simultaneous similarity of matrices abstract in this paper we solve completely and explicitly the long standing problem of classifying pairs of n n complex matrices a b under the simultaneous similarity tat 1 tbt 1 roughly speaking the classification decomposes to a finite number of steps in each step we consider an open algebraic set m 0n 2 r mn mn mn the set of n n complex valued matrices here r and are two positive integers then we construct a finite number of rational functions o1 os in the entries of a and b whose values are constant on all pairs similar in m n 2 r to a b the values of the functions oi a b i 1 s determine a finite number at most n 2 r of similarity classes in m n 2 r let sn be the subspace of complex symmetric matrices in mn for a b sn sn we consider the similarity class tatt tbtt where t ranges over all complex orthogonal matrices then the characteristic polynomial i a xb determines a finite number of similarity classes for almost all pairs a b sn sn

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

parametric integer programming in fixed dimension parametric integer programming deals with a family of integer programs that is defined by the same constraint matrix but where the right hand sides are points of a given polyhedron the question is whether all these integer programs are feasible kannan showed that this can be checked in polynomial time if the number of variables in the integer programs is fixed and the polyhedron of right hand sides has fixed affine dimension in this paper we extend this result by providing a polynomial algorithm for this problem under the only assumption that the number of variables in the integer programs is fixed we apply this result to deduce a polynomial algorithm to compute the maximum gap between the optimum values of an integer program in fixed dimension and its linear programming relaxation as the right hand side is varying over all vectors for which the integer program is feasible

circuit satisfiability and constraint satisfaction around skolem arithmetic abstract we study interactions between skolem arithmetic and certain classes of circuit satisfiability and constraint satisfaction problems csps we revisit results of glaser et al 1 in the context of csps and settle the major open question from that paper finding a certain satisfiability problem on circuits involving complement intersection union and multiplication to be decidable this we prove using the decidability of skolem arithmetic then we solve a second question left open in 1 by proving a tight upper bound for the similar circuit satisfiability problem involving just intersection union and multiplication we continue by studying first order expansions of skolem arithmetic without constants n as csps we find already here a rich landscape of problems with non trivial instances that are in p as well as those that are np complete

on isometric embeddings of graphs si g est un graphe connexe fini v e une facon standard de definir une distance d g sur g est de definir d g x y comme etant le nombre d aretes dans un chemin le plus court joignant x et y dans v on etudie la theorie des plongements isometriques de graphes dans des produits cartesiens d espaces metriques

simultaneous reduction of sets of matrices under similarity abstract let s be a set of n n matrices over a field f and a the algebra generated by s over f the problem of deciding whether the elements of s can be simultaneously reduced to block triangular form with the diagonal blocks of some specified size is considered and an account is given of various methods used to attack the problem most of the techniques use representation theory to obtain information on a the problems of simultaneous triangularization existence of common eigenvectors etc are also considered the aim of the paper is to survey the methods used to attack these problems and to give some typical results the paper does not contain many new results

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

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

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

short presburger arithmetic is hard we study the computational complexity of short sentences in presburger arithmetic short pa here by short we mean sentences with a bounded number of variables quantifiers inequalities and bo

trace preserving homomorphisms of semigroups abstract we consider those homomorphisms of semigroups of trace class operators on a hilbert space that preserve trace if is a spatially induced isomorphism on a semigroup s that is s t ts for an invertible operator t and for all s in s then clearly has this property more generally if t in the relation above is a densely defined closed injective operator with dense image still preserves trace we prove the converse of this statement under certain conditions using these results we prove simultaneous similarity theorems for semigroups of operators on finite or infinite dimensional spaces whose members are individually similar to unitary or j unitary operators

mixed integer quadratic programming is in np mixed integer quadratic programming is the problem of optimizing a quadratic function over points in a polyhedral set where some of the components are restricted to be integral in this paper we prove that the decision version of mixed integer quadratic programming is in np thereby showing that it is np complete this is established by showing that if the decision version of mixed integer quadratic programming is feasible then there exists a solution of polynomial size this result generalizes and unifies classical results that quadratic programming is in np vavasis in inf process lett 36 2 73 77 17 and integer linear programming is in np borosh and treybig in proc am math soc 55 299 304 1 von zur gathen and sieveking in proc am math soc 72 155 158 18 kannan and monma in lecture notes in economics and mathematical systems vol 157 pp 161 172 springer 9 papadimitriou in j assoc comput mach 28 765 768 15

convergence in distribution of random metric measure spaces λ coalescent measure trees we consider the space of complete and separable metric spaces which are equipped with a probability measure a notion of convergence is given based on the philosophy that a sequence of metric measure spaces converges if and only if all finite subspaces sampled from these spaces converge this topology is metrized following gromov s idea of embedding two metric spaces isometrically into a common metric space combined with the prohorov metric between probability measures on a fixed metric space we show that for this topology convergence in distribution follows provided the sequence is tight from convergence of all randomly sampled finite subspaces we give a characterization of tightness based on quantities which are reasonably easy to calculate subspaces of particular interest are the space of real trees and of ultra metric spaces equipped with a probability measure as an example we characterize convergence in distribution for the ultra metric measure spaces given by the random genealogies of the coalescents we show that the coalescent defines an infinite random metric measure space if and only if the so called dust free property holds

a structure theorem for some matrix algebras abstract let f be a field and m n f the algebra of n n matrices over f it is in general a very difficult and tedious problem to determine the structure of the subalgebra x of m n f generated by a given subset s of m n f we show that if x contains a matrix a with n distinct eigenvalues in f then x can be determined up to similarity very quickly by a graph theoretic method as a consequence we show that any such x can be generated by a pair of elements one of which can be taken to be a and the other a 0 1 matrix as an application we obtain a specht type similarity theorem

simple sentences that are hard to decide abstract it was considered to be typical for first order theories that a restriction to sentences with only a limited number of quantifier alternations leads to an exponential decrease of complexity using domino games which were treated in a previous paper to describe computations of alternating turing machines we prove that this is not always true we present a list of theories all of them decidable in c 0 atime 2 cn n for which the subclasses with bounded quantifier alternations still have alternating exponential time complexity in particular this yields non deterministic exponential time lower bounds for very simple prefix classes with 2 or 3 alternations theories with such behaviour are the theory of boolean algebras the theory of polynomial rings over finite fields the theory of idempotent rings the theory of finite sets with inclusion the theory of semilattices the theory of stone algebras the theory of distributive p algebras in the lee class b n and the theories of natural numbers with divisibility or coprimeness

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

subclasses of presburger arithmetic and the polynomial time hierarchy abstract we investigate the complexity of subclasses of presburger arithmetic i e the first order theory of natural numbers with addition the subclasses are defined by restricting the quantifier prefix to finite lists q 1 q s for all m 2 we find formula classes defined by prefixes with m 1 alternations and m 5 quantifiers which are p m respectively p m complete for m 1 the class of formulas is shown to be np complete for m 0 and for all natural numbers t the class of t formulas is known to be in p thus we have a nice characterisation of the polynomial time hierarchy by classes of presburger formulas finally the np completeness of the class is used to prove that for certain formulas there exist no equivalent quantifier free formulas of polynomial length

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

sentences over integral domains and their computational complexities letrbe a hilbertian domain and letkbe its fraction field let x1 xn y be a quantifier free arithmetical formula overr we may also take x1 xn y to be an arithmetical formula overkx1 xn and write it as y in this paper we show that ifrhas enough non units and x1 xn y x1 xn y called an n sentence is true inr then y y is true inkx1 xn also ifr kt wherekis an infinite integral domain and x1 xn y x1 xn y is true inr then y y is true inrx1 xn these results are applied to find the upper and lower bounds of the time complexities of various decision problems on diophantine equations with parameters and arithmetical sentences some of the results are 1 the decision problem of sentences and diophantine equations with parameters over the ring of integers of a global field are co np complete 2 the decision problem of sentences over the ring of integers of a global field is np complete 3 letkbe an infinite domain the time complexities of the decision problems of equations with parameters and sentences over the polynomial ringkt are polynomial time reducible to factoring polynomials overk 4 the decision problem of sentences over all algebraic integer rings is in p 5 the decision problem of sentences over all integral domains with characteristic 0 is in p 6 the time complexity of the decision problem of sentences over all integral domains is polynomial time reducible to factoring integers overzand factoring polynomials over finite fields

the computational complexity of simultaneous diophantine approximation problems simultaneous diophantine approximation in d dimensions deals with the approximation of a vector bf alpha alpha 1 cdots alpha d of d real numbers by vectors of rational numbers all having the same denominator this paper considers the computational complexity of algorithms to find good simultaneous approximations to a given vector bf alpha of d rational numbers we measure the goodness of an approximation using the sup norm we show that a result of h w lenstra jr produces polynomial time algorithms to find sup norm best approximations to a given vector bf alpha when the dimension d is fixed we show that a recent algorithm of a k lenstra h w lenstra jr and l lovasz to find short vectors in an integral lattice can be used to find a good approximation to a given vector bf alpha in d dimensions with a denominator q satisfying 1 leqq q leqq 2 d 2 n which is within a factor sqrt 5d 2 d 1 2 of the best approximation with denominato

the complexity of presburger arithmetic with bounded quantifier alternation depth abstract it is shown how the method of fischer and rabin can be extended to get good lower bounds for presburger arithmetic with a bounded number of quantifier alternations in this case the complexity is one exponential lower than in the unbounded case this situation is typical for first order theories

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

automorphism groups of algebras of triangular matrices abstract if a is an algebra of n x n matrices over an integral domain we can associate with a a graph whose edges are pairs i j such that the i j entry of every element of a is zero the graph in turn defines a group of permutations and automorphisms of a which are conjugations by permutations matrices for a suitably restricted class of algebras a we show that the automorphism group of a is the semidirect product of this group of permutation matrices with the subgroup of inner automorphisms

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

complexity of subcases of presburger arithmetic we consider formula subclasses of presburger arithmetic which have a simple structure in one sense or the other and investigate their computational complexity we also prove some results on the lower bounds of lengths of formulas which are related to questions on quantifier elimination 1 introduction the table on pp 3 4 in the monograph 3 of j ferrante and c w rachkoff shows that most of the known decidable theories are so complex that they may as well be considered as undecidable it thus makes sense to investigate subclasses of formulas of such theories which distinguish themselves e g by a simple structure or otherwise the hope is to find a formula class which is decidable in deterministic polynomial time and is of interest because interesting statements can be expressed by formulas of this class here we investigate pres burger arithmetic from this point of view we thereby work with presburger arithmetic which contains a function symbol 2 representing multiplication by 2 i e 2 x 2x the presence of 2 has no influence on complexity as far as the theory as a whole is concerned but can make a difference for certain subclasses presburger arithmetic is a decidable theory of high complexity as has been shown by m j fischer and m o rabin in 5 subclasses of high complexity are obtained by considering formulas of bounded quantifier alternation these have been investigated by m furer in 4 in order to describe some formula classes considered here let m be the set of all quantifierfree formulas we then fix a list qx qs of quantifiers and consider classes of formulas of the form 1 oi qsxs eyx ey l lem

the semigroup of metric measure spaces and its infinitely divisible probability measures a metric measure space is a complete separable metric space equipped with a probability measure that has full support two such spaces are equivalent if they are isometric as metric spaces via an isometry that maps the probability measure on the first space to the probability measure on the second the resulting set of equivalence classes can be metrized with the gromov prohorov metric of greven pfaffelhuber and winter we consider the natural binary operation on this space that takes two metric measure spaces and forms their cartesian product equipped with the sum of the two metrics and the product of the two probability measures we show that the metric measure spaces equipped with this operation form a cancellative commutative polish semigroup with a translation invariant metric there is an explicit family of continuous semicharacters that is extremely useful for inter alia establishing that there are no infinitely divisible elements and that each element has a unique factorization into prime elements we investigate the interaction between the semigroup structure and the natural action of the positive real numbers on this space that arises from scaling the metric for example we show that for any given positive real numbers a b c the trivial space is the only space that satisfies a b c we establish that there is no analogue of the law of large numbers if x1 x2 is an identically distributed independent sequence of random spaces then no subsequence of formula see text converges in distribution unless each xk is almost surely equal to the trivial space we characterize the infinitely divisible probability measures and the levy processes on this semigroup characterize the stable probability measures and establish a counterpart of the lepage representation for the latter class

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

isometric embeddings of graphs abstract we prove that any finite undirected graph can be canonically embedded isometrically into a maximum cartesian product of irreducible factors

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

sums of idempotents and logarithmic residues in zero pattern matrix algebras abstract a zero pattern algebra is a matrix subalgebra of c n n determined by a pattern of zeros the issue in this paper is under what conditions is a matrix in a zero pattern algebra a a sum of rank one idempotents in a or a logarithmic residue in a here logarithmic residues are contour integrals of logarithmic derivatives of analytic a valued functions it turns out that there is a necessary condition involving certain rank trace requirements although these requirements are generally not sufficient there are several important cases where they are

the complexity of query evaluation in indefinite temporal constraint databases abstract in previous work we have developed the scheme of indefinite l constraint databases where l the parameter is a first order constraint language this scheme extends the constraint database proposal of kanellakis et al 1990 1995 to include indefinite or uncertain information in the style of imielinski and lipski 1984 in this paper we study the complexity of query evaluation in an important instance of this abstract scheme indefinite temporal constraint databases our results indicate that the data combined complexity of query evaluation does not change when we move from queries in relational calculus over relational databases to queries in relational calculus with temporal constraints over temporal constraint databases this fact remains true even when we consider query evaluation in relational databases with indefinite information vs query evaluation in indefinite temporal constraint databases in the course of our work we provide precise bounds on the complexity of decision quantifier elimination for a subtheory of presburger arithmetic and a subtheory of real addition with order the bounds for the latter theory are original and of independent interest

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

on the geometry of metric measure spaces ii we introduce a curvature dimension condition cd k n for metric measure spaces it is more restrictive than the curvature bound underline text curv left m text d m right k introduced in sturm k t 2006 on the geometry of metric measure spaces i acta math 196 65 131 which is recovered as the borderline case cd k the additional real parameter n plays the role of a generalized upper bound for the dimension for riemannian manifolds cd k n is equivalent to text ric m left xi xi right k left xi right 2 and dim m n the curvature dimension condition cd k n is stable under convergence for any triple of real numbers k n l the family of normalized metric measure spaces m d m with cd k n and diameter l is compact condition cd k n implies sharp version of the brunn minkowski inequality of the bishop gromov volume comparison theorem and of the bonnet myers theorem moreover it implies the doubling property and local scale invariant poincare inequalities on balls in particular it allows to construct canonical dirichlet forms with gaussian upper and lower bounds for the corresponding heat kernels

short rational generating functions for lattice point problems abstract we prove that for any xed d the generating function of the projectionof the set of integer points in a rational d dimensional polytope can be computed inpolynomial time as a corollary we deduce that various interesting sets of latticepoints notably integer semigroups and minimal hilbert bases of rational cones have short rational generating functions provided certain parameters the dimensionand the number of generators are xed it follows then that many computationalproblems for such sets for example nding the number of positive integers notrepresentable as a non negative integer combination of given coprime positive integersa 1 a d admit polynomial time algorithms we also discuss a related problem ofcomputing the hilbert series of a ring generated by monomials 1 introduction and main resultsour main motivation is the following question which goes back to frobenius andsylvester 1 1 the frobenius problem let a 1 a d be positive coprime integers andlets n

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

a survey of canonical forms and invariants for unitary similarity abstract matrices a and b are said to be unitarily similar if u au b for some unitary matrix u this expository paper surveys results on canonical forms and invariants for unitary similarity the first half gives a detailed description of methods developed by several authors brenner littlewood mitchell mcrae radjavi serge chuk and benedetti and cragnolini using inductively defined reduction procedures to transform matrices to canonical form the matrix is partitioned and successive unitary similarities applied to reduce the submatrices to some nice form at each stage one refines the partition and restricts the set of permissible unitary similarities to those that preserve the already reduced blocks the process ends in a finite number of steps producing both the canonical form and the subgroup of the unitary group that preserves that form depending on the initial step various canonical forms may be defined the method can also be used to define canonical forms relative to certain subgroups of the unitary group and canonical forms for finite sets of matrices under simultaneous unitary similarity the remainder of the paper surveys results on unitary invariants and other topics related to unitary similarity such as the specht pearcy trace invariants the numerical range and unitary reducibility

integer polynomial optimization in fixed dimension we classify according to their computational complexity integer optimization problems whose constraints and objective functions are polynomials with integer coefficients and the number of variables is fixed for the optimization of an integer polynomial over the lattice points of a convex polytope we show an algorithm to compute lower and upper bounds for the optimal value for polynomials that are nonnegative over the polytope these sequences of bounds lead to a fully polynomial time approximation scheme for the optimization problem