knapsack problems a parameterized point of view abstract the knapsack problem kp is a very famous np hard problem in combinatorial optimization also its generalization to multiple dimensions named d dimensional knapsack problem d kp and to multiple knapsacks named multiple knapsack problem mkp are well known problems in this paper we give a first study on the fixed parameter tractability of these three problems the idea behind fixed parameter tractability is to split the complexity into two parts one part that depends purely on the size of the input and one part that depends on some parameter of the problem that tends to be small in practice further we consider the closely related question whether the sizes and the values can be reduced such that their bit length is bounded polynomially or even constantly in a given parameter i e the existence of kernelizations is studied we give several upper and some lower bounds on the parameterized complexity and kernel sizes for the following parameters the number of items the threshold value for the profit the sizes the profits the number d of dimensions and the number m of knapsacks we also consider the connection of parameterized knapsack problems to linear programming approximation and pseudo polynomial algorithms

computing efficiently the lattice width in any dimension we provide an algorithm for the exact computation of the lattice width of a set of points k in z 2 in linear time with respect to the size of k this method consists in computing a particular surrounding polygon from this polygon we deduce a set of candidate vectors allowing the computation of the lattice width moreover we describe how this new algorithm can be extended to an arbitrary dimension thanks to a greedy and practical approach to compute a surrounding polytope indeed this last computation is very efficient in practice as it processes only a few linear time iterations whatever the size of the set of points hence it avoids complex geometric processings

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

existence theorems for nonselfadjoint semilinear elliptic boundary value problems where e is a real elliptic linear differential operator in a bounded domain g of r with a given system of linear homogeneous conditions say bx 0 on the boundary ag of g and where n is a nemitsky type nonnecessarily linear operator we shall make use here of the alternative method and particularly we shall make use for the elliptic case of new remarks these remarks suggest that both the auxiliary and bifurcation equations can be analyzed under different topologies and by a more specific construction of the operator s y0 x0 actually some of these remarks have been already used implicitly in previous papers on the semilinear wave equation in r2 cesari and kannan 5 cesari and pucci 6 for selfadjoint elliptic problems landesman and lazer 9 proved also by the alternative method a remarkable theorem which was then extended by williams 14 by the same method and by others by different arguments later shaw 12 proved again by the alternative method that landesman s and lazer s theorem extends even to nonselfadjoint problems with equal fredholm indices and whose eigenfunctions share regions of positivity and negativity with their corresponding adjoint eigenfunctions in the present paper we definitely aim at elliptic problems which are not necessarily selfadjoint and do not necessarily satisfy shaw s requirements the sufficient conditions we obtain are more quantitative in character and concern the cases nx f t g t d x and nx f f g t x t t e g however as we show by examples for the case nx f t g t x t our sufficient conditions for existence allow a great freedom on g on which no monotonicity is required

combinatorial n fold integer programming and applications many fundamental mathsf np hard problems can be formulated as integer linear programs ilps a famous algorithm by lenstra solves ilps in time that is exponential only in the dimension of the program and polynomial in the size of the ilp that algorithm became a ubiquitous tool in the design of fixed parameter algorithms for mathsf np hard problems where one wishes to isolate the hardness of a problem by some parameter however in many cases using lenstra s algorithm has two drawbacks first the run time of the resulting algorithms is often double exponential in the parameter and second an ilp formulation in small dimension cannot easily express problems involving many different costs inspired by the work of hemmecke et al math program 137 1 2 ser a 325 341 2013 we develop a single exponential algorithm for so called combinatorialn fold integer programs which are remarkably similar to prior ilp formulations for various problems but unlike them also allow variable dimension we then apply our algorithm to many relevant problems problems like closest string swap bribery weighted set multicover and several others and obtain exponential speedups in the dependence on the respective parameters the input size or both unlike lenstra s algorithm which is essentially a bounded search tree algorithm our result uses the technique of augmenting steps at its heart is a deep result stating that in combinatorial n fold ips existence of an augmenting step implies existence of a local augmenting step which can be found using dynamic programming our results provide an important insight into many problems by showing that they exhibit this phenomenon and highlights the importance of augmentation techniques

global periodic solutions of the nonlinear wave equation on considere le probleme u tt u t g u t r 2 u t t u t u t t t t t t t ou u et sont des fonctions scalaires de r 2 et g une fonction scalaire sur r on suppose que g et sont continues et on determine des conditions suffisantes sous forme d inegalites pour garantir que ce probleme a des solutions continues

neighborhood systems for production sets with indivisibilities a production set with indivisibilities is described by an activity analysis matrix with activity levels which can assume arbitrary integral values a neighborhood system is an association with each integral vector of activity levels of a finite set of neighboring vectors the neighborhood relation is assumed to be symmetric and translation invariant each such neighborhood system can be used to define a local maximum for the associated integer programs obtained by selecting a single commodity whose level is to be maximized subject to specified factor endowments of the remaining commodities it is shown that each technology matrix subject to mild regularity assumptions has a unique minimal neighborhood system for which a local maximum is global the complexity of such minimal neighborhood systems is examined for several examples

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

integer hulls of linear polyhedra and scl in families the integer hull of a polyhedron is the convex hull of the integer points contained in it we show that the vertices of the integer hulls of a rational family of polyhedra of size o n have eventually quasipolynomial coordinates as a corollary we show that the stable commutator length of elements in a surgery family is eventually a ratio of quasipolynomials and that unit balls in the scl norm eventually quasiconverge in finite dimensional surgery families

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

κ partitioning problems for maximizing the minimum load abstract the optimization versions of the k partitioning problems are considered in this paper for the objective to maximize the minimum load of m subsets we first show that the folding algorithm has a tight worst case ratio of max 2 k 1 m then we present an algorithm called harmonic1 with a worst case ratio at least max 1 k 1 i 1 m 1 i 1 it concludes the harmonic1 is better than folding for k 2 i 1 m 1 i 1 we further show that harmonici is asymptotically optimal ordinal algorithm we also present an algorithm harmonic2 for solving the general i partitioning problem

hardness of approximating the shortest vector problem in lattices let p 1 be any fixed real we show that assuming np n rp there is no polynomial time algorithm that approximates the shortest vector problem svp in e p norm within a constant factor under the stronger assumption np n rtime 2poly log n we show that there is no polynomial time algorithm with approximation ratio 2 log n 1 2 e where n is the dimension of the lattice and e 0 is an arbitrarily small constant we first give a new randomized reduction from closest vector problem cvp to svp that achieves some constant factor hardness the reduction is based on bch codes its advantage is that the svp instances produced by the reduction behave well under the augmented tensor product a new variant of tensor product that we introduce this enables us to boost the hardness factor to 2 log n 1 2 e

bin packing in multiple dimensions inapproximability results and approximation schemes we study the following packing problem given a collection of d dimensional rectangles of specified sizes pack them into the minimum number of unit cubes we show that unlike the one dimensional case the two dimensional packing problem cannot have an asymptotic polynomial time approximation scheme aptas unless pnp on the positive side we give an aptas for the special case of packing d dimensional cubes into the minimum number of unit cubes second we give a polynomial time algorithm for packing arbitrary two dimensional rectangles into at most opt square bins with sides of length 1 where opt denotes the minimum number of unit bins required to pack these rectangles interestingly this result has no additive constant term i e is not an asymptotic result as a corollary we obtain the first approximation scheme for the problem of placing a collection of rectangles in a minimum area encasing rectangle

data complexity of the two variable fragment with counting quantifiers the data complexity of both satisfiability and finite satisfiability for the two variable fragment with counting quantifiers is np complete the data complexity of both query answering and finite query answering for the two variable guarded fragment with counting quantifiers is co np complete

mixed integer programming with convex concave constraints fixed parameter tractability and applications to multicovering and voting abstract a classic result of lenstra math oper res 1983 says that an integer linear program can be solved in fixed parameter tractable fpt time for the parameterization by the number of variables we extend this result by incorporating piecewise linear convex or concave functions to our mixed integer programs this general technique allows us to analyze the parameterized complexity of a number of classic np hard computational problems in particular we prove that weighted set multicover is in fpt when parameterized by the number of elements to cover and that there exists an fpt time approximation scheme for multiset multicover for the same parameter this is our most technical result further we use our general technique to prove that a number of problems from computational social choice e g problems related to bribery and control in elections are in fpt when parameterized by the number of candidates for bribery this resolves a nearly 10 year old family of open problems and for weighted electoral control of approval voting this improves some previously known xp memberships to fpt memberships

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

an implicit integration algorithm with a projection method for viscoplastic constitutive equations robinson s viscoplastic model a representative of the so called overstress models is integrated by use of the generalized midpoint rule the solution of the non linear system of algebraic equations arising from time discretization of the constitutive equations is determined using a projection method in combination with newton s method consistent tangent moduli are calculated and the quadratic convergence of the global newton equilibrium iteration is shown the time increment size is controlled by the convergence behaviour of the equilibrium iteration and the accuracy of the numerical integration various numerical examples are considered to demonstrate the efficiency of the methods

fast approximation schemes for two stage two dimensional bin packing we present an asymptotic fully polynomial time approximation scheme for the two dimensional generalization of bin packing which requires packing or cutting a given set of rectangles from the minimum number of square bins with the further restriction that packing the rectangles in the bins is done in two stages as is frequently the case in real world applications to the best of our knowledge this is the first approximation scheme for a nontrivial two dimensional and real world generalization of a classical one dimensional packing problem in which rectangles have to be packed in finite squares in addition to the approximability result we obtain as a byproduct of our analysis an interesting structural result namely the asymptotic optimality of the lower bound provided by the natural linear programming relaxation of the problem having one variable for each bin pattern that can be solved effectively by column generation techniques besides its practical relevance a main reason to concentrate on the two stage version of the problem is that its study continues to shed light on the general version in particular we strongly conjecture that the approximation guarantee of our algorithm for two dimensional bin packing without the two stage restriction is better than the best known ones a simplification of the method leads to an approximation scheme for the two stage version of two dimensional strip packing

sensitivity theorems in integer linear programming we consider integer linear programming problems with a fixed coefficient matrix and varying objective function and right hand side vector among our results we show that for any optimal solution to a linear program max wx ax b the distance to the nearest optimal solution to the corresponding integer program is at most the dimension of the problem multiplied by the largest subdeterminant of the integral matrixa using this we strengthen several integer programming proximity results of blair and jeroslow graver and wolsey we also show that the chvatal rank of a polyhedron x ax b can be bounded above by a function of the matrixa independent of the vectorb a result which as blair observed is equivalent to blair and jeroslow s theorem that each integer programming value function is a gomory function

super golden gates for pu 2 abstract to each of the symmetry groups of the platonic solids we adjoin a carefully designed involution yielding topological generators of pu 2 which have optimal covering properties as well as efficient navigation these are a consequence of optimal strong approximation for integral quadratic forms associated with certain special quaternion algebras and their arithmetic groups the generators give super efficient 1 qubit quantum gates and are natural building blocks for the design of universal quantum gates

on principal ideal testing in totally complex quartic fields and the determination of certain cyclotomic constants let y be any totally complex quartic field two algorithms are described for determining whether or not any given ideal in y is principal one of these algorithms is very efficient in practice but its complexity is difficult to analyze the other algorithm is computationally more elaborate but in this case a complexity analysis can be provided these ideas are applied to the problem of determining the cyclotomic numbers of order 5 for a prime p 1 mod 5 given any quadratic or quintic nonresidue of p it is shown that these cyclotomic numbers can be efficiently computed in 0 log p 3 binary operations

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 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

neighbors of the origin for four by three matrices scarf has defined a neighborhood system for families of integer programs where the right hand side is allowed to vary this system depends on a matrix a of constraint and objective function coefficients of the integer programs this paper characterizes the set of neighbors of the origin when a is four by three showing that it may be described as the set of integer vectors in a union of two dimensional polyhedra where the number of polyhedra is quadratic in the bit size of a

approximate composable truthful mechanism design abstract this paper aims to study techniques for designing truthful mechanisms for a combinatorial optimization problem that might require composition algorithms we show that the composition algorithm a b is monotone if the algorithm a and the algorithm b are both monotone we apply this technique to the two dimensional orthogonal knapsack problem with provable constant approximation bounds improving the previous non constant results in 5 moreover we show that the technique can also be applied to the heterogeneous multiple clusters scheduling problem and a truthful mechanism with provable approximation bounds was presented

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

lattice closures of polyhedra given p subset mathbb r n a mixed integer set p i p cap mathbb z t times mathbb r n t and a k tuple of n dimensional integral vectors pi 1 ldots pi k where the last n t entries of each vector is zero we consider the relaxation of p i obtained by taking the convex hull of points x in p for which pi 1 tx ldots pi t kx are integral we then define the k dimensional lattice closure of p i to be the intersection of all such relaxations obtained from k tuples of n dimensional vectors when p is a rational polyhedron we show that given any collection of such k tuples there is a finite subcollection that gives the same closure more generally we show that any k tuple is dominated by another k tuple coming from the finite subcollection the k dimensional lattice closure contains the convex hull of p i and is equal to the split closure when k 1 therefore a result of cook et al math program 47 155 174 1990 implies that when p is a rational polyhedron the k dimensional lattice closure is a polyhedron for k 1 and our finiteness result extends this to all k ge 2 we also construct a polyhedral mixed integer set with n integer variables and one continuous variable such that for any k n finitely many iterations of the k dimensional lattice closure do not give the convex hull of the set our result implies that t branch split cuts cannot give the convex hull of the set nor can valid inequalities from unbounded full dimensional convex lattice free sets

an implicit unitless error and step size control method in integrating unified viscoplastic creep ode type constitutive equations a series of numerical analyses are carried out to investigate the difficulties in numerical integration of unified viscoplastic creep constitutive equations which are normally represented as a system of ordinary differential equations odes the problems of numerically integrating the constitutive equations are identified and analysed to overcome the stiffness problems implicit methods are used for the numerical integration and a generic technique is introduced to calculate the jacobian matrix a normalization technique is introduced in the paper to convert the integration errors for each time increment to unitless errors thus a single tolerance can be used to control the accuracy and step size in integrating a set of unified viscoplastic creep constitutive equations in addition an implicit step size control method is proposed and used in the integrations this method reduces the possibility of rejection of an integration increment due to poor accuracy copyright 2007 john wiley sons ltd

on the regularity of petri net languages petri nets are known to be useful for modeling concurrent systems once modeled by a petri net the behavior of a concurrent system can be characterized by the set of all executable transition sequences which in turn can be viewed as a language over an alphabet of symbols corresponding to the transitions of the underlying petri net in this paper we study the language issue of petri nets from a computational complexity viewpoint we analyze the complexity of theregularity problem i e the problem of determining whether a given petri net defines an irregular language or not for a variety of classes of petri nets includingconflict free trap circuit normal sinkless extended trap circuit bpp andgeneralpetri nets extended trap circuit petri nets are trap circuit petri nets augmented with a specific type ofcircuits as it turns out the complexities for these petri net classes range from nl nondeterministic logspace ptime polynomial time and np nondeterministic polynomial time to expspace exponential space in the process of deriving the complexity results we develop adecomposition approachwhich we feel is interesting in its own right and might have other applications to the analysis of petri nets as well as a by product an np upper bound of the reachability problem for the class of extended trap circuit petri nets which properly contains that of trap circuit and hence conflict free and bpp nets and is incomparable with that of normal and sinkless petri nets is derived

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

exact algorithms for weighted and unweighted borda manipulation problems we derive exact combinatorial algorithms for both weighed and unweighted borda manipulation problems we derive an fpt algorithm for the unweighted borda manipulation problem with respect to the number of candidates we prove that borda manipulation with two manipulators is polynomial time solvable in single peaked elections we propose o m 2 m t 1 time and o t m 1 t t 1 m time algorithms for the weighted and the unweighted borda manipulation problems respectively where t is the number of manipulators and m is the number of candidates in particular for t 2 our algorithm for the unweighted case runs in o 3 m time affirmatively resolving one open problem posed by betzler et al 2 as a byproduct of our results we show that the unweighted borda manipulation problem admits an algorithm with running time o 2 9 m 2 log m based on an integer linear programming technique finally we prove that the unweighted borda manipulation problem is polynomial time solvable in single peaked elections in contrast to the np hardness in the general case

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

periodic solutions of nonlinear wave equations on obtient des conditions suffisantes pour l existence de solutions doublement periodiques de l equation d onde non lineaire u ts f t s u u t t s u t s u t s t ou f est une fonction continue donnee dans r 3 et f est t periodique en t et s

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

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

splitting full matrix algebras over algebraic number fields abstract let k be a fixed algebraic number field and let a be an associative algebra over k given by structure constants such that a m n k holds for some positive integer n suppose that n is bounded then an isomorphism a m n k can be constructed by a polynomial time ff algorithm an ff algorithm is a deterministic procedure which is allowed to call oracles for factoring integers and factoring univariate polynomials over finite fields as a consequence we obtain a polynomial time ff algorithm to compute isomorphisms of central simple algebras of bounded degree over k

improved branching disjunctions for branch and bound an analytic center approach abstract in classical branch and bound algorithms the branching disjunction is often based on a single variable which is a special case of the more general approach that involves multiple variables in this paper we present a new approach to generate good general branching disjunctions based on the shape of the polyhedron and interior point concepts the approach is based on approximating the feasible polyhedron using dikin s inscribed ellipsoid calculated using the analytic center from interior point methods we use the fact that the width of the ellipsoid in a given direction has a closed form expression to formulate a quadratic problem whose optimal solution is a thin direction of the ellipsoid while solving a quadratic problem at each node of the branch and bound tree is impractical we use an efficient neighborhood search heuristic for its solution we report computational results on hard mixed integer problems from the literature showing that the proposed approach leads to smaller branch and bound trees and a reduction in the computational time as compared with classical branching and strong branching as the computation of the analytic center is a bottleneck we finally test the approach within a general interior point based benders decomposition where the analytic center is readily available and show clear dominance of the approach over classical branching

the complexity of the a b c problem we present a deterministic polynomial time algorithm for the a b c problem which is the membership problem for 2 generated commutative linear semigroups over an algebraic number field we also obtain a polynomial time algorithm for the easier membership problem for 2 generated abelian linear groups furthermore we provide a polynomial sized encoding for the set of all solutions

a semilinear elliptic problem which is not selfadjoint cesari 7 i in a preceding paper established conditions sufficient that a semilinear elliptic boundary value problem possibly not selfadjoint shall have a solution

computing two dimensional integer hulls an optimal algorithm is presented for computing the smallest set of linear inequalities that define the integer hull of a possibly unbounded two dimensional convex polygon r input to the algorithm is a set of linear inequalities defining r and the integer hull computed is the convex hull of the integer points of r it is proven that the integer hull has at most o n log amax inequalities where n is the number of input inequalities and amax is the magnitude of the largest input coefficient it is shown that the algorithm presented has complexity o n log amax and that this is optimal by proving that the integer hull may have omega n log a max inequalities in the worst case

on the complexity of selecting disjunctions in integer programming the imposition of general disjunctions of the form pi x leq pi 0 vee pi x geq pi 0 1 where pi pi 0 are integer valued is a fundamental operation in both the branch and bound and cutting plane algorithms for solving mixed integer linear programs such disjunctions can be used for branching at each iteration of the branch and bound algorithm or to generate split inequalities for the cutting plane algorithm we first consider the problem of selecting a general disjunction and show that the problem of selecting an optimal such disjunction according to specific criteria described herein is mathcal np hard we further show that the problem remains mathcal np hard even for binary programs or when considering certain restricted classes of disjunctions we observe that the problem of deciding whether a given inequality is a split inequality can be reduced to one of the above problems which leads to a proof that the problem is mathcal np complete

tight bounds and 2 approximation algorithms for integer programs with two variables per inequality the problem of integer programming in bounded variables over constraints with no more than two variables in each constraint is np complete even when all variables are binary this paper deals with integer linear minimization problems inn variables subject tom linear constraints with at most two variables per inequality and with all variables bounded between 0 andu for such systems a 2 approximation algorithm is presented that runs in time o mnu2 log un2m so it is polynomial in the input size if the upper boundu is polynomially bounded the algorithm works by finding first a super optimal feasible solution that consists of integer multiples of 1 2 that solution gives a tight bound on the value of the minimum it furthermore has an identifiable subset of integer components that retain their value in an integer optimal solution of the problem these properties are a generalization of the properties of the vertex cover problem the algorithm described is in particular a 2 approximation algorithm for the problem of minimizing the total weight of true variables among all truth assignments to the 2 satisfiability problem

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

improved complexity for maximum volume inscribed ellipsoids let cal p x vert ax le b where a is an m n matrix we assume that cal p contains a ball of radius one centered at the origin and is itself contained in a ball of radius r centered at the origin we consider the problem of approximating the maximum volume ellipsoid inscribed in cal p such ellipsoids have a number of interesting applications including the inscribed ellipsoid method for convex optimization we describe an optimization algorithm that obtains an ellipsoid whose volume is at least a factor e epsilon of the maximum possible in o m 3 5 ln mr epsilon operations our result provides an alternative to a saddlepoint based approach with the same complexity developed by nemirovskii we also show that a further reduction in complexity can be obtained by first computing an approximation of the analytic center of cal p

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

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

on xml integrity constraints in the presence of dtds the article investigates xml document specifications with dtds and integrity constraints such as keys and foreign keys we study the consistency problem of checking whether a given specification is meaningful that is whether there exists an xml document that both conforms to the dtd and satisfies the constraints we show that dtds interact with constraints in a highly intricate way and as a result the consistency problem in general is undecidable when it comes to unary keys and foreign keys the consistency problem is shown to be np complete this is done by coding dtds and integrity constraints with linear constraints on the integers we consider the variations of the problem by both restricting and enlarging the class of constraints and identify a number of tractable cases as well as a number of additional np complete ones by incorporating negations of constraints we establish complexity bounds on the implication problem which is shown to be conp complete for unary keys and foreign keys

polynomial algorithms for guillotine cutting of a rectangle into small rectangles of two kinds abstract in this paper the problem of optimally guillotine cutting a rectangle a b into small rectangles of two kinds is considered rectangles of the first kind c a i i i have the same width and their heights can be various rectangles of the second kind b j d j j have the same height and their widths can be various the number of occurrences of each small rectangle in a cutting pattern is not restricted similar problems often appear in the furniture industry this cutting problem is reduced to the shortest path problem in a special rectangular grid for which a linear time algorithm is suggested this approach generalizes the approach of e girlich a g tarnowski on polynomial solvability of two multiprocessor scheduling problems mathematical methods of operations research 50 1999 27 51 a g tarnowski advanced polynomial time algorithm for guillotine generalized pallet loading problem in the international scientific collection decision making under conditions of uncertainty cutting packing problems ufa state aviation technical university 1997 pp 93 124 and allows us to construct polynomial algorithms for the guillotine cutting problem considered with a fixed number of small rectangles of two kinds

on the complexity of decoding lattices using the korkin zolotarev reduced basis upper and lower bounds are derived for the decoding complexity of a general lattice l the bounds are in terms of the dimension n and the coding gain spl gamma of l and are obtained based on a decoding algorithm which is an improved version of kannan s 1983 method the latter is currently the fastest known method for the decoding of a general lattice for the decoding of a point x the proposed algorithm recursively searches inside an n dimensional rectangular parallelepiped cube centered at x with its edges along the gram schmidt vectors of a proper basis of l we call algorithms of this type recursive cube search rcs algorithms it is shown that kannan s algorithm also belongs to this category the complexity of rcs algorithms is measured in terms of the number of lattice points that need to be examined before a decision is made to tighten the upper bound on the complexity we select a lattice basis which is reduced in the sense of korkin zolotarev 1873 it is shown that for any selected basis the decoding complexity using rcs algorithms of any sequence of lattices with possible application in communications spl gamma spl ges 1 grows at least exponentially with n and spl gamma it is observed that the densest lattices and almost all of the lattices used in communications e g barnes wall lattices and the leech lattice have equal successive minima esm for the decoding complexity of esm lattices a tighter upper bound and a stronger lower bound result are derived

prices matter for the parameterized complexity of shift bribery in the shift bribery problem we are given an election a preferred candidate p and a budget the goal is to ensure p s victory by shifting p higher in some voters preference orders however each such shift request comes at a price and we must not exceed the given budget we study the parameterized computational complexity of shift bribery for a number of parameters and several classes of price functions for the number of affected voters shift bribery is w 2 hard for borda maximin and copeland for the number of positions by which p is shifted in total the problem is fixed parameter tractable for borda and maximin and is w 1 hard for copeland for the budget the results depend on the price function class finally shift bribery tends to be tractable when parameterized by the number of voters but the results for the number of candidates are more enigmatic

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

nonlinear second order elliptic partial differential equations at resonance on etudie la resolubilite de problemes aux valeurs limites pour des equations aux derivees partielles elliptiques d ordre 2 semi lineaires de type resonance dans lesquelles la perturbation non lineaire ne satisfait pas necessairement la condition de landesman lazer ou l hypothese de monotonicite

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

efficient and accurate explicit integration algorithms with application to viscoplastic models several explicit integration algorithms with self adative time integration strategies are developed and investigated for efficiency and accuracy these algorithms involve the runge kutta second order the lower runge kutta method of orders one and two and the exponential integration method the algorithms are applied to viscoplastic models put forth by freed and verrilli and bodner and partom for thermal mechanical loadings including tensile relaxation and cyclic loadings the large amount of computations performed showed that for comparable accuracy the efficiency of an integration algorithm depends significantly on the type of application loading however in general for the aforementioned loadings and viscoplastic models the exponential integration algorithm with the proposed self adaptive time integration strategy worked more or comparably efficiently and accurately than the other integration algorithms using this strategy for integrating viscoplastic models may lead to considerable savings in computer time better efficiency without adversely affecting the accuracy of the results this conclusion should encourage the utilization of viscoplastic models in the stress analysis and design of structural components

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

complexity analysis of a parallel lattice basis reduction algorithm lattice basis reduction is an important problem in geometry of numbers with applications in combinatorial optimization computer algebra and cryptography the well known sequential lll algorithm finds a short vector in o n4 log b arithmetic operations on integers having binary length o n log b where n denotes the dimension of the lattice and b denotes the maximum l2 norm of the initial basis vectors in this paper a new analysis of the parallel algorithm of roch and villard is presented it is shown that on an n x n mesh it needs o n2 log b arithmetic operations on integers having binary length o n log b this improves the previous analysis and shows that an asymptotical speedup of n2 is possible using n2 processors

factorization properties of lattices over the integers abstract let a be a nonsingular n n matrix over the integers l l a denotes the lattice whose elements are combinations with integer coefficients of the rows of a l is cyclic if it can be defined in the modular form l x x i a i x i 0 mod d where the a i s and d are integers and 0 a i d let l l 1 l 2 b be lattices over the integers l l 1 l 2 is a factorization of l if every element of l is a combination of the rows of b such that the vector of combination coefficients is in l 1 and b is a nonsingular n n matrix the following results are proved every lattice can be expressed as a product of cyclic factors in polynomial time every cyclic lattice can be factored into simple term explained in the text factors in polynomial time every simple lattice can be factored into prime factors in polynomial time if a prime factorization of the determinant of its basis is given in addition we provide polynomial algorithms for the following problems transform a cyclic lattice given by a basis into a modular form and vice versa find a basis of a finite modular lattice given in modular form

the flatness theorem for nonsymmetric convex bodies via the local theory of banach spaces let l be a lattice in rn and k a convex body disjoint from l the classical flatness theorem asserts that then w k l the l width of k does not exceed some bound depending only on the dimension n this fact was later found relevant to questions in integer programming kannan and lovasz 1988 showed that under the above assumptions w k l cn2 where c is a universal constant banaszczyk 1996 proved that w k l cn 1 log n if k has a centre of symmetry in the present paper we show that w k l cn3 2 for an arbitrary k it is conjectured that the exponent 3 2 may be replaced by 1 perhaps at the cost of a logarithmic factor we prove that for some naturally arising classes of bodies

variable metric relaxation methods part ii the ellipsoid method the deepest or least shallow cut ellipsoid method is a polynomial time and space method which finds an ellipsoid representable by polynomial space integers such that the maximal ellipsoidal distance relaxation method using this fixed ellipsoid is polynomial this is equivalent to finding a linear transforming such that the maximal distance relaxation method of agmon motzkin and schoenberg in this transformed space is polynomial if perfect arithmetic is used then the sequence of ellipsoids generated by the method converges to a set of ellipsoids which share some of the properties of the classical hessian at an optimum point of a function and thus the ellipsoid method is quite analogous to a variable metric quasi newton method

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 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

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

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

minimizing cubic and homogeneous polynomials over integers in the plane we complete the complexity classification by degree of minimizing a polynomial over the integer points in a polyhedron in a real vector space of dimension two previous work shows that optimizing a quadratic polynomial over the integer points in a polyhedral region in a real vector space of dimension two can be done in polynomial time whereas optimizing a quartic polynomial in the same type of region is np hard we close the gap by showing that this problem can be solved in polynomial time for cubic polynomials furthermore we show that the problem of minimizing a homogeneous polynomial of any fixed degree over the integer points in a bounded polyhedron in a real vector space of dimension two is solvable in polynomial time we show that this holds for polynomials that can be translated into homogeneous polynomials even when the translation vector is unknown we demonstrate that such problems in the unbounded case can have smallest optimal solutions of exponential size in the size of the input thus req

typechecking top down xml transformations fixed input or output schemas typechecking consists of statically verifying whether the output of an xml transformation always conforms to an output type for documents satisfying a given input type in this general setting both the input and output schema as well as the transformation are part of the input for the problem however scenarios where the input or output schema can be considered to be fixed are quite common in practice in the present work we investigate the computational complexity of the typechecking problem in the latter setting

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