las vegas algorithms for linear and integer programming when the dimension is small this paper gives an algorithm for solving linear programming problems for a problem with n constraints and d variables the algorithm requires an expected o d2n logn o d d 2 o 1 o d4 n logn arithmetic operations as n the constant factors do not depend on d also an algorithm is given for integer linear programming let 4 bound the number of bits required to specify the rational numbers defining an input constraint or the objective function vector let n and d be as before then the algorithm requires expected o 2ddn 8dd n l n n ln n do d 4 ln n operations on numbers with do 1 4 bits as n where the constant factors do not depend on d or 4 to other convex programming problems for example an algorithm for finding the smallest sphere enclosing a set of n points in ed has the same time bound

application of the arnoldi algorithm to the solution of the advection dispersion equation the solution to the finite element matrix differential equations resulting from the discretization of the contaminant transport equation is normally carried out by a finite difference approximation to the time derivative the total computational effort in simulating a contaminant plume is then directly related to the number of unknowns and the number of time steps required to obtain accurate and stable solutions an alternative method is the arnoldi algorithm which uses orthogonal matrix transformations to reduce the finite element equations to a much smaller upper hessenberg system of first order differential equations this new system can be solved by a standard crank nicolson algorithm with very little computational effort a matrix vector multiplication is then used to obtain the original solution at desired time steps the algorithm is used to simulate accurately the contaminant plumes for a strip source areal aquifer a cross sectional problem and the borden landfill in ontario canada the arnoldi algorithm shows an impressive 613 increase in speed over the conventional crank nicolson scheme for this latter case the method affords an efficient means of solving large problems particularly when time durations are long

a new o n log n algorithm for computing the intersection of convex polygons abstract we present a new o n log n algorithm for computing the intersection of a set of arbitrary possibly unbounded polygons where n is the total number of edges in the polygons an interesting property of this algorithm is that if the intersection is empty then the algorithm finds a minimal set of at most three polygons whose intersection is empty the algorithm is based on a simple technique for detecting a redundant inequality among a set of inequalities of two variables

linear programming in linear time when the dimension is fixed it is demonstrated that the linear programming problem in d variables and n constraints can be solved in o n time when d is fixed this bound follows from a multidimensional search technique which is applicable for quadratic programming as well there is also developed an algorithm that is polynomial in both n and d provided d is bounded by a certain slowly growing function of n

on the diameter of an ideal abstract we begin the study of the notion of diameter of an ideal i s k x 1 x n an invariant measuring the distance between the minimal primes of i we provide large classes of hirsch ideals i e ideals satisfying diam i height i such as quadratic radical ideals such that s i is gorenstein and height i 4 or ideals admitting a square free complete intersection initial ideal

a quasi polynomial bound for the diameter of graphs of polyhedra the diameter of the graph of a d dimensional polyhedron with n facets is at most nlog d 2 let p be a convex polyhedron the graph of p denoted by g p is an abstract graph whose vertices are the extreme points of p and two vertices u and v are adjacent if the interval v u is an extreme edge 1 dimensional face of p the diameter of the graph of p is denoted by p let d n be the maximal diameter of the graphs of d dimensional polyhedra p with n facets a facet is a d 1 dimensional face thus p is the set of solutions of n linear inequalities in d variables it is an old standing problem to determine the behavior of the function d n the value of d n is a lower bound for the number of iterations needed for dantzig s simplex algorithm for linear programming with any pivot rule in 1957 hirsch conjectured 2 that d n n d klee and walkup 6 showed that the hirsch conjecture is false for unbounded polyhedra they proved that for n 2d d n n d d 5 this is the best known lower bound for d n the statement of the hirsch conjecture for bounded polyhedra is still open for a recent survey on the hirsch conjecture and its relatives see 5 in 1967 barnette proved 1 3 that d n n3 an improved upper bound d n n2 was proved in 1970 by larman 7 barnette s and larman s bounds are linear in n but exponential in the dimension d in 1990 the first author 4 proved a subexponential bound d n 2 n d the purpose of this paper is to announce and to give a complete proof of a quasipolynomial upper bound for d n such a bound was proved by the first author in march 1991 the proof presented here is a substantial simplification that was subsequently found by the second author see 4 for the original proof and related results the existence of a polynomial or even linear upper bound for d n is still open recently the first author found a randomized pivot rule for linear programming which requires an expected n d or less arithmetic operations for every linear programming problem with d variables and n constraints 1991 mathematics subject classification primary 52a25 90c05 received by the editors july 1 1991 the first author was supported in part by a bsf grant by a gif grant the second author was supported by an afosr grant c 1992 american mathematical society 0273 0979 92 1 00 25 per page

continuous location of dimensional structures abstract a natural extension of point facility location problems are those problems in which facilities are extensive i e those that cannot be represented by isolated points but as some dimensional structures such as straight lines line segments polygonal curves or circles in this paper a review of the existing work on the location of extensive facilities in continuous spaces is given gaps in the knowledge are identified and suggestions for further research are made

on connectedness a solution based on oscillatory correlation a long standing problem in neural computation has been the problem of connectedness first identified by minsky and papert 1969 this problem served as the cornerstone for them to establish analytically that perceptrons are fundamentally limited in computing geometrical topological properties a solution to this problem is offered by a different class of neural networks oscillator networks to solve the problem the representation of oscillatory correlation is employed whereby one pattern is represented as a synchronized block of oscillators and different patterns are represented by distinct blocks that desynchronize from each other oscillatory correlation emerges from legion locally excitatory globally inhibitory oscillator network whose architecture consists of local excitation and global inhibition among neural oscillators it is further shown that these oscillator networks exhibit sensitivity to topological structure which may lay a neurocomputational foundation for explaining the psychophysical phenomenon of topological perception

fuzzy versions of the covering circle problem abstract in this paper we consider two fuzzy versions of the well known problem of determining the smallest circle center and radius that would cover a given finite set of points in the plane when the locations of points are not precise but fuzzy the first is modeled as a possibility constrained mathematical program while the second is modeled as a necessity constrained one polynomial algorithms are presented for both the versions also a numerical example is included for one case these models and solutions are of interest in both theoretical and practical contexts

a randomized algorithm for fixed dimensional linear programming we give a las vegas randomized algorithm for linear programming in a fixed dimensiond for which the expected computation time is o d 3 varepsilon d d n where limd e d 0 this improves the corresponding worst case complexity o 3 d 2 n the method is based on a recent idea of clarkson two variations on the algorithm are examined briefly

a two resource allocation problem solvable in linear time the allocation problem discussed here is as follows the processing time of a task is a linear function t ax by of amounts x y of resources allocated to it where t a b are positive constants given the amounts x and y of available resources we wish to allocate them among n independent tasks so as to minimize the maximal completion time we solve this problem in linear time

space efficient planar convex hull algorithms a space efficient algorithm is one in which the output is given in the same location as the input and only a small amount of additional memory is used by the algorithm we describe four space efficient algorithms for computing the convex hull of a planar point set

approximations of one dimensional digital signals under the l infty norm approximation of digital signals by means of continuous time functions is often required in many tasks of digital to analog conversion signal processing and coding in many cases the approximation is performed based on an l2 optimality criterion in this paper we study approximations of one dimensional signals under the linfin norm we first introduce approximations in linear spaces for which linear programming methods are known for the particular case of linear approximations i e first order polynomials we propose a geometric solution that is shown to be computationally more efficient than the linear programming approach then we study the problem of piecewise approximations i e dividing the domain into intervals and approximating the signal in linear spaces within every segment independently so as to reach an optimal noncontinuous approximation given an error bound delta we establish a strategy to determine the minimum number k of segments for which the approximation is guaranteed to produce an error within delta we then show how to find the optimal partition that gives the piecewise linfin optimal solution with k segments the computational complexity of the algorithms is studied showing that in many practical situations the number of operations is o n with n being the number of samples

an improved algorithm for the solution of the regularization path of support vector machine this paper describes an improved algorithm for the numerical solution to the support vector machine svm classification problem for all values of the regularization parameter c the algorithm is motivated by the work of hastie and follows the main idea of tracking the optimality conditions of the svm solution for ascending value of c it differs from hastie s approach in that the tracked path is not assumed to be 1 d instead a multidimensional feasible space for the optimality condition is used to solve the tracking problem such a treatment allows the algorithm to properly handle data sets which hastie s approach fails these data sets are characterized by the presence of linearly dependent points in the kernel space duplicate points or nearly duplicate points such data sets are quite common among many real world data especially those with nominal features other contributions of this paper include a unifying formulation of the tracking process in the form of a linear programming problem update formula for the linear programs considerations that guard against accumulation of errors resulting from the use of incremental updates and routines to speed up the algorithm the algorithm is implemented under the matlab environment and is available for download experiments with several data sets including data set having up to several thousand data points are reported

an optimal parallel algorithm for digital curve segmentation first we give an optimal erew pram algorithm that finds an unknown discrete monotone function with domain and range of size n in o log n time using o n independent threshold queries of kind x y here independent means that simultaneous queries always refer to mutually disjoint values x and y this is used for solving within the same resources a certain segmentation problem for words over semigroups the classical problem of partitioning a digital curve into a minimum number of digital line segments which is of interest in digital image processsing turns out to be a special case of this and can therefore be solved in o log n time using o n work on an erew pram this strengthens and generalizes all known algorithmic results about digital curve segmentation as a further prerequisite we use the dorst smeulders parametrization of digital line segments

highly efficient oscillation free solution of the transport equation over long times and large spaces transport problems with sharp transitions or high convection often stress even the most advanced computational methods beyond feasibility limits one promising line of attack involves transforming problems to a moving deforming coordinate system mdcs using mdcs as recently developed one may concentrate computational attention when and where it is most appropriate also possibly greatly reducing the effective magnitude of terms which cause trouble in principle mdcs may be used with finite difference finite element or other numerical methods according to preference in the particular finite element system advanced here mesh movement may be variable in time and space so that physically fixed boundaries may remain fixed in the transformed coordinate system a simple finite difference system is used in time when the method is applied to the standard transport equation test case great efficiency and accuracy are obtained in early time even very steep fronts may be represented well with no oscillations using space and time step sizes well beyond conventional peclet courant number constraints more important problems are solved involving transport over much longer times and larger spaces when diffusion has some chance to operate and hence a parabolic model is more warranted the same accuracy and stability are achieved as during early time using no more nodes and time steps overall than during early time runs this constitutes orders of magnitude savings in effort over what the application of conventional constraints would require relative efficiency from application of mdcs to other problems would depend on the particulars of each case

area universal and constrained rectangular layouts a rectangular layout is a partition of a rectangle into a finite set of interior disjoint rectangles these layouts are used as rectangular cartograms in cartography as floorplans in building architecture and vlsi design and as graph drawings often areas are associated with the rectangles of a rectangular layout and it is desirable for one rectangular layout to represent several area assignments a layout is area universal if any assignment of areas to rectangles can be realized by a combinatorially equivalent rectangular layout we identify a simple necessary and sufficient condition for a rectangular layout to be area universal a rectangular layout is area universal if and only if it is one sided we also investigate similar questions for perimeter assignments the adjacency requirements for the rectangles of a rectangular layout can be specified in various ways most commonly via the dual graph of the layout we show how to find an area universal layout for a given set of adjacency requirements whe

on the average number of steps of the simplex method of linear programming the goal is to give some theoretical explanation for the efficiency of the simplex method of george dantzig fixing the number of constraints and using dantzig s self dual parametric algorithm we show that the number of pivots required to solve a linear programming problem grows in proportion to the number of variables on the average

locating least distant lines in the plane in this paper we deal with locating a line in a plane given a set of existing facilities represented by points in the plane our objective is to find a straight line l minimizing the sum of weighted distances to the existing facilities or minimizing the maximum weighted distance to the existing facilities respectively we show that for all distance measures derived from norms one of the lines minimizing the sum objective contains at least two of the existing facilities for the center objective we always get an optimal line which is at maximum distance from at least three of the existing facilities if all weights are equal there is an optimal line which is parallel to one facet of the convex hull of the existing facilities

simple extensions of polytopes we introduce the simple extension complexity of a polytope p p as the smallest number of facets of any simple i e non degenerate in the sense of linear programming polytope which can be projected onto p p we devise a combinatorial method to establish lower bounds on the simple extension complexity and show for several polytopes that they have large simple extension complexities these examples include both the spanning tree and the perfect matching polytopes of complete graphs uncapacitated flow polytopes for non trivially decomposable directed acyclic graphs hypersimplices and random 0 1 polytopes with vertex numbers within a certain range on our way to obtain the result on perfect matching polytopes we generalize a result of padberg and rao s on the adjacency structures of those polytopes in addition to the material in the extended abstract kaibel and walter in integer programming and combinatorial optimization lecture notes in computer science vol 8494 springer berlin 2014 we include omitted proofs supporting figures and an analysis of known upper bounding techniques

motion segmentation based on motion brightness integration and oscillatory correlation a segmentation method based on the integration of motion and brightness is proposed for image sequences the method is composed of two parallel pathways that process motion and brightness respectively inspired by the visual system the motion pathway has two stages the first stage estimates local motion at locations with reliable information the second stage performs segmentation based on local motion estimates in the brightness pathway the input scene is segmented into regions based on brightness distribution subsequently segmentation results from the two pathways are integrated to refine motion estimates the final segmentation is performed in the motion network based on refined estimates for segmentation locally excitatory globally inhibitory oscillator network legion architecture is employed whereby the oscillators corresponding to a region of similar motion brightness oscillate in synchrony and different regions attain different phases results on synthetic and real image sequences are provided and comparisons with other methods are made

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

parallel linear programming in fixed dimension almost surely in constant time for any fixed dimension d thelinear programming problem with n inequality constraints can be solved on a probabilistic crcw pram with o n processors almost surely in constant time the algorithm always findsthe correct solution with nd log 2 d processors the probability that the algorithm will not finish within o d 2 log 2 d time tends to zero exponentially with n authors abstract

bounds for the computational power and learning complexity of analog neural nets it is shown that high order feedforward neural nets of constant depth with piecewise polynomial activation functions and arbitrary real weights can be simulated for boolean inputs and outputs by neural nets of a somewhat larger size and depth with heaviside gates and weights from 1 0 1 this provides the first known upper bound for the computational power of the former type of neural nets it is also shown that in the case of first order nets with piecewise linear activation functions one can replace arbitrary real weights by rational numbers with polynomially many bits without changing the boolean function that is computed by the neural net in order to prove these results we introduce two new methods for reducing nonlinear problems about weights in multilayer neural nets to linear problems for a transformed set of parameters these transformed parameters can be interpreted as weights in a somewhat larger neural net as another application of our new proof technique we show that neural nets with piecewise polynomial activation functions and a constant number of analog inputs are probably approximately correct pac learnable in valiant s model for pac learning comm assoc comput mach 27 1984 pp 1134 1142

transportation problems and simplicial polytopes that are not weakly vertex decomposable provan and billera defined the notion of weak k decomposability for pure simplicial complexes in the hopes of bounding the diameter of convex polytopes they showed the diameter of a weakly k decomposable simplicial complex is bounded above by a polynomial function of the number of k faces in and its dimension for weakly 0 decomposable complexes this bound is linear in the number of vertices and the dimension in this paper we exhibit the first examples of non weakly 0 decomposable simplicial polytopes our examples are in fact polar to certain transportation polytopes

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

linear time algorithms for linear programming in r 3 and related problems linear time algorithms for linear programming in r 2 and r 3 are presented the methods used are applicable for other graphic and geometric problems as well as quadratic programming for exam

the multi service center problem abstract we propose a new type of network location problem for placing multiple but distinct facilities called the p service center problem in this problem we are to locate p facilities in the graph each of which provides distinct service required by all vertices for each vertex its p service distance is the summation of its weighted distances to the p facilities the objective is to minimize the maximum value among the p service distances of all vertices in this paper we show that the p service center problem on a general graph is np hard and propose a simple approximation algorithm of factor p c for any integer constant c moreover we study the basic case p 2 on paths and trees when the underlying network is a path we solve the 2 service center problem in o n time where n is the number of vertices when the underlying network is a tree we give an o n 3 time algorithm for the case of nonnegative weights an o n log n time algorithm for the case of positive weights and an o n time algorithm for the case of uniform weights

learnability and the vapnik chervonenkis dimension valiant s learnability model is extended to learning classes of concepts defined by regions in euclidean space e n the methods in this paper lead to a unified treatment of some of valiant s results along with previous results on distribution free convergence of certain pattern recognition algorithms it is shown that the essential condition for distribution free learnability is finiteness of the vapnik chervonenkis dimension a simple combinatorial parameter of the class of concepts to be learned using this parameter the complexity and closure properties of learnable classes are analyzed and the necessary and sufficient conditions are provided for feasible learnability

many projectively unique polytopes we construct an infinite family of 4 polytopes whose realization spaces have dimension smaller or equal to 96 this in particular settles a problem going back to legendre and steinitz whether and how the dimension of the realization space of a polytope is determined bounded by its f vector from this we derive an infinite family of combinatorially distinct 69 dimensional polytopes whose realization is unique up to projective transformation this answers a problem posed by perles and shephard in the sixties moreover our methods naturally lead to several interesting classes of projectively unique polytopes among them projectively unique polytopes inscribed to the sphere the proofs rely on a novel construction technique for polytopes based on solving cauchy problems for discrete conjugate nets in s d a new alexandrov van heijenoort theorem for manifolds with boundary and a generalization of lawrence s extension technique for point configurations

simulation of contaminant transport in three dimensions 1 the alternating direction galerkin technique the alternating direction galerkin technique for the simulation of advective dispersive transport in three dimensions is developed the technique is designed for high efficiency in handling the large and detailed grids that are often required in simulations of natural groundwater systems it is comparable to an alternating direction finite difference scheme except for two advantages first the capability to handle certain curvilinear grids that may conform to either a flow net geometry or a natural stratigraphy and second the option to enhance solution accuracy through a choice between a finite difference or a finite element representation of the time derivative criteria for controlling numerical dispersion and accuracy can be applied easily three options for the time stepping algorithm are developed and analyzed for stability and their accuracy is investigated for simple flow systems the technique is shown to be far more efficient than a conventional three dimensional finite element model in a companion paper an application to a field scale contaminant transport problem is described

parallel computation in spiking neural nets abstract numerical quantities can be represented as phase differences between equiperiodic oscillating subsystems in a spiking neural net it is then possible to represent integer variables and the increment and decrement operations x x 1 x x 1 it is possible to represent the if construction the while construction and some other programming language constructions including variants of the seq par and alt constructors which were used in occam we give a general purpose parallel programming language with integer variables which can be systematically implemented in spiking neural networks addition subtraction and multiplication are done albeit inefficiently as examples

hirsch polytopes with exponentially long combinatorial segments in their paper proving the hirsch bound for flag normal simplicial complexes adiprasito and benedetti math oper res 39 4 1340 1348 2014 define the notion of combinatorial segment the study of the maximal length of these objects provides the upper bound o n2 d for the diameter of any normal pure simplicial complex of dimension d with n vertices and the hirsch bound n d 1 if the complexes are moreover flag in the present article we propose a formulation of combinatorial segments which is equivalent but more local by introducing the notions of monotonicity and conservativeness of dual paths in pure simplicial complexes we use these definitions to investigate further properties of combinatorial segments besides recovering the two stated bounds we show a refined bound for banner complexes and study the behavior of the maximal length of combinatorial segments with respect to two usual operations namely join and one point suspension finally we show the limitations of combinatorial segments by constructing pure normal simplicial complexes in which all combinatorial segments between two particular facets achieve the length omega n2 d this includes vertex decomposable therefore hirsch polytopes

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

polyhedral graph abstractions and an approach to the linear hirsch conjecture we introduce a new combinatorial abstraction for the graphs of polyhedra the new abstraction is a flexible framework defined by combinatorial properties with each collection of properties taken providing a variant for studying the diameters of polyhedral graphs one particular variant has a diameter which satisfies the best known upper bound on the diameters of polyhedra another variant has superlinear asymptotic diameter and together with some combinatorial operations gives a concrete approach for disproving the linear hirsch conjecture

finding transversals for sets of simple geometric figures abstract a straight line that intersects all members of a set s of objects in the real plane is called a transversal of s geometric transforms are described that reduce transversal problems for various types of objects to convex hull problems for points these reductions lead to efficient algorithms for finding transversals which are also described applications of the algorithms are found in computer graphics reproduce the line displayed by a collection of pixels and in statistics find the line that minimizes the maximum distance from a collection of weighted points in the plane

measure of circularity for parts of digital boundaries and its fast computation this paper focuses on the design of an effective method that computes the measure of circularity of a part of a digital boundary an existing circularity measure of a set of discrete points which is used in computational metrology is extended to the case of parts of digital boundaries from a single digital boundary two sets of points are extracted so that the circularity measure computed from these sets is representative of the circularity of the digital boundary therefore the computation consists of two steps first the inner and outer sets of points are extracted from the input part of a digital boundary using digital geometry tools next the circularity measure of these sets is computed using classical tools of computational geometry it is proved that the algorithm is linear in time in the case of convex parts thanks to the specificity of digital data and is in o nlogn otherwise experiments done on synthetic and real images illustrate the interest of the properties of the circularity measure

fuzzy disk for covering fuzzy points abstract in this paper we consider an important fuzzy version of the well known smallest covering circle problem which is also called the euclidean 1 center problem here we are given a set of points in the plane whose locations are crisp however the requirement for coverage of each point is fuzzy as represented by its grade of membership as such we qualify the points as fuzzy in the above framework we wish to find a fuzzy disk with minimum fuzzy area for covering the given fuzzy points after developing a general model certain special cases are investigated in detail polynomial algorithms are presented for the special cases

an alternating direction galerkin technique for simulation of contaminant transport in complex groundwater systems a general contaminant transport simulation technique based on a galerkin finite element representation in space and an alternating direction time stepping scheme is developed the formulation which is in terms of natural coordinates following the principal directions of hydraulic conductivity of the medium decouples the spatial components of the equations leading to tridiagonal matrices the numerical accuracy can be easily controlled in each principal direction through the well established peclet and courant numbers results are found to be comparable to those from the classic finite element scheme but the cost of computation is generally much less because the computational effort varies only linearly with the grid size the technique is thus well suited for transport simulations with complex layered anisotropic groundwater systems requiring large and detailed grids

reconstructing polygons from scanner data a range finding scanner can collect information about the shape of an unknown polygonal room in which it is placed suppose that a set of scanners returns not only a set of points but also additional information such as the normal to the plane when a scan beam detects a wall we consider the problem of reconstructing the floor plan of a room from different types of scan data in particular we present algorithmic and hardness results for reconstructing two dimensional polygons from point wall pairs point normal pairs and visibility polygons the polygons may have restrictions on topology e g to be simply connected or geometry e g to be orthogonal we show that this reconstruction problem is np hard under most models but that some restrictive assumptions do allow polynomial time reconstruction algorithms

quadratic diameter bounds for dual network flow polyhedra both the combinatorial and the circuit diameter of polyhedra are of interest to the theory of linear programming for their intimate connection to a best case performance of linear programming algorithms we study the diameters of dual network flow polyhedra associated to b flows on directed graphs g v e g v e and prove quadratic upper bounds for both of them the minimum of v 1 cdot e v 1 e and frac 1 6 v 3 16 v 3 for the combinatorial diameter and frac v cdot v 1 2 v v 1 2 for the circuit diameter previously bounds on these diameters have only been known for bipartite graphs the situation is much more involved for general graphs in particular we construct a family of dual network flow polyhedra with members that violate the circuit diameter bound for bipartite graphs by an arbitrary additive constant

the efficiency of the simplex method a survey the linear programming problem is by far the most widely used optimization model its impact on economic and government modeling is immense the simplex method for solving the linear programming lp problem due to george dantzig has been an extremely efficient computational tool for almost four decades the method has been the subject of intense investigations for many years but some major aspects of its behavior are not fully understood yet the purpose of this paper is to survey the body of knowledge on the efficiency of the simplex method from both practical and theoretical points of view adopting the number of iterations pivot steps as the yardstick for efficiency we survey four aspects of the issue 1 reports on practical experience of the method s performance on real life lp problems 2 results on controlled monte carlo experiments solving lp problems which were randomly generated according to some predetermined distributions 3 complexity results including theoretical analyses on both upper and lower bounds for the performance of the simplex as well as non simplex algorithms for lp 4 results of recent theoretical studies using probabilistic analysis to derive bounds on the average behavior of the simplex method we discuss the consequences and limitations of the various studies special emphasis is given to open questions

dual consistent systems of linear inequalities and cardinality constrained polytopes we introduce a concept of dual consistency of systems of linear inequalities with full generality we show that a cardinality constrained polytope is represented by a certain system of linear inequalities if and only if the systems of linear inequalities associated with the cardinalities are dual consistent typical dual consistent systems of inequalities are those which describe polymatroids generalized polymatroids and dual greedy polyhedra with certain choice functions we show that the systems of inequalities for cardinality constrained ordinary bipartite matching polytopes are not dual consistent in general and give additional inequalities to make them dual consistent moreover we show that ordinary systems of inequalities for the cardinality constrained poly matroid intersection are not dual consistent which disproves a conjecture of maurras spiegelberg and stephan about a linear representation of the cardinality constrained polymatroid intersection

is binary encoding appropriate for the problem language relationship abstract it is proved that there exist encoding schemes which are arbitrarily as efficient as the binary encoding in terms of compactness and arithmetic operations with respect to which khachiyan s algorithm for linear programming is exponential this constitutes an objection to the standard translation of problems into languages via the binary encoding

the hirsch conjecture for the fractional stable set polytope the edge formulation of the stable set problem is defined by two variable constraints one for each edge of a graph g g expressing the simple condition that two adjacent nodes cannot belong to a stable set we study the fractional stable set polytope i e the polytope defined by the linear relaxation of the edge formulation even if this polytope is a weak approximation of the stable set polytope its simple geometrical structure provides deep theoretical insight as well as interesting algorithmic opportunities exploiting a graphic characterization of the bases we first redefine pivots in terms of simple graphic operations that turn a given basis into an adjacent one these results lead us to prove that the combinatorial diameter of the fractional stable set polytope is at most the number of nodes of the given graph as a corollary the hirsch bound holds for this class of polytopes

the inverse fermat weber problem abstract given n points in the plane with nonnegative weights the inverse fermat weber problem consists in changing the weights at minimum cost such that a prespecified point in the plane becomes the euclidean 1 median the cost is proportional to the increase or decrease of the corresponding weight in case that the prespecified point does not coincide with one of the given n points the inverse fermat weber problem can be formulated as linear program we derive a purely combinatorial algorithm which solves the inverse fermat weber problem with unit cost using o n greedy like iterations where each of them can be done in constant time if the points are sorted according to their slopes if the prespecified point coincides with one of the given n points it is shown that the corresponding inverse problem can be written as convex problem and hence is solvable in polynomial time to any fixed precision

a geometric transversal approach to analyzing track coverage in sensor networks this paper presents a new coverage formulation addressing the quality of service of sensor networks that cooperatively detect targets traversing a region of interest the problem of track coverage consists of finding the positions of n sensors such that a lebesgue measure on the set of tracks detected by at least k sensors is optimized this paper studies the geometric properties of the network addressing a deterministic track coverage formulation and binary sensor models it is shown that the tracks detected by a network of heterogeneous omnidirectional sensors are the geometric transversals of non translates families of circles a novel methodology based on cone theory is presented for representing and measuring sets of transversals in closed form then the solution of the track coverage problem can be formulated as a nonlinear program nlp the numerical results show that this approach can improve track coverage by up to two orders of magnitude compared to grid and random deployments also it can be used to reduce the number of sensors required to achieve a desired detection performance by up to 50 and to optimally replenish or reposition existing sensor networks

fixed rank rayleigh quotient maximization by an mpsk sequence certain optimization problems in communication systems such as limited feedback constant envelope beamforming or noncoherent m ary phase shift keying mpsk sequence detection result in the maximization of a fixed rank positive semidefinite quadratic form over the mpsk alphabet this form is a special case of the rayleigh quotient of a matrix and in general its maximization by an mpsk sequence is mathcal np hard however if the rank of the matrix is not a function of its size then the optimal solution can be computed with polynomial complexity in the matrix size in this work we develop a new technique to efficiently solve this problem by utilizing auxiliary continuous valued angles and partitioning the resulting continuous space of solutions into a polynomial size set of regions each of which corresponds to a distinct mpsk sequence the sequence that maximizes the rayleigh quotient is shown to belong to this polynomial size set of sequences thus efficiently reducing the size of the feasible set from exponential to polynomial based on this analysis we also develop an algorithm that constructs this set in polynomial time and show that it is fully parallelizable memory efficient and rank scalable the proposed algorithm compares favorably with other solvers for this problem that have appeared recently in the literature

estimation of dispersion and first order rate coeft by numerical routing the engineer is often required to estimate dispersion and first order reaction coefficients from data obtained in natural streams one way to accomplish this is by use of routing methods in which the coefficients are determined such that the convective dispersion transport equation most nearly describes the observed concentrations in the past analytic solutions to the transport equation have generally been used with the routing method especially when determining dispersion coefficients because numerical solutions often contained too much numerical dispersion for the results to be realistic it is shown that a lagrangian model can be used to determine realistic estimates of dispersion and reaction coefficients by use of the routing method the numerical routing procedure is tested using data obtained analytically under steady flow in a large and small river in a river with unsteady flow and in a river with steady but highly nonuniform flow conditions the numerical routing procedure based on a lagrangian solution scheme performed well in all cases while offering the flexibility that only a numerical solution scheme can offer

the diameters of network flow polytopes satisfy the hirsch conjecture we solve a problem in the combinatorics of polyhedra motivated by the network simplex method we show that the hirsch conjecture holds for the diameter of the graphs of all network flow polytopes in particular the diameter of a network flow polytope for a network with n nodes and m arcs is never more than m n 1 a key step to prove this is to show the same result for classical transportation polytopes

linear programming approaches to the convex hull problem in rm in this paper two linear programming formulations of the convex hull problem are presented each formulation is the dual of the other linear programming problems that identify a face of the convex hull are also discussed an efficient algorithm is developed computational results obtained on an ibm 3090 are presented

improved asymptotic analysis of the average number of steps performed by the self dual simplex algorithm in this paper we analyze the average number of steps performed by the self dual simplex algorithm for linear programming under the probabilistic model of spherical symmetry the model was proposed by smale consider a problem ofn variables withm constraints smale established that for every number of constraintsm there is a constantc m such that the number of pivot steps of the self dual algorithm m n is less thanc m lnn m m 1 we improve upon this estimate by showing that m n is bounded by a function ofm only the symmetry of the function inm andn implies that m n is in fact bounded by a function of the smaller ofm andn

random linear programs with many variables and few constraints we extend and simplify smale s work on the expected number of pivots for a linear program with many variables and few constraints our analysis applies to new versions of the simplex algorithm and to new random distributions

the d step conjecture and its relatives the d step conjecture arose from an attempt to understand the computational complexity of edge following algorithms for linear programming such as the simplex algorithm it can be stated in terms of diameters of graphs of convex polytopes in terms of the existence of nonrevisiting paths in such graphs in terms of an exchange process for simplicial bases of a vector space and in terms of matrix pivot operations first formulated by w m hirsch in 1957 the conjecture remains unsettled though it has been proved in many special cases and counterexamples have been found for slightly stronger conjectures if the conjecture is false as we believe to be the case then finding a counterexample will be merely a small first step in the line of investigation related to the conjecture this report summarizes what is known about the d step conjecture and its relatives a considerable amount of new material is included but it does not seem to come close to settling the conjecture of special interest is the first example of a polytope that is not vertex decomposable showing that a certain natural approach to the conjecture will not work also significant are the quantitative relations among the lengths of paths associated with various forms of the conjecture

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

minimizing weighted earliness tardiness and due date cost with unit processing time jobs abstract the classical single machine scheduling and due date assignment problem of panwalker et al panwalker s s smith m l seidmann a 1982 common due date assignment to minimize total penalty for the one machine scheduling problem operations research 30 2 1982 391 399 is the following all n jobs share a common due date which is to be determined jobs completed prior to or after the due date are penalized according to a cost function which is linear and job independent the objective is to minimize the total earliness tardiness and due date cost we study a generalized version of this problem in which i the earliness and tardiness costs are allowed to be job dependent and asymmetric and ii jobs are processed on parallel identical machines we focus on the case of unit processing time jobs the problem is shown to be solved in polynomial o n4 time then we study the special case with no due date cost a classical problem known in the literature as twet we introduce an o n3 solution for this case finally we study the minmax version of the problem i e the objective is to minimize the largest cost incurred by any of the jobs which is shown to be solved in polynomial time as well

polynomial fixed parameter algorithms a case study for longest path on interval graphs we study the design of fixed parameter algorithms for problems already known to be solvable in polynomial time the main motivation is to get more efficient algorithms for problems with unattractive polynomial running times here we focus on a fundamental graph problem longest path that is given an undirected graph find a maximum length path in g longest path is np hard in general but known to be solvable in o n4 time on n vertex interval graphs we show how to solve longest path on interval graphs parameterized by vertex deletion number k to proper interval graphs in o k9n time notably longest path is trivially solvable in linear time on proper interval graphs and the parameter value k can be approximated up to a factor of 4 in linear time from a more general perspective we believe that using parameterized complexity analysis may enable a refined understanding of efficiency aspects for polynomial time solvable problems similarly to what classical parameterized complexity analysis does for np hard problems

new lower bounds for convex hull problems in odd dimensions we show that in the worst case omega n ceil d 2 1 n log n sidedness queries are required to determine whether the convex hull of n points in real d is simplicial or to determine the number of convex hull facets this lower bound matches known upper bounds in any odd dimension our result follows from a straightforward adversary argument a key step in the proof is the construction of a quasi simplicial n vertex polytope with omega n ceil d 2 1 degenerate facets while it has been known for several years that d dimensional convex hulls can have omega n floor d 2 facets the previously best lower bound for these problems is only omega n log n using similar techniques we also obtain simple and correct proofs of erickson and seidel s lower bounds for detecting affine degeneracies in arbitrary dimensions and circular degeneracies in the plane as a related result we show that detecting simplicial convex hulls in real d is ceil d 2 sum hard in the sense of gajentaan and overmars

a counterexample to the hirsch conjecture the hirsch conjecture 1957 stated that the graph of a d dimensional polytope with n facets cannot have combinatorial diameter greater than n d that is any two vertices of the polytope can be connected by a path of at most n d edges this paper presents the rst counterexample to the conjecture our polytope has dimension 43 and 86 facets it is obtained from a 5 dimensional polytope with 48 facets that violates a certain generalization of the d step conjecture of klee and walkup

a lagrangian porous media mass transport model the mathematical simulation of advective dispersive contaminant transport in groundwater involving large peclet numbers is subject to numerical difficulties for most numerical models the computational cost and computer core requirements escalate as the peclet number increases a two dimensional galerkin finite element model for flow and lagrangian mass transport in porous media has been developed to alleviate numerical and computational difficulties the solution methodology involves a linear triangular mesh which tracks along streamlines calculated from a flow equation having the stream function as the dependent variable a comparison of this model to eulerian or fixed coordinate type models showed this model to be accurate and stable at a reduced computational effort for an actual field contamination problem which contains large vertical concentration gradients the simulated results compared with observed data

the hirsch conjecture holds for normal flag complexes using an intuition from metric geometry we prove that any flag normal simplicial complex satisfies the nonrevisiting path conjecture as a consequence the diameter of its facet ridge graph is smaller than the number of vertices minus the dimension as in the hirsch conjecture this proves the hirsch conjecture for all flag polytopes and more generally for all connected flag homology manifolds

strongly polynomial time and nc algorithms for detecting cycles in periodic graphs this paper is concerned with the problem of recognizing in a graph with rational vector weights associates with the edges the existence of a cycle whose total weight is the zero vector this problem is known to be equivalent to the problem of recognizing the existence of cycles in periodic dynamic graphs and to the validity of systems of recursive formulas it was previously conjectured that combinatorial algorithms exist for the cases of two and three dimensional vector weights it is shown that strongly polynomial algorithms exist for any fixed dimension d moreover these algorithms also establish membership in the class nc on the other hand it is shown that when the dimension of the weights is not fixed the problem is equivalent to the general linear programming problem under strongly polynomial and logspace reductions

minmax regret 1 facility location on uncertain path networks abstract let p be an undirected path graph of n vertices each edge of p has a positive length and a constant capacity every vertex has a nonnegative supply which is an unknown value but is known to be in a given interval the goal is to find a point on p to build a facility and move all vertex supplies to the facility such that the maximum regret is minimized the previous best algorithm solves the problem in o n log 2 n time and o n log n space in this paper we present an o n log n time and o n space algorithm and our approach is based on new observations and algorithmic techniques

the many facets of linear programming we examine the history of linear programming from computational geometric and complexity points of view looking at simplex ellipsoid interior point and other methods

an o n algorithm for the multiple choice knapsack linear program an algorithm for solving the linear program associated with the multiple choice knapsack problem is described the algorithm is shown to work in time linear in the number of variables this improves the previously known best bound for this problem and is optimal to within a constant factor