spirit to recent research on purely combinatorial problems that has investigated systems of defining linear inequalities or facets of the underlying polytope at least two numerical studies have validated this line of research computationally and the advances in the problem solving capabilities are considerable we expect that similar gains are possible for the mixed zero one problem more precisely we consider the mixed integer programs whose feasible region x is composed of i a simple additive constraint in the continuous variables xj for j 1 2 n and ii constraints 0 xj mjyj defined by binary variables yj for j 1 2 n this type of feasible region arises in a variety of mixed integer problems and particularly in network problems with fixed charges on the arcs we derive two classes of facet defining linear inequalities of the convex hull of x and show that the second of these classes gives a complete description of the convex hull when mj m for all j we also develop methods to detect violated inequalities from these classes so that these facets can be used as cutting planes to strengthen the formulations of certain mixed integer problems

on the h optimal sensitivity problem for systems with delays in this paper we extend some of the results of ieee trans automat control ac 31 1986 pp 763 766 to more general delay systems in particular we analyze the effect of the interaction of delays and nonminimum phase zeros on the h infty optimal weighted sensitivity

recognizing badly presented z modules finitely generated z modules have canonical decompositions when such modules are given in a finitely presented form there is a classical algorithm for computing a canonical decomposition this is the algorithm for computing the smith normal form of an integer matrix we discuss algorithms for smith normal form computation and present practical algorithms which give excellent performance for modules arising from badly presented abelian groups we investigate such issues as congruential techniques sparsity considerations pivoting strategies for gauss jordan elimination lattice basis reduction and computational complexity our results which are primarily empirical show dramatically improved performance on previous methods

implementing the hamiltonian test for the h norm in linear continuous time periodic systems by the floquet similarity transformations of finite dimensional linear continuous time periodic fdlcp systems the time domain hamiltonian test for the h norm is first interpreted in terms of toeplitz operators of the system matrices based on this novel frequency domain interpretation implementing the time domain hamiltonian test can be accomplished by working on its frequency domain counterpart via truncations this gives us a bisection algorithm for evaluating the h norm of a class of fdlcp systems through finite dimensional linear time invariant continuous time models the finite dimensional hamiltonian test is necessary and sufficient in the asymptotic sense and claimed only via fourier coefficients of the system matrices without the transition matrix of the fdlcp system concerned there are numerical examples to illustrate the suggested algorithm

the l diversity problem tractability and approximability publishing personal data without giving up privacy is becoming an increasingly important problem in different fields in the last years different interesting approaches have been proposed i e k anonymity and l diversity given an input table these approaches partition its rows so that the computed partition satisfies some constraint in order to prevent the inference of the individuals the data belong to then the rows in a same set of the partition are related to the same rows by suppressing some of their entries here we focus on the l diversity problem where the attributes of the input table are distinguished in sensitive attributes and quasi identifier attributes the goal is to partition the rows of the input table so that each set c of the partition contains at most 1l c rows having a specific value in the sensitive attribute and the number of suppressions is minimized in this paper we investigate the approximation and parameterized complexity ofl diversity first we prove that the problem is not approximable within factor clnl for some constant c 0 even if the input table consists of only two columns and that the problem is apx hard even if l 4 and the input table contains exactly three columns then we give an approximation algorithm of factor m where m 1 is the number of columns in the input table when the sensitive attribute ranges over an alphabet of constant size concerning the parameterized complexity we prove that the problem is w 1 hard when parameterized by the cost bound by l and by the size of the alphabet then we prove that the problem admits a fixed parameter algorithm when both the maximum number of different values in a column and the number of columns are parameters

integer programming with a fixed number of variables it is shown that the integer linear programming problem with a fixed number of variables is polynomially solvable the proof depends on methods from geometry of numbers

analysis of pslq an integer relation finding algorithm let k be either the real complex or quaternion number system and let o k be the corresponding integers let x x 1 x n be a vector in k n the vector x has an integer relation if there exists a vector m m 1 m n o k n m 0 such that m 1 x 1 m 2 x 2 m n x n 0 in this paper we define the parameterized integer relation construction algorithm pslq where the parameter can be freely chosen in a certain interval beginning with an arbitrary vector x x 1 x n k n iterations of pslq will produce lower bounds on the norm of any possible relation for x thus pslq can be used to prove that there are no relations for x of norm less than a given size let m x be the smallest norm of any relation for x for the real and complex case and each fixed parameter in a certain interval we prove that pslq constructs a relation in less than o n 3 n 2 logm x iterations

computing quasi lti robustness margins in sampled data systems the robustness test for sampled data systems with slowly time varying perturbations is known to be infinite dimensional in nature the note develops computationally explicit upper and lower bounds for the corresponding stability radius presenting them in terms of linear matrix inequalities lmis given by state space formulas derived the upper bound is shown to converge monotonically to the stability radius and so can be systematically tightened at the cost of increased computational effort the lower bound is monotonically increasing and is conjectured to also converge to the stability radius

hermite normal form computation using modulo determinant arithmetic this paper describes a new class of hermite normal form solution procedures which perform modulo determinant arithmetic throughout the computation this class of procedures is shown to possess a polynomial time complexity bound which is a function of the length of the input string computational results are also given

algorithms for computing a hermite reduction of a matrix with polynomial coefficients resume nous etudions dans cet article des algorithmes de calcul de la reduction de hermite d une matrice a coefficients polynomiaux qui evitent l explosion de la taille des objets intermediaires et ont une bonne complexite sequentielle le deuxieme et le troisieme algorithmes generalisent la methode des sous resultants pour le calcul du pgcd de deux polynomes le dernier de ces algorithmes semble optimal en ce sens qu il calcule une reduction de hermite avec une matrice de passage de degre minimal en utilisant une methode progressive et economique nous ramenons le calcul de la reduite de hermite a un probleme de triangulation progressive des matrices dont les tailles sont bien controlees et dont les entrees sont des coefficients de polynomes donnes en entree l algorithme est donc faisable des qu on dispose d une methode de triangulation en temps polynomial pour les matrices dans le corps des coefficients il revient au meme de dire que le calcul des determinants est en temps polynomial dans le corps en question pour le codage choisi les resultats theorqiues sont meilleurs que pour les algorithmes connus precedemment et les resultats partiques sont encourageants

the complexity of three way statistical tables multiway tables with specified marginals arise in a variety of applications in statistics and operations research we provide a comprehensive complexity classification of three fundamental computational problems on tables existence counting and entry security one outcome of our work is that each of the following problems is intractable already for slim 3 tables with constant number 3 of rows 1 deciding existence of 3 tables with specified 2 marginals 2 counting all 3 tables with specified 2 marginals 3 deciding whether a specified value is attained in a specified entry by at least one of the 3 tables having the same 2 marginals as a given table this implies that a characterization of feasible marginals for such slim tables sought by much recent research is unlikely to exist another consequence of our study is a systematic efficient way of embedding the set of 3 tables satisfying any given 1 marginals and entry upper bounds in a set of slim 3 tables satisfying suitable 2 marginals with no entry bounds this provides a valuable tool for studying multi index transportation problems and multi index transportation polytopes remarkably it enables us to automatically recover a famous example due to vlach of a real feasible integer infeasible collection of 2 marginals for 3 tables of smallest possible size 3 4 6

quasi finite rank approximation of compression operators in sampled data systems and time delay systems this article studies an approximation problem of compression operators that play important roles in the operator theoretic approach to sampled data systems and time delay systems taking account of the ease in the use of approximated operators in such an approach we confine ourselves to a special form of approximation that we call quasi finite rank approximation we first introduce a suboptimal method for such approximation by means of what we call the fast lifting approach with which the problem can be reduced to a finite dimensional one in an approximate sense we further introduce another method that somehow applies the first method in a nested fashion for better overall performance we also discuss the asymptotic exactness of the suboptimal optimisation methods as the integer parameter m in the fast lifting approach tends to infinity as well as the associated monotonic properties as to the reduction of the deviation from the optimal approximation numerical examples are given to demonstrate the eff

lifting properties of maximal lattice free polyhedra we study the uniqueness of minimal liftings of cut generating functions obtained from maximal lattice free polyhedra we prove a basic invariance property of unique minimal liftings for general maximal lattice free polyhedra this generalizes a previous result by basu et al math oper res 37 2 346 355 2012 for simplicial maximal lattice free polytopes thus completely settling this fundamental question about lifting for maximal lattice free polyhedra we further give a very general iterative construction to get maximal lattice free polyhedra with the unique lifting property in arbitrary dimensions this single construction not only obtains all previously known polyhedra with the unique lifting property but goes further and vastly expands the known list of such polyhedra finally we extend characterizations from basu et al 2012 about lifting with respect to maximal lattice free simplices to more general polytopes these nontrivial generalizations rely on a number of results from discrete geometry including the venkov alexandrov mcmullen theorem on translative tilings and characterizations of zonotopes in terms of central symmetry of their faces

branching on general disjunctions this paper considers a modification of the branch and cut algorithm for mixed integer linear programming where branching is performed on general disjunctions rather than on variables we select promising branching disjunctions based on a heuristic measure of disjunction quality this measure exploits the relation between branching disjunctions and intersection cuts in this work we focus on disjunctions defining the mixed integer gomory cuts at an optimal basis of the linear programming relaxation the procedure is tested on instances from the literature experiments show that for a majority of the instances the enumeration tree obtained by branching on these general disjunctions has a smaller size than the tree obtained by branching on variables even when variable branching is performed using full strong branching

two efficient algorithms for the computation of ideal sums in quadratic orders this paper deals with two different asymptotically fast algorithms for the computation of ideal sums in quadratic orders if the class number of the quadratic number field is equal to 1 these algorithms can be used to calculate the gcd in the quadratic order we show that the calculation of an ideal sum in a fixed quadratic order can be done as fast as in z up to a constant factor i e in o n logn where n bounds the size of the operands and n denotes an upper bound for the multiplication time of n bit integers using schonhage strassen s asymptotically fast multiplication for n bit integers we achieve n o n log n log log n

frequency domain analysis of linear time periodic systems in this paper we study convergence of truncated representations of the frequency response operator of a linear time periodic system the frequency response operator is frequently called the harmonic transfer function we introduce the concepts of input output and skew roll off these concepts are related to the decay rates of elements in the harmonic transfer function a system with high input and output roll off may be well approximated by a low dimensional matrix function a system with high skew roll off may be represented by an operator with only few diagonals furthermore the roll off rates are shown to be determined by certain properties of taylor and fourier expansions of the periodic systems finally we clarify the connections between the different methods for computing the harmonic transfer function that are suggested in the literature

on the string consensus problem and the manhattan sequence consensus problem abstract we study the manhattan sequence consensus problem msc problem in which we are given k integer sequences each of length l and we are to find an integer sequence x of length l called a consensus sequence such that the maximum manhattan distance of x from each of the input sequences is minimized a related problem with hamming distance instead of manhattan distance is called hamming string consensus hsc also known under the names of string center problem or closest string problem for binary sequences manhattan distance coincides with hamming distance hence in this case hsc is a special case of msc we design a practically efficient o l time algorithm solving msc for k 5 sequences it improves upon the quadratic algorithm by amir et al 2012 1 for hsc for k 5 binary strings similarly as in the algorithm of amir et al we use a column based framework we replace the implied general integer linear programming by its easy special cases due to combinatorial properties of msc for k 5 practicality of our algorithms has been verified experimentally we also show that for a general parameter k any instance can be reduced in linear time to a kernel of size k so the problem is fixed parameter tractable nevertheless for k 4 this is still too large for any naive solution to be feasible in practice this is a full version of an article published at spire 2014 15

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

on the hardness of approximating stopping and trapping sets we prove that approximating the size of stopping and trapping sets in tanner graphs of linear block codes and more restrictively the class of low density parity check ldpc codes is np hard the ramifications of our findings are that methods used for estimating the height of the error floor of moderate and long length ldpc codes based on stopping and trapping set enumeration cannot provide accurate worst case performance predictions for most codes

on the weighted sensitivity minimization problem for delay systems we consider the h optimal sensitivity problem for delay systems in particular we consider computation of inf w q q h jr where w s is any function in rh jr and in h jr is any inner function we derive a new explicit solution in the pure delay case where e sh h 0

h 0e sensitivity minimization for delay systems this paper announces results on the problem of feedback compensator design for x norm weighted sensitivity minimization when the plant contains a delay in the input a complete solution is presented for the case of one pole zero weighting function and a single input single output plant for stable minimum phase rational part generalizations and proofs will be published elsewhere

how to compute the volume in high dimension in some areas of theoretical computer science we feel that randomized algorithms are better and in some others we can prove that they are more efficient than the deterministic ones

exponential lower bounds on the lengths of some classes of branch and cut proofs we examine the complexity of branch and cut proofs in the context of 0 1 integer programs we establish an exponential lower bound on the length of branch and cut proofs that use 0 1 branching and lift and project cuts called simple disjunctive cuts by some authors gomory chvatal cuts and cuts arising from the n0 matrix cut operator of lovasz and schrijver a consequence of the lower bound result in this paper is that branch and cut methods of the type described above have exponential running time in the worst case

an lmi solution to the robust synthesis problem for multi rate sampled data systems in this paper we address the asynchronous multi rate sampled data h synthesis problem necessary and sufficient conditions are given for the existence of a controller achieving the desired performance and the problem is shown to be equivalent to a convex optimization problem expressed in the form of linear operator inequalities in the case where the sample and hold rates are synchronous these operator inequalities reduce to linear matrix inequalities for which standard numerical software is available

minkowski s convex body theorem and integer programming the paper presents an algorithm for solving integer programming problems whose running time depends on the number n of variables as non this is done by reducing an n variable problem to 2n5i 2 problems in n i variables for some i greater than zero chosen by the algorithm the factor of on5 2 per variable improves the best previously known factor which is exponential in n minkowski s convex body theorem and other results from geometry of numbers play a crucial role in the algorithm several related algorithms for lattice problems are presented the complexity of these problems with respect to polynomial time reducibilities is studied

a general framework for linear periodic systems with applications to h sup infinity sampled data control the authors present a framework for dealing with continuous time periodic systems the main tool is a lifting technique which provides a strong correspondence between continuous time periodic systems and certain types of discrete time time invariant systems with infinite dimensional input and output spaces despite the infinite dimensionality of the input and output spaces a lifted system has a finite dimensional state space if the original system does this fact permits rather constructive methods for analyzing these systems as a demonstration of the utility of this framework the authors use it to describe the continuous time i e intersample behavior of sampled data systems and to obtain a complete solution to the problem of parameterizing all controllers that constrain the l sup 2 induced norm of a sampled data system to within a certain bound

the intractability of resolution abstract we prove that for infinitely many disjunctive normal form propositional calculus tautologies the length of the shortest resolution proof of cannot be bounded by any polynomial of the length of the tautologies we use were introduced by cook and reckhow 1979 and encode the pigeonhole principle extended resolution can furnish polynomial length proofs of these formulas

a primal all integer algorithm based on irreducible solutions this paper introduces an exact primal augmentation algorithm for solving general linear integer programs the algorithm iteratively substitutes one column in a tableau by other columns that correspond to irreducible solutions of certain linear diophantine inequalities we prove that various versions of our algorithm are finite it is a major concern in this paper to show how the subproblem of replacing a column can be accomplished effectively an implementation of the presented algorithms is given computational results for a number of hard 0 1 integer programs from the miplib demonstrate the practical power of the method

an application of the hermite normal form in integer programming abstract a new algorithm for solving integer programming problems is developed it is based on cuts which are generated from the hermite normal form of the basis matrix at a continuous extreme point

hard equality constrained integer knapsacks we consider the following integer feasibility problem given positive integer numbersa0 a1 a n with gcd a1 a n 1 anda a1 a n does there exist a vectorx z n 0satisfyinga x a0 we prove that if the coefficientsa1 a nhave a certain decomposable structure then the frobenius number associated witha1 a n i e the largest value ofa0 for whicha x a0 does not have a nonnegative integer solution is close to a known upper bound in the instances we consider we takea0 to be the frobenius number furthermore we show that the decomposable structure ofa1 a nmakes the solution of a lattice reformulation of our problem almost trivial since the number of lattice hyperplanes that intersect the polytope resulting from the reformulation in the direction of the last coordinate is going to be very small for branch and bound such instances are difficult to solve since they are infeasible and have large values ofa0 a i 1 i n we illustrate our results by some computational examples

fast lll type lattice reduction we modify the concept of lll reduction of lattice bases in the sense of lenstra lenstra lovasz factoring polynomials with rational coefficients math ann 261 1982 515 534 towards a faster reduction algorithm we organize lll reduction in segments of the basis our slll bases approximate the successive minima of the lattice in nearly the same way as lll bases for integer lattices of dimension n given by a basis of length 2 o n slll reduction runs in o n 5 e bit operations for every e 0 compared to o n 7 e for the original lll and to o n 6 e for the lll algorithms of schnorr a more efficient algorithm for lattice reduction journal of algorithm 9 1988 47 62 and storjohann faster algorithms for integer lattice basis reduction tr 249 swiss federal institute of technology eth zurich department of computer science zurich switzerland july 1996 we present an even faster algorithm for slll reduction via iterated subsegments running in o n 3log n arithmetic steps householder reflections are shown to provide better accuracy than gram schmidt for orthogonalizing lll bases in floating point arithmetic

the multiple subset sum problem in the em multiple subset sum problem mssp items from a given ground set are selected and packed into a given number of identical bins such that the sum of the item weights in every bin does not exceed the bin capacity and the total sum of the weights of the items packed is as large as possible this problem is a relevant special case of the multiple knapsack problem for which the existence of a polynomial time approximation scheme ptas is an important open question in the field of knapsack problems one main result of the present paper is the construction of a ptas for mssp for the bottleneck case of the problem where the minimum total weight contained in any bin is to be maximized we describe a 2 3 approximation algorithm and show that this is the best possible approximation ratio moreover ptass are derived for the special cases in which either the number of bins or the number of different item weights is constant we finally show that even for the case of only two bins no fully ptas exists for both versions of the problem

a cutting plane algorithm for the linear ordering problem the linear ordering problem is an np hard combinatorial optimization problem with a large number of applications including triangulation of input output matrices archaeological senation minimizing total weighted completion time in one machine scheduling and aggregation of individual preferences in a former paper we have investigated the facet structure of the 0 1 polytope associated with the linear ordering problem here we report on a new algorithm that is based on these theoretical results the main part of the algorithm is a cutting plane procedure using facet defining inequalities this procedure is combined with various heuristics and branch and bound techniques our computational results compare favorably with the results of existing codes in particular we could triangulate all input output matrices of size up to 60 60 available to us within acceptable time bounds

corrections to bisection algorithm for computing the frequency response gain of sampled data systems infinite dimensional congruent transformation approach this paper derives a bisection algorithm for computing the frequency response gain of sampled data systems with their intersample behavior taken into account the properties of the infinite dimensional congruent transformation i e the schur complement arguments and the sylvester law of inertia play a key role in the derivation specifically it is highlighted that counting up the numbers of the negative eigenvalues of self adjoint operators is quite important for the computation of the frequency response gain this contrasts with the well known arguments on the related issue of the sampled data h sub spl infin problem where the key role is played by the positivity of operators and the loop shifting technique the effectiveness of the derived algorithm is demonstrated through a numerical example

finite rank criteria for h infty control of infinite dimensional systems this paper studies the hinfin control of infinite dimensional systems whose transfer matrices are expressible as a series connection of a rational transfer matrix and a scalar possibly irrational inner function this class of systems is adequate for describing many control problems in practice when weighting functions are rational and plants have at most finitely many unstable modes we show that this problem can be reduced to solving two matrix valued riccati equations and an additional rank condition furthermore the obtained controller structure is characterized by the inner function and controllers for the finite dimensional part this result provides us with a solution that is easily implementable without much prior knowledge on infinite dimensional control theory a numerical example is given to illustrate the result

computing hermite and smith normal forms of triangular integer matrices abstract this paper considers the problem of transforming a triangular integer input matrix to canonical hermite and smith normal form we provide algorithms and prove deterministic running times for both transformation problems that are optimal in the matrix dimension the algorithms are easily implemented assume standard integer arithmetic and admit excellent performance in practice the results presented here lead to a faster algorithm for computing the smith normal form of an arbitrary i e non triangular input matrix

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

asymptotically fast triangularization of matrices over rings this paper considers the problems of triangularizing and diagonalizing matrices over rings with particular emphasis on the integral case it begins with a description of fast algorithms for the computation of hermite and smith normal forms of integer matrices then it shows how to apply fast matrix multiplication techniques to the problem of triangularizing a matrix over a ring using elementary column operations these general results lead to an algorithm for triangularizing integer matrices that has a faster running time than the known hermite normal form algorithms the triangular matrix that is computed has small entries like the hermite normal form and will suffice for many applications

mixed integer reformulations of integer programs and the affine tu dimension of a matrix we study the reformulation of integer linear programs by means of a mixed integer linear program with fewer integer variables such reformulations can be solved efficiently with mixed integer linear programming techniques we exhibit examples that demonstrate how integer programs can be reformulated using far fewer integer variables to this end we introduce a generalization of total unimodularity called the affine tu dimension of a matrix and study related theory and algorithms for determining the affine tu dimension of a matrix we also present bounds on the number of integer variables needed to represent certain integer hulls