refined proximity and sensitivity results in linearly constrained convex separable integer programming abstract we define a new measure of the structure of a linear constraint matrix and establish some propertie we then use the measure to refine proximity and sensitivity results in the literature in particular we refine bounds on the distance between continuous and discrete optima of linearly constrained convex separable programming problems and on the size of the feasible region for a linear program we apply the results to improve the computational complexity of hochbaum and shanthikumar s algorithm for integer nonlinear programming reducing the computational effort for certain problems from exponential to polynomial

computational complexity of a cost allocation approach to a fixed cost spanning forest problem we present a computational analysis of a game theoretic approach to a cost allocation problem arising from a graph optimization problem referred to as the fixed cost spanning forest fcsf problem the customers in the fcsf problem represented by nodes in a graph g are in need of service that can be produced at some facilities yet to be constructed the cost allocation problem is concerned with the fair distribution of the cost of providing the service among customers we formulate this cost allocation problem as a cooperative game referred to as the fcsf game in general the core of a fcsf game may be empty however for the case when g is a tree it is shown that the core is not empty moreover we prove that in this case core points can be generated in strongly polynomial time we further provide a nonredundant characterization of the core of the fcsf game defined over a tree in the special case when all nodes are communities this is shown to lead in some instances to a strongly polynomial algorithm for computing the nucleolus

a simple variant of the mizuno todd ye predictor corrector algorithm and its objective function free complexity in this paper we propose a simple variant of the mizuno todd ye predictor corrector algorithm for linear programming our variant employs a simple and a natural scaling invariant search directio

reciprocal relations for the weight factors arising in the series expansion of backbone percolation functions the authors consider several properties of the backbone of percolation clusters the percolation model on a graph g is defined as follows each element vertex or edge of g is either open or closed the element e being open with probability pe independently of all other elements the u v backbone in any configuration is the subgraph consisting of all elements which belong to at least one open self avoiding path from a vertex u to a vertex v they show that the expected value of any u v backbone random variable z which depends only on the elements of the u v backbone e g z may be the number of edges or the length of the shortest path of the backbone is given by e z sigma k contained in p 1 mod k mod 1zk pi e in g k pe where p is the set of all self avoiding paths from u to v on g g k is the subgraph formed by all elements which belong to at least one self avoiding path of k and the weightzk is given by zk sigma j contained in k 1 mod jmod 1 zj where zj is the value of z when the backbone is g j certain reciprocal relations are found to hold between the weights and the backbone variables in some important cases using these results series expansions for the various properties can be obtained

exact enumeration approach to fractal properties of the percolation backbone and 1 σ expansion an exact numeration approach is developed for the backbone fractal of the incipient infinite cluster at the percolation threshold the authors use this approach to calculate exactly the first low density expansion of lbb p for arbitrary system dimensionality d where lbb p is the mean of backbone bonds and p is the bond occupation probability standard series extrapolation methods provide estimates of the fractal dimension of the backbone for all d these disagree with the sierpinski gasket model of the backbone they also calculate the first low density expansions of lmin p and lred p which are respectively the mean number of bonds in the minimum path between i and j and the mean number of singly connected red bonds

on the castelnuovo mumford regularity of rings of polynomial invariants we show that when a group acts on a polynomial ring over a eld the ring of invariants has castelnuovo mumford regularity at most zero as a consequence we prove a well known conjecture that the invariants are always generated in degrees at most n jgj 1 where n 1 is the number of polynomial generators andjgj 1 is the order of the group we also prove some other related conjectures in invariant theory

a strongly polynomial algorithm to solve combinatorial linear programs khachiyan and recently karmarkar gave polynomial algorithms to solve the linear programming problem these algorithms have a small theoretical drawback namely the number of arithmetic steps depends on the size of the input numbers we present a polynomial linear programming algorithm whose number of arithmetic steps depends only on the size of the numbers in the constraint matrix but is independent of the size of the numbers in the right hand side and objective vectors in particular it gives a polynomial algorithm for the minimum cost flow and multicommodity flow problems in which the number of arithmetic steps is independent of the size of the costs and capacities the algorithm makes use of an existing polynomial linear programming algorithm the problem of whether any algorithm has a running time that is independent even of the size of the numbers in the constraint matrix remains open

on some balanced totally balanced and submodular delivery games this paper studies a class of delivery problems associated with the chinese postman problem and a corresponding class of delivery games a delivery problem in this class is determined by a connected graph a cost function defined on its edges and a special chosen vertex in that graph which will be referred to as the post office it is assumed that the edges in the graph are owned by different individuals and the delivery game is concerned with the allocation of the traveling costs incurred by the server who starts at the post office and is expected to traverse all edges in the graph before returning to the post office a graph g is called chinese postman submodular or for short cp submodular cp totally balanced cp balanced respectively if for each delivery problem in which g is the underlying graph the associated delivery game is submodular totally balanced balanced respectively for undirected graphs we prove that cp submodular graphs and cp totally balanced graphs are weakly cyclic graphs and conversely an undirected graph is shown to be cp balanced if and only if it is a weakly euler graph for directed graphs cp submodular graphs can be characterized by directed weakly cyclic graphs further it is proven that any strongly connected directed graph is cp balanced for mixed graphs it is shown that a graph is cp submodular if and only if it is a mixed weakly cyclic graph finally we note that undirected directed and mixed weakly cyclic graphs can be recognized in linear time

towards a genuinely polynomial algorithm for linear programming a linear programming algorithm is called genuinely polynomial if it requires no more than p m n arithmetic operations to solve problems of order m times n where p is a polynomial it is not known whether such an algorithm exists we present a genuinely polynomial algorithm for the simpler problem of solving linear inequalities with at most two variables per inequality the number of operations required is o mn 3 log text m the technique used was developed in a previous paper where a novel binary search idea was introduced

on the complexity of quadratic programming in real number models of computation abstract the complexity of linearly constrained nonconvex quadratic programming is analyzed within the framework of real number models namely the one of blum shub and smale and its modification recently introduced by koiran weak bss model in particular we show that this problem is not np complete in the koiran setting applications to the full bss model are discussed

up and downgrading the 1 center in a network abstract we study budget constrained network improvement and degrading problems based on the vertex 1 center problem on graphs given a graph with vertex weights and edge lengths the task is to decrease and increase the vertex weights within certain limits such that the optimal 1 center objective value with respect to the new weights is minimized and maximized respectively the upgrading improvement problem is shown to be solvable in o n 2 time provided that the distance matrix is given the downgrading 1 center problem is shown to be strongly np hard on general graphs but can be solved in o n 2 time on trees as byproduct we suggest an algorithm that solves the problem of minimizing over the upper envelope of n piecewise linear functions in o k n time where k is the total number of breakpoints

inapproximability and a polynomially solvable special case of a network improvement problem abstract we consider a network improvement problem in which we wish to spend as little as possible to ensure that the distances from a given vertex to all other vertices are within given bounds we first prove that achieving an approximation ratio 1 is np hard for some constant 0 then we show that the problem can be solved by a combinatorial algorithm running in o v log v time when the network is restricted to a tree

backbone and elastic backbone of percolation clusters obtained by the new method of burning we introduce a method of obtaining information about percolation clusters in which the clusters are burned in different ways this method yields the backbone and the elastic backbone their number of loops shortest path and other quantities it works in any dimension and can be applied also to other clusters we obtain accurate estimates for the backbone exponents in two and three dimensions

all linear and integer programs are slim 3 way transportation programs we show that any rational convex polytope is polynomial time representable as a 3 way line sum transportation polytope of slim r c 3 format this universality theorem has important consequences for linear and integer programming and for confidential statistical data disclosure we provide a polynomial time embedding of arbitrary linear programs and integer programs in such slim transportation programs and in bitransportation programs our construction resolves several standing problems on 3 way transportation polytopes for example it demonstrates that unlike the case of 2 way contingency tables the range of values an entry can attain in any slim 3 way contingency table with specified 2 margins can contain arbitrary gaps our smallest such example has format 6 4 3 our construction provides a powerful automatic tool for studying concrete questions about transportation polytopes and contingency tables for example it automatically provides new proofs for some classical results including a well known real feasible but integer infeasible 6 4 3 transportation polytope of m vlach and bitransportation programs where any feasible bitransportation must have an arbitrarily large prescribed denominator

polynomial algorithms for lp over a subring of the algebraic integers with applications to lp with circulant matrices we show that a modified variant of the interior point method can solve linear programs lps whose coefficients are real numbers from a subring of the algebraic integers by defining the encoding size of such numbers to be the bit size of the integers that represent them in the subring we prove the modified algorithm runs in time polynomial in the encoding size of the input coefficients the dimension of the problem and the order of the subring we then extend the tardos scheme to our case obtaining a running time which is independent of the objective and right hand side data as a consequence of these results we are able to show that lps with real circulant coefficient matrices can be solved in strongly polynomial time finally we show how the algorithm can be applied to lps whose coefficients belong to the extension of the integers by a fixed set of square roots

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

lower bounds in a parallel model without bit operations we define a natural and realistic model of parallel computation called the pram model without bit operations it is like the usual pram model the main difference being that no bit operations are provided it encompasses virtually all known parallel algorithms for weighted combinatorial optimization and algebraic problems in this model we prove that for some large enough constant b the mincost flow problem for graphs with n vertices cannot be solved deterministically or with randomization in sqrt n b expected time using 2 sqrt n b processors this is so even if we restrict every cost and capacity to be an integer nonnegative if it is a capacity of bitlength at most an for some large enough constant a a similar lower bound is also proved for the max flow problem it follows that these problems cannot be solved in our model deterministically or with randomization in omega n c expected time with 2 omega n c processors where c is an appropriate positive constant and n is the total bitlength of the input since these problems were known to be p complete this provides concrete support for the belief that p completeness implies high parallel complexity and for the p not nc conjecture our lower bounds also extend to the pram model with limited bit operations which provides instructions for parity and left or right shift by one bit our proof is based on basic algebraic geometry so we investigate if the algebrogeometric approach could also work for the p versus nc problem our results support this possibility and a close analysis of the limitation of our technique in this context suggests that such a proof of p not nc should somehow use geometric invariant theory in a deep way

the noether bound in invariant theory of finite groups abstract let r be a commutative ring v a finitely generated free r module and g glr v a finite group acting naturally on the graded symmetric algebra a sym v let ag denote the minimal number m such that the ring ag of invariants can be generated by finitely many elements of degree at most m furthermore let h g be a normal subgroup such that the index g h is invertible in r in this paper we prove the inequality a g a h g h for h 1 and g invertible in r we obtain noether s bound ag g which so far had been shown for arbitrary groups only under the assumption that the factorial of the group order g is invertible in r

a cooperative location game based on the 1 center location problem abstract in this paper we introduce and analyze new classes of cooperative games related to facility location models defined on general metric spaces the players are the customers demand points in the location problem and the characteristic value of a coalition is the cost of serving its members specifically the cost in our games is the service radius of the coalition we call these games the minimum radius location games mrlg we study the existence of core allocations and the existence of polynomial representations of the cores of these games focusing on network spaces i e finite metric spaces induced by undirected graphs and positive edge lengths and on the lp metric spaces defined over r d

descent direction algorithm with multicommodity flow problem for signal optimization and traffic assignment jointly network managers wish to optimize control parameters such as signal setting which are very related to the traffic assignment models on the other hand traffic assignment patterns as an important instrument for predicting the amount of flow on network links are dependent to control decisions according to the significance of this concept some important papers about this mutually relation are reviewed in this paper then we implement a nonlinear algorithm on a minimal cost multicommodity flow mcmf problem to optimize some control policies subject to optimal flows although we take signal times into account but this approach has a far more reaching application in urban network control and design we employ a hybrid intelligent algorithm integrating decent direction algorithm and an interior point algorithm in a mutually consistent scheme for obtaining optimal signals and equilibrium flows an example is given to illustrate the effectiveness of our scheme

series expansion analysis of the backbone properties of two dimensional percolation clusters low density series expansions for the backbone properties of two dimensional bond percolation clusters are derived and analysed expansions for most of the 14 properties considered are new and are obtained to order on the square lattice and order on the triangular lattice earlier series work was confined to three properties of the square lattice and was to order the fractal dimension of the bonds or sites in the backbone is estimated to be and is intermediate between a previously conjectured field theory value and the latest monte carlo results the union intersection and length of the longest self avoiding paths are found to have the same fractal dimension which is close to and consistent with the field theory conjecture for on the other hand the union intersection and length of the shortest paths are found to have different dimensions and in the case of the intersection the triangular and square lattices are found to have sigificantly different dimensions the fractal dimension of the shortest path also appears to be non universal and we find for the square lattice and for the triangular lattice critical amplitude ratios are considered and found to be in agreement with theoretical inequalities

random walks totally unimodular matrices and a randomised dual simplex algorithm we discuss the application of random walks to generating a random basis of a totally unimodular matrix and to solving a linear program with such a constraint matrix we also derive polynomial upper bounds on the combinatorial diameter of an associated polyhedron

a strongly polynomial algorithm for linear systems having a binary solution this paper proposes a strongly polynomial algorithm which either finds a solution of a linear system ax b 0 x 1 or correctly decides that the system has no 0 1 solutions the algorithm can be used as the basis for the construction of a polynomial algorithm for linear programming

on some network flow games we analyze three subclasses of cooperative games arising from network optimization problems in which the resources such as arcs or nodes in the network are controlled by individuals who have conflicting objectives the first subclass of cooperative games is induced by network optimization problems over directed augmented trees we show that for this subclass of games the kernel coincides with the nucleolus and that the nucleolus can be characterized as the unique revenue allocation vector in which every pair of arc owners who are adjacent in the tree are located symmetrically with respect to their bargaining range we further give a linear characterization of the core of this subclass of games which is then used to provide a more explicit representation of the nucleolus and to construct a strongly polynomial algorithm for generating it the second subclass of cooperative games is induced by maximum flow problems in simple undirected networks we provide a useful parametric representation of the core of this subclass of games which is used to characterize the nucleolus and the intersection of the core and the kernel explicitly we show that the intersection of the core and the kernel consists of all revenue allocations in the core which assign equal payoffs to any pair of unseparated arc owners we further demonstrate that among the core vectors the nucleolus is the unique revenue allocation vector in which the smallest allocations are maximized in a lexicographical sense the third cooperative game that we study is the assignment game introduced by shapley and shubik 1972 this game is induced by the assignment problem which can be cast as a network optimization problem we investigate the relationship between the kernel and the core of the assignment game and provide a necessary and sufficient condition for the core to be contained in the kernel we further show that in general the intersection of the kernel and the core is not a convex set we also exhibit that under certain conditions the nucleolus has a simple characterization as the unique vector in the core in which the smallest revenue allocations are maximized in a lexicographical sense finally we consider the horse market example of bohm bawerk 1923 for which it is shown that the core is contained in the kernel and that the nucleolus is the midpoint of the core

the invariants of modular indecomposable representations of mathbb z _ p 2 we consider the invariant ring for an indecomposable representation of a cyclic group of order p 2 over a field mathbb f of characteristic p we describe a set of mathbb f algebra generators of this ring of invariants and thus derive an upper bound for the largest degree of an element in a minimal generating set for the ring of invariants this bound as a polynomial in p is of degree two

linear programming the simplex algorithm and simple polytopes in the first part of the paper we survey some far reaching applications of the basic facts of linear programming to the combinatorial theory of simple polytopes in the second part we discuss some recent developments concerning the simplex algorithm we describe subexponential randomized pivot rules and upper bounds on the diameter of graphs of polytopes 1997 the mathematical programming society inc published by elsevier science b v

on the algorithmic inversion of the discrete radon transform the present paper deals with the computational complexity of the discrete inverse problem of reconstructing finite point sets and more general functionals with finite support that are accessible only through some of the values of their discrete radon transform it turns out that this task behaves quite differently from its well studied companion problem involving 1 dimensional x rays concentrating on the case of coordinate hyperplanes in rd and on functionals zd d with d 0 1 r n0 for some arbitrary but fixed r we show in particular that the problem can be solved in polynomial time if information is available for m such hyperplanes when m d 1 but is np hard for m d and d 0 1 r however for d n0 a case that is relevant in the context of contingency tables the problem is still in p similar results are given for the task of determining the uniqueness of a given solution and for a related counting problem

on generators of the hilbert ideal for cyclic groups in modular invariant theory abstract let g be a cyclic group of order p 2 and v be a faithful indecomposable representation of g over a field f of characteristic p we show that the hilbert ideal of the invariant ring is generated by polynomials of degree at most g whenever dim v 4 p or dim v p 2 2 p proving a conjecture of derksen and kemper in this particular case

cluster structure near the percolation threshold derives exact relations that allow one to describe unambiguously and quantitatively the structure of clusters near the percolation threshold pc in particular the author proves the relations p dpij dp lambda ij where p is the bond density pij is the pair connectedness function and lambda ij is the average number of cutting bonds between i and j from this relation it follows that the average number of cutting bonds between two points separated by a distance of the order of the connectedness length xi diverges as mod p pc mod 1 the remaining multiply connected bonds in the percolating backbone which lump together in blobs diverge with a dimensionality dependent exponent he also shows that in the cell renormalisation group of reynolds et al 1978 1980 the thermal eigenvalue is simply related to the average number of cutting bonds in the spanning cluster he discusses a percolation model in which the blobs can be controlled by varying a parameter and study the influence on the critical exponents

strongly polynomial algorithms for the quadratic transportation problem with a fixed number of sources the transportation problem with a linear objective function is known to be solvable in strongly polynomial time whereas instances with a convex nonquadratic objective function are not even for cases with the integrality constraints relaxed we present a linear time algorithm for the continuous quadratic transportation problem qtp with two source nodes further we show how problem instances with a fixed number of source or destination nodes k could be solved in strongly polynomial time onk 1 using an algebraic tree computation model the algorithms exploit a relationship between the transportation problem and the resource allocation problem the strong polynomiality of the algorithms presented here imply the existence of strongly polynomial algorithms for integer quadratic transportation problems with a fixed number of source nodes as well

invariants of finite groups over fields of characteristicp abstract let v denote a finite dimensional k vector space and let g denote a finite group of k linear automorphisms of v let v m denote the direct sum of m copies of v and let g act on the symmetric algebra k v m of v m by the diagonal action on v m a result of noether implies that if char k 0 then k v m g can be generated as a k algebra by polynomials whose degrees are g no matter how large m is this paper proves that this result no longer holds when the characteristic of k divides g more precisely it is proved in this case that there is a positive number depending only on g and char k such that every set of k algebra generators of k v m g contains a generator whose degree is m

a note on the hilbert ideals of a cyclic group of prime order the hilbert ideal is the ideal generated by positive degree invariant polynomials of a finite group for a cyclic group of prime order p we show that the image of the transfer lie in the ideal generated by invariants of degree at most p 1 consequently we show that the hilbert ideal corresponding to an indecomposable representation is generated by polynomials of degree at most p confirming a conjecture of harm derksen and gregor kemper for this case

on vector invariants over finite fields abstract let f denote a finite field and let s m n f denote a set of generators of the invariants of sl n f acting on mn component vectors this paper proves that if m n 1 then s m n f must contain a generator whose degree is greater than or equal to m n 2 f 1 similar results are obtained for the vector invariants of other groups of matrices with entries in f

vector invariants ofu2 fp a proof of a conjecture of richman abstract in this paper we prove a conjecture of d r richman concerning the vector invariants of the group u 2 f p let v be a vector space of dimension 2 with basis x y over the field f p and let f p x y be the symmetric algebra of v over f p if denotes a generator of u 2 f p then we may assume x x and y x y let a n be the symmetric algebra of v n we obtain an automorphism of a n of order p by using the diagonal action of extended to the whole of a n the subalgebra of polynomials left invariant by this action is called the ring of vector invariants of u 2 f p richman conjectured that these rings of invariants have certain sets of generators and gave a proof in the case p 2 we prove his conjecture for all primes