growth rates and explosions in sandpiles we study the abelian sandpile growth model where n particles are added at the origin on a stable background configuration in zd any site with at least 2d particles then topples by sending one particle to each neighbor we find that with constant background height h 2d 2 the diameter of the set of sites that topple has order n1 d this was previously known only for h d our proof uses a strong form of the least action principle for sandpiles and a novel method of background modification we can extend this diameter bound to certain backgrounds in which an arbitrarily high fraction of sites have height 2d 1 on the other hand we show that if the background height 2d 2 is augmented by 1 at an arbitrarily small fraction of sites chosen independently at random then adding finitely many particles creates an explosion a sandpile that never stabilizes

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

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

fast algorithms for n dimensional restrictions of hard problems let m be a parallel ram with p processors and arithmetic operations addition and subtraction recognizing l n n in t steps inputs for m are given integer by integer not bit by bit then l can be recognized by a sequential linear search algorithm lsa in o n 4 log n t log p steps thus many n dimensional restrictions of np complete problems binary programming traveling salesman problem etc and even that of the uniquely optimum traveling salesman problem which is p 2 complete can be solved in polynomial time by an lsa this result generalizes the construction of a polynomial lsa for the n dimensional restriction of the knapsack problem previously shown by the author and destroys the hope of proving nonpolynomial lower bounds on lsas for any problem that can be recognized by a pram as above with 2 poly n processors in poly n time

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

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

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

actuator design for parabolic distributed parameter systems with the moment method in this paper we model and solve the problem of designing in an optimal way actuators for parabolic partial differential equations settled on a bounded open connected subset omega of r n we optimize not only the location but also the shape of actuators by finding what is the optimal distribution of actuators in omega over all possible such distributions of a given measure using the moment method we formulate a spectral optimal design problem which consists of maximizing a criterion corresponding to an average over random initial data of the largest l 2 energy of controllers since we choose the moment method to control the pde our study mainly covers one dimensional parabolic operators but we also provide several examples in higher dimensions we consider two types of controllers either internal controls modeled by characteristic functions or lumped controls that are tensorized functions in time and space under appropriate spectral assumptions we prove existence and uniqueness of an optimal actuator distribution and we provide a simple computation procedure numerical simulations illustrate our results

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

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

long time behavior of a two phase optimal design for the heat equation we consider a two phase isotropic optimal design problem within the context of the transient heat equation the objective is to minimize the average of the dissipated thermal energy during a fixed time interval 0 t the time independent material properties are taken as design variables a full relaxation for this problem was established in a munch p pedregal and f periago j math pures appl 9 89 2008 pp 225 247 by using the homogenization method in this paper we study the asymptotic behavior as t goes to infinity of the solutions of the relaxed problem and prove that they converge to an optimal relaxed design of the corresponding two phase optimization problem for the stationary heat equation next we study necessary optimality conditions for the relaxed optimization problem under the transient heat equation and use those to characterize the microstructure of the optimal designs which appears in the form of a sequential laminate of rank at most n the spatial dimension an asymptotic analysis of the optimality conditions lets us prove that for t large enough the order of lamination is in fact of at most n 1 several numerical experiments in two dimensions complete our study

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

topological lower bounds on algebraic random access machines we prove general lower bounds for set recognition on random access machines rams that operate on real numbers with algebraic operations times as well as rams that use the operations times lfloor rfloor we do it by extending a technique formerly used with respect to algebraic computation trees in the case of algebraic computation trees the complexity was related to the number of connected components of the set w to be recognized for rams four similar results apply to the number of connected components of w circ the topological interior of w two results use overline w circ the interior of the topological closure of w we present theorems that can be applied to a variety of problems and obtain lower bounds many of them tight for the following models 1 a ram which operates on real numbers using integers to address memory and either the operations times or times lfloor rfloor 2 a ram of each of the above instruction sets extended by allowing arbitrary real numbers to be used as memory addresses and adding a test for integer instruction 3 a ram of each of the above instruction sets which can compute with arbitrary real numbers as well as use them for memory addressing while the input is restricted to the integers for one result on this model we require that all program constants be rational

lower bounds for solving linear diophantine equations on random access machines the problem of recognizing the language l n l n k of solvable diophantine linear equations with n variables and solutions from o k n is considered the languages nen l n nen l n l the knapsack problem are np complete the o n 2 lower bound for l n 1 on linear search algorithms due to dobkin and lipton is generalized to an o n 2 log k 1 lower bound for l n k the method of klein and meyer auf der heide is further improved to carry over the o n 2 lower bound for l n 1 to random access machines rams in such a way that it holds for a large class of problems and for very small input sets by this method lower bounds that depend on the input size as is necessary for l n are proved thereby an o n 2 log k 1 lower bound is obtained for rams recognizing l n or l n k for inputs from 0 nk 0 n 2 n

a polynomial linear search algorithm for the n dimensional knapsack problem presentation d un algorithme de recherche lineaire qui reconnait le probleme de sac a dos a n dimensions par etapes de 2n 4 logn o n 3

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

self organized critical state of sandpile automaton models we study a general bak tang wiesenfeld type automaton model of self organized criticality in which the toppling conditions depend on local height but not on its gradient we characterize the critical state and determine its entropy for an arbitrary finite lattice in any dimension the two point correlation function is shown to satisfy a linear equation the spectrum of relaxation times describing the approach to the critical state is also determined exactly

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

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

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

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

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

bounds on functions biorthogonal to sets of complex exponentials control of damped elastic systems abstract in this article we obtain explicit bounds on the norms of biorthogonal functions to sets of complex exponentials exp k t where the k belong to a sector of the positive real axis the bounds obtained are uniform within a class of sequences which exhibit similar growth and separation properties the dependence of the bounds upon the growth and separation parameters is carefully examined the main theorem then leads to some results regarding the solvability of moment problems of the form 0 t f t exp k t dt c k which arise in control problems associated with damped or dissipative systems as an application we establish an exact null controllability result for boundary control of a rectangular structurally damped plate

threshold state and a conjecture of poghosyan poghosyan priezzhev and ruelle we prove a precise relationship between the threshold state of the fixed energy sandpile and the stationary state of dhar s abelian sandpile in the limit as the initial condition s0 tends to infty the former is obtained by size biasing the latter according to burst size an avalanche statistic the question of whether and how these two states are related has been a subject of some controversy since 2000

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

on the limits of computations with the floor function abstract up to now few models of computation with the power of evaluating discontinuous functions have been analyzed and few of their lower bounds or results on the decidability of languages are known in this paper we present a model of an analytic computation tree act these trees operate on real numbers and are able to compare real numbers to evaluate functions on real numbers and to evaluate certain discontinuous functions like the floor function this model generalizes the model of algebraic computation trees introduced by ben or we show by topological arguments that by acts one cannot decide certain classes of languages examples of which are q n and the set of tuples x1 xn r n that have components which are z linearly or algebraically dependent

optimal shape and location of sensors for parabolic equations with random initial data in this article we consider parabolic equations on a bounded open connected subset omega of mathbb r n we model and investigate the problem of optimal shape and location of the observation domain having a prescribed measure this problem is motivated by the question of knowing how to shape and place sensors in some domain in order to maximize the quality of the observation for instance what is the optimal location and shape of a thermometer we show that it is relevant to consider a spectral optimal design problem corresponding to an average of the classical observability inequality over random initial data where the unknown ranges over the set of all possible measurable subsets of omega of fixed measure we prove that under appropriate sufficient spectral assumptions this optimal design problem has a unique solution depending only on a finite number of modes and that the optimal domain is semi analytic and thus has a finite number of connected components this result is in strong contrast with hyperbolic conservative equations wave and schrodinger studied in privat et al j eur math soc 2015 for which relaxation does occur we also provide examples of applications to anomalous diffusion or to the stokes equations in the case where the underlying operator is any positive possible fractional power of the negative of the dirichlet laplacian we show that surprisingly enough the complexity of the optimal domain may strongly depend on both the geometry of the domain and on the positive power the results are illustrated with several numerical simulations

linear quadratic optimal actuator location in control of vibrations diffusion and many other problems governed by partial differential equations there is freedom in the choice of actuator location the actuator location should be chosen to optimize performance objectives in this paper we consider linear quadratic performance two types of cost are considered the choice depends on whether the response to the worst initial condition is to be minimized or whether the initial condition is regarded as random in practice approximations are used in controller design and thus in selection of the actuator locations the optimal cost and location of the approximating sequence should converge to the exact optimal cost and location in this work conditions for this convergence are given in the case of linear quadratic control examples are provided to illustrate that convergence may fail when these conditions are not satisfied

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

simulating probabilistic by deterministic algebraic computation trees a probabilistic algebraic computation tree probabilistic act which recognizes l rn in expected time t and which gives the wrong answer with probability 12 can be simulated by a deterministic act in o t2n steps the same result holds for linear search algorithms lsas the result for acts establishes a weaker version of results previously shown by the author for lsas namely that lsas can only be slightly sped up by their nondeterministic versions this paper shows that acts can only be slightly sped up by their probabilistic versions the result for lsas solves a problem posed by snir 1983 he found an example where probabilistic lsas are faster than deterministic ones and asked how large this gap can be

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

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

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

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

patterns formed by addition of grains to only one site of an abelian sandpile abstract we study the patterns formed by adding n sand grains at a single site in the abelian sandpile model on an infinite square lattice when in the initial configuration all sites have same height when this height is 2 stable heights being 0 3 we show that the perturbed region is a square whose length increases as n for large n if all lengths are rescaled by n the pattern tends to a limiting pattern in which the locally averaged height takes piece wise constant rational values we study the structure of these patterns and also other initial configurations we introduce a toppling function that fully describes these we show that it is piecewise quadratic and determine its form as well as the relation with the height configurations of the pattern

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

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

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

scaling limit of the odometer in divisible sandpiles in a recent work levine et al ann henri poincare 17 1677 1711 2016 https doi org 10 1007 s00023 015 0433 x prove that the odometer function of a divisible sandpile model on a finite graph can be expressed as a shifted discrete bilaplacian gaussian field for the discrete torus they suggest the possibility that the scaling limit of the odometer may be related to the continuum bilaplacian field in this work we show that in any dimension the rescaled odometer converges to the continuum bilaplacian field on the unit torus

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 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

generalized finite automata over real and complex numbers in a recent work gandhi khoussainov and liu 7 introduced and studied a generalized model of finite automata able to work over arbitrary structures as one relevant area of research for this model the authors identify studying such automata over particular structures such as real and algebraically closed fields in this paper we start investigations into this direction we prove several structural results about sets accepted by such automata and analyze decidability as well as complexity of several classical questions about automata in the new framework our results show quite a diverse picture when compared to the well known results for finite automata over finite alphabets

lower time bounds for solving linear diophantine equations on several parallel computational models we consider parallel random access machines pram s with p processors and distributed systems of random access machines dram s with p processors being partially joint by wires according to a communication graph for these computational models we prove lower bounds for testing the solvability of linear diophantine equations and related problems including the knapsack problem these bounds are achieved by generalizing and simplifying a lower bound for parallel computation trees due to yao introducing a new type of computation trees which models computations of dram s and by generalizing a technique used by paul and simon and klein and meyer auf der heide to carry over lower bounds from computation trees to ram s thereby we prove that for many problems p processors cannot speed up a computation by a factor o p but only by a factor o log p 1 and in the case of dram s whose communication network has degree c by a factor o log c 1 only

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

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

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

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

regularity issues for the null controllability of the linear 1 d heat equation the fact that the heat equation is controllable to zero in any bounded domain of the euclidean space any timet 0 and from any open subset of the boundary is well known on the other hand numerical experiments show the ill posedness of the problem in this paper we develop a rigorous analysis of the 1 d problem which provides a sharp description of this ill posedness to be more precise to each initial data y 0 2 l 2 0 1 of the 1 d linear heat equation it corresponds a boundary control of minimal l 2 0 t norm which drives the state to zero in time t 0 this control is given by a solution of the homogeneous adjoint equation with some initial data b 0 minimizing a suitable quadratic cost our aim is to study the relation between the regularity of y 0 and that of b 0 we show that there are regular data y 0 for which the corresponding b 0 are highly irregular not belonging to any negative exponent sobolev space moreover the class of such initial data y 0 is dense in l 2 0 1 this explains the severe ill posedness of the numerical algorithms developed for the approximation of the minimal l 2 0 t norm control of y 0 based on the computation of b 0 the lack of polynomial convergence rates for tychono regularization processes is a consequence of this phenomenon too

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

root system chip firing i interval firing jim propp recently introduced a variant of chip firing on a line where the chips are given distinct integer labels hopkins mcconville and propp showed that this process is confluent from some but not all initial configurations of chips we recast their set up in terms of root systems labeled chip firing can be seen as a root firing process which allows the moves for alpha in phi whenever langle lambda alpha vee rangle 0 where phi is the set of positive roots of a root system of type a and lambda is a weight of this root system we are thus motivated to study the exact same root firing process for an arbitrary root system actually this central root firing process is the subject of a sequel to this paper in the present paper we instead study the interval root firing processes determined by for alpha in phi whenever langle lambda alpha vee rangle in k 1 k 1 or langle lambda alpha vee rangle in k k 1 for any k ge 0 we prove that these interval firing processes are always confluent from any initial weight we also show that there is a natural way to consistently label the stable points of these interval firing processes across all values of k so that the number of weights with given stabilization is a polynomial in k we conjecture that these ehrhart like polynomials have nonnegative integer coefficients