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

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

arctic boundaries of the ice model on three bundle domains in this paper we consider the six vertex model at ice point on an arbitrary three bundle domain which is a generalization of the domain wall ice model on the square or equivalently of a uniformly random alternating sign matrix we show that this model exhibits the arctic boundary phenomenon whose boundary is given by a union of explicit algebraic curves this was originally predicted by colomo and sportiello j stat phys 164 1488 1523 2016 as one of the initial applications of a general heuristic that they introduced for locating arctic boundaries called the geometric tangent method our proof uses a probabilistic analysis of non crossing directed path ensembles to provide a mathematical justification of their tangent method heuristic in this case which might be of independent interest

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

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

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

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

generalized design of low complexity block diagonalization type precoding algorithms for multiuser mimo systems block diagonalization bd based precoding techniques are well known linear transmit strategies for multiuser mimo mu mimo systems by employing bd type precoding algorithms at the transmit side the mu mimo broadcast channel is decomposed into multiple independent parallel single user mimo su mimo channels and achieves the maximum diversity order at high data rates the main computational complexity of bd type precoding algorithms comes from two singular value decomposition svd operations which depend on the number of users and the dimensions of each user s channel matrix in this work low complexity precoding algorithms are proposed to reduce the computational complexity and improve the performance of bd type precoding algorithms we devise a strategy based on a common channel inversion technique qr decompositions and lattice reductions to decouple the mu mimo channel into equivalent su mimo channels analytical and simulation results show that the proposed precoding algorithms can achieve a comparable sum rate performance as bd type precoding algorithms substantial bit error rate ber performance gains and a simplified receiver structure while requiring a much lower complexity

moderate deviations for empirical measures of markov chains lower bounds we obtain upper bounds for moderate deviations of empirical measures of markov chains with general state space in the topology under the minimal assumption of geometric ergodicity the regeneration split chain method and a result on projective systems are employed in the proof

the range of stable random walks limit theorems are proved for the range of d dimensional random walks in the domain of attraction of a stable process of index p3 the range rn is the number of distinct sites of zd visited by the random walk before time n our results depend on the value of the ratio 3 d the most interesting results are obtained for 2 3 1 which can only occur if d 1 we prove the convergence in distribution of rn e rn toward some constant times the lebesgue measure of the range of the limiting stable process some of our results require regularity assumptions on the characteristic function of x

functional central limit theorem for a class of negatively dependent heavy tailed stationary infinitely divisible processes generated by conservative flows we prove a functional central limit theorem for partial sums of symmetric stationary long range dependent heavy tailed infinitely divisible processes the limiting stable process is particularly interesting due to its long memory which is quantified by a mittag leffler process induced by an associated harris chain at the discrete time level previous results in owada and samorodnitsky ann probab 43 2015 240 285 dealt with positive dependence in the increment process whereas this paper derives the functional limit theorems under negative dependence the negative dependence is due to cancellations arising from gaussian type fluctuations of functionals of the associated harris chain the new types of limiting processes involve stable random measures due to heavy tails mittag leffler processes due to long memory and brownian motions due to the gaussian second order cancellations along the way we prove a function central limit theorem for fluctuations of functionals of harris chains which is of independent interest as it extends a result of chen probab theory related fields 116 2000 89 123

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 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

lattice reduction based approximate map detection with bit wise combining and integer perturbed list generation for iterative detection and decoding idd in multiple input multiple output mimo systems the log likelihood ratio llr of each coded bit can be found by an optimal bit wise maximum a posteriori probability map detector however since this map detector requires a prohibitively high computational complexity low complexity suboptimal detectors are desirable in this paper lattice reduction lr based mimo detection is investigated to derive a low complexity detector that can achieve near map performance for idd in order to approximate llr values incorporating the extrinsic information provided by a soft input soft output siso decoder bit wise lr based minimum mean square error mmse filters are derived furthermore in order to minimize the performance degradation due to quantization or rounding errors in the lr based detection a low complexity integer perturbed list generation method is proposed where no tree search is used by taking advantage of a near orthogonal channel basis obtained by lr through a complexity analysis and simulations it is shown that the proposed approach achieves near optimal performance while the complexity is comparable with that of the mmse soft cancellation method which is known to be computationally efficient as a bit wise detector a parallel implementation of the proposed method would be straightforward which lowers the detection delay

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

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

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

arctic curves of the twenty vertex model with domain wall boundaries we use the tangent method to compute the arctic curve of the twenty vertex 20v model with particular domain wall boundary conditions for a wide set of integrable weights to this end we extend to the finite geometry of domain wall boundary conditions the standard connection between the bulk 20v and 6v models via the kagome lattice ice model this allows to express refined partition functions of the 20v model in terms of their 6v counterparts leading to explicit parametric expressions for the various portions of its arctic curve the latter displays a large variety of shapes depending on the weights and separates a central liquid phase from up to six different frozen phases a number of numerical simulations are also presented which highlight the arctic curve phenomenon and corroborate perfectly the analytic predictions of the tangent method we finally compute the arctic curve of the quarter turn symmetric holey aztec domino tiling qthadt model a problem closely related to the 20v model and whose asymptotics may be analyzed via a similar tangent method approach again results for the qthadt model are found to be in perfect agreement with our numerical simulations

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

boosted kz and lll algorithms there exist two issues among popular lattice reduction algorithms that should cause our concern the first one is korkine zolotarev kz and lenstra lenstra lovasz lll algorithms may increase the lengths of basis vectors the other is kz reduction suffers worse performance than minkowski reduction in terms of providing short basis vectors despite its superior theoretical upper bounds to address these limitations we improve the size reduction steps in kz and lll to set up two new efficient algorithms referred to as boosted kz and lll for solving the shortest basis problem with exponential and polynomial complexity respectively both of them offer better actual performance than their classic counterparts and the performance bounds for kz are also improved we apply them to designing integer forcing if linear receivers for multi input multioutput communications our simulations confirm their rate and complexity advantages

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

how often does a harris recurrent markov chain recur let x n n geq 0 be a harris recurrent markov chain with state space e mathscr e transition probability p x a and invariant measure pi given a nonnegative pi integrable function f on e the exact asymptotic order is given for the additive functionals sum k 1 n f x k quad n 1 2 dotsin the forms of both weak and strong convergences in particular the frequency of x n n geq 0 visiting a given set a in mathscr e with 0 pi a infty is determined by taking f i a under the regularity assumption the limits in our theorems are identified the one and two dimensional random walks are taken as the examples of applications

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

reduced and fixed complexity variants of the lll algorithm for communications the lenstra lenstra lovasz lll algorithm is a popular lattice reduction algorithm in communications in this paper variants of the lll algorithm with either reduced or fixed complexity are proposed and analyzed specifically the use of effective lll reduction for lattice decoding is presented where size reduction is only performed for pairs of consecutive basis vectors its average complexity measured by the number of floating point operations and averaged over i i d standard normal lattice bases is shown to be o n 3 log n where n is the lattice dimension this average complexity is an order lower than previously thought to address the issue of variable complexity of the lll algorithm two fixed complexity approximations are proposed one is fixed complexity effective lll for which the first vector of the basis is proven to be bounded in length the other is fixed complexity lll with deep insertion which is shown to be closely related to the well known v blast algorithm such fixed complexity structures are much desirable in hardware implementation since they allow straightforward constant throughput implementation

fixed complexity lll algorithm a common technique to perform lattice basis reduction is the lenstra lenstra lovasz lll algorithm an implementation of this algorithm in real time systems suffers from the problem of variable run time and complexity this correspondence proposes a modification of the lll algorithm the signal flow is altered to follow a deterministic structure which promises to obtain an easier implementation as well as a fixed execution time known in advance in the case of a maximum number of iterations as it is likely in real time systems our modification clearly outperforms the original lll algorithm as far as the quality of the reduced lattice basis is concerned

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

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

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

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

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

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

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

lattice reduction aided broadcast precoding a precoding scheme for multiuser broadcast communications is described which fills the gap between the low complexity tomlinson harashima precoding and the sphere decoder based system of peel et al simulation results show that replacing the closest point search with the babai approximation the full diversity order supported by the channel is available to each user as in the system of peel et al and unlike tomlinson harashima precoding which suffers some diversity penalty the complexity of the scheme is similar to that of tomlinson harashima precoding

numerical study of the f model with domain wall boundaries we perform a numerical study of the f model with domain wall boundary conditions various exact results are known for this particular case of the six vertex model including closed expressions for the partition function for any system size as well as its asymptotics and leading finite size corrections to complement this picture we use a full lattice multicluster algorithm to study equilibrium properties of this model for systems of moderate size up to l 512 we compare the energy to its exactly known large l asymptotics we investigate the model s infinite order phase transition by means of finite size scaling for an observable derived from the staggered polarization in order to test the method put forward in our recent joint work with duine and barkema in addition we analyze local properties of the model our data are perfectly consistent with analytical expressions for the arctic curves we investigate the structure inside the temperate region of the lattice confirming the oscillations in vertex densities that were first observed by syljuasen and zvonarev and recently studied by lyberg et al we point out anti ferroelectric oscillations close to the corresponding frozen regions as well as higher order oscillations forming an intricate pattern with saddle point like features

on the limit laws of the second order for additive functionals of harris recurrent markov chains let xn n 0 be a harris recurrent markov chain with state space e and invariant measure the law of the iterated logarithm and the law of weak convergence are given for the additive functionals of the form

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