efficient algorithms for the inverse spanning tree problem the inverse spanning tree problem is to modify edge weights in a graph so that a given treet is a minimum spanning tree the objective is to minimize the cost of the deviation the cost of deviation is typically a convex function we propose algorithms here that are faster than all known algorithms for the problem our algorithm s run time for any convex inverse spanning tree problem iso nm log 2n for a graph onn nodes andm edges plus the time required to find the minima of then convex deviation functions this not only improves on the complexity of previously known algorithms for the general problem but also for algorithms devised for special cases for the case when the objective is weighted absolute deviation sokkalingam et al 1999 devised an algorithm with run timeo n2m log nc forc the maximum edge weight for the sum of absolute deviations our algorithm runs in timeo n2 logn matching the run time of a recent ahuja and orlin 2000 improvement for this case a new algorithm for bipartite matching in path graphs is reported here with complexity ofo n1 5 logn this leads to a second algorithm for the sum of absolute deviations running ino n1 5 logn logc steps

linear programming for the 0 1 quadratic knapsack problem abstract in this paper we consider the quadratic knapsack problem which consists in maximizing a positive quadratic pseudo boolean function subject to a linear capacity constraint we propose a new method for computing an upper bound this method is based on the solution of a continuous linear program constructed by adding to a classical linearization of the problem some constraints rebundant in 0 1 variables but nonredundant in continuous variables the obtained upper bound is better than the bounds given by other known methods we also propose an algorithm for computing a good feasible solution this algorithm is an elaboration of the heuristic methods proposed by chaillou hansen and mahieu and by gallo hammer and simeone the relative error between this feasible solution and the optimum solution is generally less than 1 we show how these upper and lower bounds can be efficiently used to determine the values of some variables at the optimum finally we propose a branch and bound algorithm for solving the quadratic knapsack problem and report extensive computational tests

analytic torsion of a bounded generalized cone we compute the analytic torsion of a bounded generalized cone by generalizing the computational methods of m spreafico and using the symmetry in the de rham complex as established by m lesch we evaluate our result in lower dimensions and further provide a separate computation of analytic torsion of a bounded generalized cone over s 1 since the standard cone over the sphere is simply a flat disc

object recognition by sequential figure ground ranking we present an approach to visual object class segmentation and recognition based on a pipeline that combines multiple figure ground hypotheses with large object spatial support generated by bottom up computational processes that do not exploit knowledge of specific categories and sequential categorization based on continuous estimates of the spatial overlap between the image segment hypotheses and each putative class we differ from existing approaches not only in our seemingly unreasonable assumption that good object level segments can be obtained in a feed forward fashion but also in formulating recognition as a regression problem instead of focusing on a one vs all winning margin that may not preserve the ordering of segment qualities inside the non maximum non winning set our learning method produces a globally consistent ranking with close ties to segment quality hence to the extent entire object or part hypotheses are likely to spatially overlap the ground truth we demonstrate results beyond the current state of the art for image classification object detection and semantic segmentation in a number of challenging datasets including caltech 101 ethz shape as well as pascal voc 2009 and 2010

cheeger muller theorem on manifolds with cusps we prove equality between the renormalized ray singer analytic torsion and the intersection r torsion on a witt manifold with cusps up to an error term determined explicitly by the betti numbers of the cross section of the cusp and the intersection r torsion of a model cone in the first step of the proof we compute explicitly the renormalized ray singer analytic torsion of a model cusp in general dimension and without the witt condition in the second step we establish a gluing formula for renormalized ray singer analytic torsion on a general class of non compact manifolds in any dimension that includes witt manifolds with cusps but also scattering manifolds with asymptotically conical ends in the final step a cheeger muller theorem on cusps follows by a combination of the previous explicit computation and the gluing formula

intersection r torsion and analytic torsion for pseudomanifolds pour une pseudovariete x n on definit un invariant i p la r torsion intersection parametrisee par la perversite p cet invariant est analogue a la torsion de reidemeister et est un invariant combinatoire

robust output feedback model predictive control of constrained linear systems this paper provides a solution to the problem of robust output feedback model predictive control of constrained linear discrete time systems in the presence of bounded state and output disturbances the proposed output feedback controller consists of a simple stable luenberger state estimator and a recently developed robustly stabilizing tube based model predictive controller the state estimation error is bounded by an invariant set the tube based controller ensures that all possible realizations of the state trajectory lie in a simple uncertainty tube the center of which is the solution of a nominal disturbance free system and the cross section of which is also invariant satisfaction of the state and input constraints for the original system is guaranteed by employing tighter constraint sets for the nominal system the complexity of the resultant controller is similar to that required for nominal model predictive control

submodular function minimization submodular functions often arise in various fields of operations research including discrete optimization game theory queueing theory and information theory in this survey paper we give overview on the fundamental properties of submodular functions and recent algorithmic devolopments of their minimization

beyond the flow decomposition barrier we introduce a new approach to the maximum flow problem this approach is based on assigning arc lengths based on the residual flow value and the residual arc capacities our approach leads to an o min n 2 3 m 1 2 m log n 2 m log u time bound for a network with n vertices m arcs and integral arc capacities in the range 1 u this is a fundamental improvement over the previous time bounds we also improve bounds for the gomory hu tree problem the parametric flow problem and the approximate s t cut problem

cpmc automatic object segmentation using constrained parametric min cuts we present a novel framework to generate and rank plausible hypotheses for the spatial extent of objects in images using bottom up computational processes and mid level selection cues the object hypotheses are represented as figure ground segmentations and are extracted automatically without prior knowledge of the properties of individual object classes by solving a sequence of constrained parametric min cut problems cpmc on a regular image grid in a subsequent step we learn to rank the corresponding segments by training a continuous model to predict how likely they are to exhibit real world regularities expressed as putative overlap with ground truth based on their mid level region properties then diversify the estimated overlap score using maximum marginal relevance measures we show that this algorithm significantly outperforms the state of the art for low level segmentation in the voc 2009 and 2010 data sets in our companion papers 1 2 we show that the algorithm can be used successfully in a segmentation based visual object category recognition pipeline this architecture ranked first in the voc2009 and voc2010 image segmentation and labeling challenges

an index formula for the intersection euler characteristic of an infinite cone the aim of this article is to establish an index formula for the intersection euler characteristic of a cone the main actor is the model witten laplacian on the infinite cone first we study its spectral properties and establish a mckean singer type formula we also give an explicit formula for the zeta function of the model witten laplacian in a second step we apply local index techniques to the model witten laplacian by combining these two steps we express the absolute and relative intersection euler characteristic of the cone as a sum of two terms a term which is local and a second term which is the cheeger invariant

faster algorithms for the quickest transshipment problem a transshipment problem with demands that exceed network capacity can be solved by sending flow in several waves how can this be done in the minimum number of waves this is the question tackled in the quickest transshipment problem hoppe and tardos math oper res 25 2000 pp 36 62 describe the only known polynomial time algorithm to solve this problem they actually solve the significantly harder problem in which it takes a prespecified amount of time for flow to travel from one end of an arc to the other their algorithm repeatedly calls an oracle for submodular function minimization we present an algorithm that finds a quickest transshipment with a polynomial number of maximum flow computations and a faster algorithm that also uses minimum cost flow computations when there is only one sink we show how the algorithm can be sped up to return a solution using o k maximum flow computations where k is the number of sources hajek and ogier networks 14 1984 pp 457 487 describe an algorithm that finds a fractional solution to the single sink quickest transshipment problem on a network with n nodes and m arcs using o n maximum flow computations they actually solve the universally quickest transshipment a flow over time that minimizes the amount of supply left in the network at every moment of time in this paper we show how to solve the universally quickest transshipment in o mnlog n2 m time the same asymptotic time as a push relabel maximum flow computation

the pseudoflow algorithm a new algorithm for the maximum flow problem we introduce the pseudoflow algorithm for the maximum flow problem that employs only pseudoflows and does not generate flows explicitly the algorithm solves directly a problem equivalent to the minimum cut problem the maximum blocking cut problem once the maximum blocking cut solution is available the additional complexity required to find the respective maximum flow is o m log n a variant of the algorithm is a new parametric maximum flow algorithm generating all breakpoints in the same complexity required to solve the constant capacities maximum flow problem the pseudoflow algorithm has also a simplex variant pseudoflow simplex that can be implemented to solve the maximum flow problem one feature of the pseudoflow algorithm is that it can initialize with any pseudoflow this feature allows it to reach an optimal solution quickly when the initial pseudoflow is close to an optimal solution the complexities of the pseudoflow algorithm the pseudoflow simplex and the parametric variants of pseudoflow and pseudoflow simplex algorithms are all o mn log n on a graph with n nodes and m arcs therefore the pseudoflow simplex algorithm is the fastest simplex algorithm known for the parametric maximum flow problem the pseudoflow algorithm is also shown to solve the maximum flow problem on s t tree networks in linear time where s t tree networks are formed by joining a forest of capacitated arcs with nodes s and t adjacent to any subset of the nodes

structural and algorithmic properties for parametric minimum cuts we consider the minimum s t cut problem in a network with parametrized arc capacities following the seminal work of gallo et al siam j comput 18 1 30 55 1989 classes of this parametric problem have been shown to enjoy the nice structural property that minimum cuts are nested and the nice algorithmic property that all minimum cuts can be computed in the same asymptotic time as a single minimum cut by using a clever flow update step to move from one value of the parameter to the next we present a general framework for parametric minimum cuts that extends and unifies such results we define two conditions on parametrized arc capacities that are necessary and sufficient for strictly decreasing differences of the parametric cut function known results in parametric submodular optimization then imply the structural property we show how to construct appropriate flow updates in linear time under the above conditions implying that the algorithmic property also holds under these conditions we then consider other classes of parametric minimum cut problems without decreasing differences for which we establish the structural and or the algorithmic property as well as other cases where nested minimum cuts arise

lagrangean methods for the 0 1 quadratic knapsack problem abstract it is well known that the lagrangean decomposition provides better bounds than the lagrangean relaxation does nevertheless the lagrangean decomposition bound is harder to compute than the lagrangean relaxation bound thus one might wonder what is the best lagrangean method to use in a branch and bound algorithm in this paper we give an answer to such a question for the 0 1 quadratic knapsack problem we first study the lagrangean decomposition for this problem and give new necessary optimality conditions for the dual problem which allow us to elaborate a heuristic method for solving the lagrangean decomposition dual problem we then introduce this method in chaillou hansen mahieu s branch and bound algorithm where upper bounds were computed by lagrangean relaxation

dynamic graph cuts for efficient inference in markov random fields in this paper we present a fast new fully dynamic algorithm for the st mincut max flow problem we show how this algorithm can be used to efficiently compute map solutions for certain dynamically changing mrf models in computer vision such as image segmentation specifically given the solution of the max flow problem on a graph the dynamic algorithm efficiently computes the maximum flow in a modified version of the graph the time taken by it is roughly proportional to the total amount of change in the edge weights of the graph our experiments show that when the number of changes in the graph is small the dynamic algorithm is significantly faster than the best known static graph cut algorithm we test the performance of our algorithm on one particular problem the object background segmentation problem for video it should be noted that the application of our algorithm is not limited to the above problem the algorithm is generic and can be used to yield similar improvements in many other cases that involve dynamic change

packing spanning trees we given an algorithm for packing spanning trees in a graph g v e with capacities on the edges the problem reduces to o v 2 maximum flow computations the algorithm is based on nash williams s proof of a min max relation for this problem

a fast parametric maximum flow algorithm and applications the classical maximum flow problem sometimes occurs in settings in which the arc capacities are not fixed but are functions of a single parameter and the goal is to find the value of the parameter such that the corresponding maximum flow or minimum cut satisfies some side condition finding the desired parameter value requires solving a sequence of related maximum flow problems in this paper it is shown that the recent maximum flow algorithm of goldberg and tarjan can be extended to solve an important class of such parametric maximum flow problems at the cost of only a constant factor in its worst case time bound faster algorithms for a variety of combinatorial optimization problems follow from the result

earliest arrival flows with multiple sources earliest arrival flows capture the essence of evacuation planning given a network with capacities and transit times on the arcs a subset of source nodes with supplies and a sink node the task is to send the given supplies from the sources to the sink as quickly as possible the latter requirement is made more precise by the earliest arrival property which requires that the total amount of flow that has arrived at the sink is maximal for all points in time simultaneously it is a classical result from the 1970s that for the special case of a single source node earliest arrival flows do exist and can be computed by essentially applying the successive shortest path algorithm for min cost flow computations although it has previously been observed that an earliest arrival flow still exists for multiple sources the problem of computing one efficiently has been open for many years we present an exact algorithm for this problem whose running time is strongly polynomial in the input plus output size of the problem

analytic torsion for group actions the reidemeister torsion is a classical topological invariant for nonsimply connected manifolds 14 let m be a closed oriented smooth manifold with the fundamental group m nontrivial let p x m o n be an orthogonal representation of nx m if the twisted real cohomology h m ep vanishes then one can define the reidemeister torsion p gr which is a homeomorphism invariant of m the original interest of p was that it is not a homotopy invariant and so can distinguish spaces which are homotopy equivalent but are not homeomorphic ray and singer asked whether as for many other topological quantities one can compute p by analytic methods 17 given a riemannian metric g on m they defined an analytic torsion t e r as a certain combination of the eigenvalues of the laplacian acting on twisted differential forms they showed that under the above acyclicity condition t is independent of the metric g and they conjectured that the analytic expression t equals the combinatorial expression this conjecture was proven to be true independently by cheeger 4 and m ller 15 one can look at the above situation in the following way the group n m acts freely on the universal cover m and so one has an invariant for free group actions a natural question is whether the reidemeister torsion can be extended to an invariant for more general group actions for a finite group acting not necessarily freely on a closed oriented pl manifold x a reidemeister torsion was defined algebraically by rothenberg 19 one can then ask whether there is a corresponding analytic torsion when x is smooth and whether the analytic torsion equals the combinatorial torsion in 11 we define the analytic torsion tp for a finite group action and show that if the relevant cohomology groups vanish then it is independent of the g invariant metric used in its definition this was shown previously in unpublished work by cheeger 5 the analysis involved to

on strongly polynomial dual simplex algorithms for the maximum flow problem several pivot rules for the dual network simplex algorithm that enable it to solve a maximum flow problem on an n node m arc network in at most 2nm pivots and o n 2 m time are presented these rules are based on the concept of a preflow and depend upon the use of node labels which are either the lengths of a shortest pseudoaugmenting path from those nodes to the sink node or valid underestimates of those lengths extended versions of our algorithms are shown to solve an important class of parametric maximum flow problems with no increase in the worst case pivot and time bounds of these algorithms

a continuation gmres method for fast computation of nonlinear receding horizon control in this paper a fast numerical algorithm for nonlinear receding horizon control is proposed the control input is updated by a differential equation to trace the solution of an associated state dependent two point boundary value problem a linear equation involved in the differential equation is solved by the generalized minimum residual method one of the krylov subspace methods with jacobians approximated by forward differences the error in the entire algorithm is analyzed and is shown to be bounded under some conditions the proposed algorithm is applied to a two link arm whose dynamics is highly nonlinear simulation results show that the proposed algorithm is faster than the conventional algorithms

parametric maximum flow algorithms for fast total variation minimization this report studies the global minimization of anisotropically discretized total variation tv energies with an l p in particular l 1 and l 2 fidelity term using parametric maximum flow algorithms to minimize s t cut representations of these energies the tv l 2 model also known as the rudin osher fatemi rof model is suitable for restoring images contaminated by gaussian noise while the tv l 1 model is able to remove impulsive noise from grayscale images and perform multiscale decompositions of them preliminary numerical results on large scale two dimensional ct and three dimensional brain mr images are presented to illustrate the effectiveness of these approaches

an efficient algorithm for image segmentation markov random fields and related problems problems of statistical inference involve the adjustment of sample observations so they fit some a priori rank requirements or order constraints in such problems the objective is to minimize the deviation cost function that depends on the distance between the observed value and the modify value in markov random field problems there is also a pairwise relationship between the objects the objective in markov random field problem is to minimize the sum of the deviation cost function and a penalty function that grows with the distance between the values of related pairs separation function we discuss markov random fields problems in the context of a representative application the image segmentation problem in this problem the goal is to modify color shades assigned to pixels of an image so that the penalty function consisting of one term due to the deviation from the initial color shade and a second term that penalizes differences in assigned values to neighboring pixels is minimized we present here an algorithm that solves the problem in polynomial time when the deviation function is convex and separation function is linear and in strongly polynomial time when the deviation cost function is linear quadratic or piecewise linear convex with few pieces where few means a number exponential in a polynomial function of the number of variables and constraints the complexity of the algorithm for a problem on n pixels or variables m adjacency relations or constraints and range of variable values colors u is o t n m n log u where t n m is the complexity of solving the minimum s t cut problem on a graph with n nodes and m arcs furthermore other algorithms are shown to solve the problem with convex deviation and convex separation in running time o mn log n log nu and the problem with nonconvex deviation and convex separation in running time o t nu mu the nonconvex separation problem is np hard even for fixed value of u for the family of problems with convex deviation functions and linear separation function the algorithm described here runs in polynomial time which is demonstrated to be fastest possible

the resolvent expansion for second order regular singular operators abstract the paper gives a scheme for computing the asymptotics of tr e tl as t 0 where l is an elliptic operator of the form l d 2 x 2 a x and a x is a family of operators satisfying appropriate ellipticity and smoothness conditions a principle example is the laplace operator for a manifold with an asymptotically conic singularity the expansion has the usual terms away from the singularity appropriately regularized at x 0 plus singular contributions determined by the function of a 0 1 4 1 2 applications to index theorems are given in a subsequent paper

eigenvalue estimates and differential form laplacians on alexandrov spaces we give upper bounds on the eigenvalues of the differential form laplacian on a compact riemannian manifold the proof uses alexandrov spaces with curvature bounded below we also construct differential form laplacians on alexandrov spaces under a local bilipschitz assumption on the alexandrov space which is conjecturally always satisfied we show that the differential form laplacian has a compact resolvent we identify its kernel with an intersection homology group

fast algorithms for bipartite network flow we discuss network flow in a bipartite graph g with node sets s and t we show that karzanov s algorithm runs in time o s 2 t on g and that the mpm algorithm can be modified to run in the same bound for the common case that the degree of each node in t is bounded by a constant we show that a modified version of the mpm algorithm runs in time o s 3 s t on g the importance of these results comes from the common occurrence of problems that can be modeled and solved using network flow on bipartite graphs where s ll t in many applications involving bounded degree on the nodes in t t theta s 2 and our results reduce the best bounds on bipartite flow from o s 6 to o s 3 we use our results on bipartite flow to obtain the fastest known bounds on several combinatorial problems and mention several additional problems involving bipartite flow where s is much smaller than t

operator assignment and the parametric preflow algorithm this note addresses a problem of assigning human operators to operations in a manufacturing system the problem involves a set of operations a work load for each operation and a set of human operators each skilled at a subset of the operations the task is to assign the operators to the operations so as to maximize the capacity of the system robustly the problem is formulated as a network flow problem with a lexicographic objective it is then shown how to solve this problem very efficiently as a parametric sequence of network maximum flow problems by application of the parametric preflow algorithm of gallo grigoriadis and tarjan

polynomial time algorithms for ratio regions and a variant of normalized cut in partitioning clustering and grouping problems a typical goal is to group together similar objects or pixels in the case of image processing at the same time another goal is to have each group distinctly dissimilar from the rest and possibly to have the group size fairly large these goals are often combined as a ratio optimization problem one example of such a problem is a variant of the normalized cut problem another is the ratio regions problem we devise here the first polynomial time algorithms solving optimally the ratio region problem and the variant of normalized cut as well as a few other ratio problems the algorithms are efficient and combinatorial in contrast with nonlinear continuous approaches used in the image segmentation literature which often employ spectral techniques such techniques deliver solutions in real numbers which are not feasible to the discrete partitioning problem furthermore these continuous approaches are computationally expensive compared to the algorithms proposed here the algorithms presented here use as a subroutine a minimum s t cut procedure on a related graph which is of polynomial size the output consists of the optimal solution to the respective ratio problem as well as a sequence of nested solutions with respect to any relative weighting of the objectives of the numerator and denominator

spectral geometry of singular riemannian spaces on considere la trace du noyau de la chaleur et l application de la methode de l equation de la chaleur au theoreme d indice pour la caracteristique d euler et des complexes signatures

self adjusting binary search trees the splay tree a self adjusting form of binary search tree is developed and analyzed the binary search tree is a data structure for representing tables and lists so that accessing inserting and deleting items is easy on an n node splay tree all the standard search tree operations have an amortized time bound of o log n per operation where by amortized time is meant the time per operation averaged over a worst case sequence of operations thus splay trees are as efficient as balanced trees when total running time is the measure of interest in addition for sufficiently long access sequences splay trees are as efficient to within a constant factor as static optimum search trees the efficiency of splay trees comes not from an explicit structural constraint as with balanced trees but from applying a simple restructuring heuristic called splaying whenever the tree is accessed extensions of splaying give simplified forms of two other data structures lexicographic or multidimensional search trees and link cut trees

a new upper bound for the 0 1 quadratic knapsack problem the 0 1 quadratic knapsack problem qkp consists in maximizing a positive quadratic pseudo boolean function subject to a linear capacity constraint we present in this paper a new method based on lagrangian decomposition for computing an upper bound of qkp we report computational experiments which demonstrate the sharpness of the bound relative error very often less than 1 for large size instances up to 500 variables

proportional equity flow problem for terminal arcs the proportional equity flow problem extends a class of problems referred to as equity flow problems whose objective is to equitably distribute flow among the arcs in a flow circulation network the proportionally bounded flow circulation problem places lower and upper bounds on each arc flow that are nondecreasing continuous functions of the flow through one special arc and the objective is to maximize the flow through the special arc the proportional equity flow problem for terminal arcs problem ta is then defined as a special case where all the proportional arcs enter a sink vertex applications of both the general problem and problem ta are given two optimality conditions for problem ta are developed as well as an algorithm that is polynomially bounded for many types of nondecreasing continuous proportional bounding functions specifically the algorithm is shown to be polynomially bounded if the largest root of an equation can be found in polynomial time

optimal attack and reinforcement of a network in a nonnegative edge weighted network the weight of an edge represents the effort required by an attacker to destroy the edge and the attacker derives a benefit for each new component created by destroying edges the attacker may want to minimize over subsets of edges the difference between or the ratio of the effort incurred and the benefit received this idea leads to the definition of the strength of the network a measure of the resistance of the network to such attacks efficient algorithms for the optimal attack problem the problem of computing the strength and the problem of finding a minimum cost reinforcement to achieve a desired strength are given these problems are also solved for a different model in which the attacker wants to separate vertices from a fixed central vertex

the quickest transshipment problem a dynamic network consists of a graph with capacities and transit times on its edges the quickest transshipment problem is de2ned by a dynamic network with several sources and sinks each source has a speci2ed supply and each sink has a specified demand the problem is to send exactly the right amount of 6ow out of each source and into each sink in the minimum overall time variations of the quickest transshipment problem have been studied extensively the special case of the problem with a single sink is commonly used to model building evacuation similar dynamic network flow problems have numerous other applications in some of these the capacities are small integers and it is important to find integral flows there are no polynomial time algorithms known for most of these problems in this paper we give the first polynomial time algorithm for the quickest transshipment problem our algorithm provides an integral optimum flow previously the quickest transshipment problem could only be solved efficiently in the special case of a single source and single sink

diagnosing infeasibilities in network flow problems we consider the problem of finding a feasible flow in a directed networkg n a in which each nodei n has a supplyb i and each arc i j a has a zero lower bound on flow and an upper bounduij it is well known that this feasibility problem can be transformed into a maximum flow problem it is also well known that there is no feasible flow in the networkg if and only if there is a subsets of nodes such that the net supplies of the nodes ins exceeds the capacity of the arcs emanating froms such a sets is called a witness of infeasibility or simply awitness of the network flow problem in the case that there are many different witnesses for an infeasible problem a small cardinality witness may be preferable in practice because it is generally easier for the user to assimilate and may provide more guidance to the user on how to identify the cause of the infeasibility here we show that the problem of finding a minimum cardinality witness is np hard we also consider the problem of determining aminimal witness that is a witnesss such that no proper subset ofs is also a witness in this paper we show that we can determine a minimal witness by solving a sequence of at mostn maximum flow problems moreover if we use the preflow push algorithm to solve the resulting maximum flow problems and organize computations properly then the total time taken by the algorithm is comparable to that of solving a single maximum flow problem this approach determines a minimal cardinality witness in o n2m1 2 time using simple data structures and in o nm logn time using the standard implementation of the dynamic tree data structures we also show that the recognition version of the minimal witness problem is equivalent to a recognition version of a related problem known as theminimum rooted cut problem 1998 the mathematical programming society inc published by elsevier science b v

regularizing infinite sums of zeta determinants we present a new multiparameter resolvent trace expansion for elliptic operators polyhomogeneous in both the resolvent and auxiliary variables for elliptic operators on closed manifolds the expansion is a simple consequence of the parameter dependent pseudodifferential calculus as an additional nontrivial toy example we treat here sturm liouville operators with separated boundary conditions as an application we give a new formula in terms of regularized sums for the zeta determinant of an infinite direct sum of sturm liouville operators the laplace beltrami operator on a surface of revolution decomposes into an infinite direct sum of sturm louville operators parametrized by the eigenvalues of the laplacian on the cross section mathbb s 1 we apply the polyhomogeneous expansion to equate the zeta determinant of the laplace beltrami operator as a regularized sum of zeta determinants of the sturm liouville operators plus a locally computable term from the polyhomogeneous resolvent trace asymptotics this approach provides a completely new method for summing up zeta functions of operators and computing the meromorphic extension of that infinite sum to s 0 we expect our method to extend to a much larger class of operators

analytic torsion on manifolds with edges abstract let m g be an odd dimensional incomplete compact riemannian singular space with a simple edge singularity we study the analytic torsion on m and in particular consider how it depends on the metric g if g is an admissible edge metric we prove that the torsion zeta function is holomorphic near s 0 hence the torsion is well defined but possibly depends on g in general dimensions we prove that the torsion depends only on the asymptotic structure of g near the singular stratum of m when the dimension of the edge is odd we prove that the analytic torsion is independent of the choice of even admissible edge metrics the main tool is the construction via the methodology of geometric microlocal analysis of the heat kernel for the friedrichs extension of the hodge laplacian in all degrees in this way we obtain detailed asymptotics of this heat kernel and its trace

solving integer programs over monotone inequalities in three variables a framework for half integrality and good approximations abstract we define a class of monotone integer programs with constraints that involve up to three variables each a generic constraint in such integer program is of the form ax by z c where a and b are nonnegative and the variable z appears only in that constraint we devise an algorithm solving such problems in time polynomial in the length of the input and the range of variables u the solution is obtained from a minimum cut on a graph with o nu nodes and o mu arcs where n is the number of variables of the types x and y and m is the number of constraints our algorithm is also valid for nonlinear objective functions nonmonotone integer programs are optimization problems with constraints of the type ax by z c without restriction on the signs of a and b such problems are in general np hard we devise here an algorithm relying on a transformation to the monotone case that delivers half integral superoptimal solutions in polynomial time such solutions provide bounds on the optimum value that can only be superior to bounds provided by linear programming relaxation when the half integral solution can be rounded to an integer feasible solution this is a 2 approximate solution in that the technique is a unified 2 approximation technique for a large class of problems the results apply also for general integer programming problems with worse approximation factors that depend on a quantifier measuring how far the problem is from the class of problems we describe the algorithm described here has a wide array of problem applications an additional important consequence of our results is that nonmonotone problems in the framework are max snp hard and at least as hard to approximate as vertex cover problems that are amenable to the analysis provided here are easily recognized the analysis itself is entirely technical and involves manipulating the constraints and transforming them to a totally unimodular system while losing no more than a factor of 2 in the integrality

an extension of a theorem by cheeger and muller to spaces with isolated conical singularities the aim of this article is to extend a theorem by cheeger and muller to spaces with isolated conical singularities by generalizing the proof of bismut and zhang to the singular setting the main tools in this approach are the witten deformation and local index techniques

fractional covers for forests and matchings we give polynomial algorithms for the fractional covering problems for forests andb matchings min 1 y ya w y 0 wherea is a matrix whose rows are the incidence vectors of forests b matchings respectively it is shown that each problem can be solved by a series of max flow min cut calculations and hence the use of the ellipsoid algorithm to guarantee a polynomial algorithm can be avoided

efficient active set optimization in triple mode mpc an active set algorithm tailored to quadratically constrained quadratic programming in model predictive control mpc is presented it enables efficient use of augmented ellipsoidal invariant sets in conjunction with polytopic constraints in triple mode mpg the algorithm gives improved optimality and larger stabilizable initial condition sets than conventional quadratic programming mpc algorithms of comparable online computational burden

generalized ray singer conjecture i a manifold with a smooth boundary this paper is devoted to a proof of a generalized ray singer conjecture for a manifold with boundary the dirichlet and the neumann boundary conditions are independently given on each connected component of the boundary and the transmission boundary condition is given on the interior boundary the ray singer conjecture rs claims that for a closed manifold the combinatorial and the analytic torsion norms on the determinant of the cohomology are equal for a manifold with boundary the ratio between the analytic torsion and the combinatorial torsion is computed some new general properties of the ray singer analytic torsion are found the proof does not use any computation of eigenvalues and their asymptotic expansions or explicit expressions for the analytic torsions of any special classes of manifolds

constrained 0 1 quadratic programming basic approaches and extensions abstract we describe the simplest technique to tackle 0 1 quadratic programs with linear constraints among those that turn out to be successful in practice this method is due to and familiar to the quadratic assignment experts even if it took some time to realize that most approaches to the problem could be interpreted in these terms whereas it does not appear to be widely known outside this community since the technique is completely general and is by far the most successful one in several other cases such as quadratic knapsack we illustrate it in its full generality pointing out its relationship to lagrangian and linear programming relaxations and discussing further extensions we believe that this method should be in the background of every practitioner in combinatorial optimization

towards equitable distributions via proportional equity constraints we propose a new approach to the formulation of models for solving problems of equitable distribution equity is achieved by the use of proportional equity constraints which require that each recipient must receive ashare of the total distribution which falls within a prespecified range the algorithmic implications of this new approach are illustrated using two mathematical models we consider the problem of maximizing total flow in a multiterminal network with proportional equity constraints an efficient algorithm for solving this problem is provided as a further application we show that the introduction of such constraints still permits easy solutions for the linear knapsack problem

heat kernels and functional determinants on the generalized cone we consider zeta functions and heat kernel expansions on the bounded generalized cone in arbitrary dimensions using an improved calculational technique the specific case of a global monopole is analysed in detail and some restrictions thereby placed on thea 5 2 coefficient the computation of functional determinants is also addressed general formulas are given and known results are incidentally and rapidly reproduced

background foreground modeling based on spatiotemporal sparse subspace clustering background estimation and foreground segmentation are important steps in many high level vision tasks many existing methods estimate background as a low rank component and foreground as a sparse matrix without incorporating the structural information therefore these algorithms exhibit degraded performance in the presence of dynamic backgrounds photometric variations jitter shadows and large occlusions we observe that these backgrounds often span multiple manifolds therefore constraints that ensure continuity on those manifolds will result in better background estimation hence we propose to incorporate the spatial and temporal sparse subspace clustering into the robust principal component analysis rpca framework to that end we compute a spatial and temporal graph for a given sequence using motion aware correlation coefficient the information captured by both graphs is utilized by estimating the proximity matrices using both the normalized euclidean and geodesic distances the low rank component must be able to efficiently partition the spatiotemporal graphs using these laplacian matrices embedded with the rpca objective function these laplacian matrices constrain the background model to be spatially and temporally consistent both on linear and nonlinear manifolds the solution of the proposed objective function is computed by using the linearized alternating direction method with adaptive penalty optimization scheme experiments are performed on challenging sequences from five publicly available datasets and are compared with the 23 existing state of the art methods the results demonstrate excellent performance of the proposed algorithm for both the background estimation and foreground segmentation

analytic torsion and r torsion for unimodular representations let m be a closed c manifold of dimension n both r torsion and analytic torsion are smooth invariants of acyclic orthogonal or unitary representations p of the fundamental group 7i m the reidemeister franz torsion or r torsion tm p of p is defined in terms of the combinatorial structure of m given by its smooth triangulations the analytic torsion tm p was introduced by ray and singer rs as an analytic counterpart of r torsion in order to define the analytic torsion one has to choose a riemannian metric on m then tm p is a certain weighted alternating product of regularized determinants of the laplacians on differential qforms of m with values in the flat bundle e defined by p it was conjectured by ray and singer rs that tm p trm p for all acyclic orthogonal or unitary representations p this conjecture was proved independently by cheeger c and the author mu the restriction to orthogonal or unitary representations is certainly a limitation of the applicability of this result if 7i m is infinite because an infinite discrete group will have in general many nonorthogonal finite dimensional representations it is the purpose of the present paper to remove this limitation we call a representation p r m gl e on a finite dimensional real or complex vector space e unimodular if i det p y i 1 for all y e i m then we define r torsion and analytic torsion for unimodular representations and the main result is that for odd dimensional manifolds m the equality of the two torsions extends to all unimodular representations

on size constrained minimum s t cut problems and size constrained dense subgraph problems in some application cases the solutions of combinatorial optimization problems on graphs should satisfy an additional vertex size constraint in this paper we consider size constrained minimum s t cut problems and size constrained dense subgraph problems we introduce the minimum s t cut with at least k vertices problem the minimum s t cut with at most k vertices problem and the minimum s t cut with exactly k vertices problem we prove that they are np complete thus they are not polynomially solvable unless p np on the other hand we also study the densest at least k subgraph problem dalks and the densest at most k subgraph problem damks introduced by andersen and chellapilla 1 we present a polynomial time algorithm for dalks when k is bounded by some constant c we also present two approximation algorithms for damks the first approximation algorithm for damks has an approximation ratio of n 1 k 1 where n is the number of vertices in the input graph the second approximation algorithm for damks has an approximation ratio of o n for some 1 3

zeta function determinant of the laplace operator on the d dimensional ball we present a direct approach for the calculation of functional determinants of the laplace operator on balls dirichlet and robin boundary conditions are considered using this approach formulas for any value of the dimension d of the ball can be obtained quite easily explicit results are presented here for dimensionsd 2 3 4 5 and 6

least majorized elements and generalized polymatroids we prove that a bounded generalized polymatroid has a least weakly submajorized supermajorized vector such a vector simultaneously minimizes every nondecreasing nonincreasing symmetric and quasi convex function defined on the generalized polymatroid the same result holds also for the set of integer vectors of a bounded integral generalized polymatroid we then extend these results to more general sets and discuss several computational aspects

about strongly polynomial time algorithms for quadratic optimization over submodular constraints we present new strongly polynomial algorithms for special cases of convex separable quadratic minimization over submodular constraints the main results are an o nm log n 2 m algorithm for the problemnetwork defined on a network onm arcs andn nodes an o n logn algorithm for thetree problem onn variables an o n logn algorithm for thenested problem and a linear time algorithm for thegeneralized upper bound problem these algorithms are the best known so far for these problems the status of the general problem and open questions are presented as well

efficient non linear model based predictive control for a large number of degrees of freedom and or large dimension systems non linear model based predictive control algorithms based on dual mode control can become intractable this paper proposes an alternative which deploys the closed loop paradigm that has proved to be very effective for the case of linear time varying or uncertain systems the various attributes and computational advantages of the approach are shown to carry over to the non linear case

analytic and topological torsion for manifolds with boundary and symmetry letg be a finite group acting on a riemannian manifoldm by isometries we introduce analytic torsion an m m1 v r z repr g pl torsion pl m m1 v k1 rg poincare torsion pd m m1 v k1 rg and euler characteristic m m1 v repr g for m the disjoint union of m1 and m2 and v an equivariant coefficient system the analytic torsionis defined in terms of the spectrum of the laplace operator the pl torsionis based on the cellular chain complex and poincare torsionmeasures the failure of equivariant poincare duality in the pl setting which does hold in the analytic context denote by repr g the subgroup of repr g generated by the irreducible representations of real or complex type we define an isomorphism

the bipartite rationing problem in the bipartite rationing problem a set of agents share a single resource available in different types each agent has a claim over only a subset of the resource types and these claims overlap in arbitrary fashion the goal is to divide fairly the various types of resources between the claimants when resources are in short supply with a single type of resource this is the standard rationing problem o neill b 1982 a problem of rights arbitration from the talmud math soc sci 2 4 345 371 of which the three benchmark solutions are the proportional uniform gains and uniform losses methods we extend these methods to the bipartite context imposing the familiar consistency requirement the division is unchanged if we remove an agent respectively a resource and take away at the same time his share of the various resources respectively reduce the claims of the relevant agents the uniform gains and uniform losses methods have infinitely many consistent extensions but the proportional me

a polynomial time algorithm for rayleigh ratio on discrete variables replacing spectral techniques for expander ratio normalized cut and cheeger constant a general form of minimizing the rayleigh ratio on discrete variables is shown here for the first time to be polynomial time solvable this is significant because major problems in clustering partitioning and imaging can be presented as the rayleigh ratio minimization on discrete variables and an orthogonality constraint these challenging problems are modeled as the normalized cut problem the graph expander ratio problem the cheeger constant problem or the conductance problem all of which are np hard these problems have traditionally been solved heuristically using the spectral technique a unified framework is provided here whereby all these problems are formulated as a constrained minimization form of a quadratic ratio referred to here as the rayleigh ratio the quadratic ratio is to be minimized on discrete variables and a single sum constraint that we call the balance or orthogonality constraint when the discreteness constraints on the variables are disregarded the resulting continuous relaxation is solved by the spectral method it is shown here that the rayleigh ratio minimization subject to the discreteness constraints requiring each variable to assume one of two values in b 1 is solvable in strongly polynomial time equivalent to a single minimum s t cut algorithm on a graph of same size as the input graph for any nonnegative value of b this discrete form for the rayleigh ratio problem was often assumed to be np hard not only is it shown here that the discrete rayleigh ratio problem is polynomial time solvable but also the algorithm is more efficient than the spectral algorithm furthermore an experimental study demonstrates that the new algorithm provides in practice an improvement often dramatic on the quality of the results of the spectral method both in terms of approximating the true optimum of the rayleigh ratio problem on both the discrete variables and the balance constraint and in terms of the subjective partition quality a further contribution here is the introduction of a problem the quantity normalized cut generalizing all the rayleigh ratio problems the discrete version of that problem is also solved with the efficient algorithm presented this problem is shown in a companion paper to enable the modeling of features essential to clustering that are valuable in practical applications

a fast parametric submodular intersection algorithm for strong map sequences this paper presents a fast algorithm to solve the intersection problem for a pair of nondecreasing and nonincreasing strong map sequences of submodular systems the worst case time bound is the same as that of the push relabel algorithm for a single intersection problem this extends the gallo grigoriadis tarjan ggt method for parametric maximum flow problems and reveals an algorithmic significance of the concept of strong maps for submodular systems

parametric combinatorial computing and a problem of program module distribution en calcul parametrique certaines parties de la description du probleme sont donnees comme des fonctions d un ou plusieurs parametres pouvant varier sur un grand nombre de valeurs des valeurs fixes des parametres specifient un probleme particulier l objectif du calcul parametrique etant de resoudre de maniere efficace de tels problemes en fonction des valeurs des parametres on propose une methode de calcul parametrique generale efficace pour une large classe de problemes combinatoires la methode est liee a une methode developpee par megiddo pour la resolution de problemes d optimisation de ratios on traite un exemple de distribution de modules de programme d ordinateur entre deux processeurs

how to compute least infeasible flows it is well known how to use maximum flow to decide when a flow problem with demands lower bounds and upper bounds is infeasible less well known is how to compute a flow that is least infeasible this paper considers many possible ways to define least infeasible and shows how to compute optimal flows for each definition for each definition it also gives a dual characterization in terms of cuts a polynomial routine for recognizing that type of least infeasible flow and relates that definition to dual cut canceling min cost flow network algorithms

a new approach to the maximum flow problem all previously known efficient maximum flow algorithms work by finding augmenting paths either one path at a time as in the original ford and fulkerson algorithm or all shortest length augmenting paths at once using the layered network approach of dinic an alternative method based on the preflow concept of karzanov is introduced a preflow is like a flow except that the total amount flowing into a vertex is allowed to exceed the total amount flowing out the method maintains a preflow in the original network and pushes local flow excess toward the sink along what are estimated to be shortest paths the algorithm and its analysis are simple and intuitive yet the algorithm runs as fast as any other known method on dense graphs achieving an o n 3 time bound on an n vertex graph by incorporating the dynamic tree data structure of sleator and tarjan we obtain a version of the algorithm running in o nm log n 2 m time on an n vertex m edge graph this is as fast as any known method for any graph density and faster on graphs of moderate density the algorithm also admits efficient distributed and parallel implementations a parallel implementation running in o n 2 log n time using n processors and o m space is obtained this time bound matches that of the shiloach vishkin algorithm which also uses n processors but requires o n 2 space

a parametric maximum flow algorithm for bipartite graphs with applications abstract suppose we are given a capacitated bipartite network g with node sets s and t in network g the arc capacities are not fixed values but are functions of a single parameter where is a continuous real variable our problem is to determine the minimum value of such that the maximum flow value in the corresponding network equals a given threshold for this problem an algorithm of time complexity o nm 2 log m n is presented where n is the number of nodes in s m is the number of nodes in t and n m examples are then given to show how to use this parametric algorithm to solve practical problems

energy optimal schedules for jobs with multiple active intervals in this paper we study the scheduling problem of jobs with multiple active intervals each job in the problem instance has n n 1 disjoint active time intervals where it can be executed and a workload characterized by the required number of cpu cycles previously people studied multiple interval job scheduling problem where each job must be assigned enough cpu cycles in one of its active intervals we study a different practical version where the partial work done by the end of an interval remains valid and each job is considered finished if total cpu cycles assigned to it in all its active intervals reach the requirement the goal is to find a feasible schedule that minimizes energy consumption by adapting the algorithm for single interval jobs proposed in yao demers and shenker 1995 1 one can still obtain an optimal schedule however the two phases in that algorithm critical interval finding and scheduling the critical interval can no longer be carried out directly we present polynomial time algorithms to solve the two phases for jobs with multiple active intervals and therefore can still compute the optimal schedule in polynomial time

almost tight bounds and existence theorems for single commodity confluent flows a flow of a commodity is said to be confluent if at any node all the flow of the commodity leaves along a single edge in this article we study single commodity confluent flow problems where we need to route given node demands to a single destination using a confluent flow single and multi commodity confluent flows arise in a variety of application areas most notably in networking in fact most flows in the internet are multi commodity confluent flows since internet routing is destination based we present near tight approximation algorithms hardness results and existence theorems for minimizing congestion in single commodity confluent flows the maximum edge congestion of a single commodity confluent flow occurs at one of the incoming edges of the destination therefore finding a minimum congestion confluent flow is equivalent to the following problem given a directed graph g with k sinks and non negative demands on all the nodes of g determine a confluent flow that routes every node demand to some sink such that the maximum congestion at a sink is minimized the main result of this article is a polynomial time algorithm for determining a confluent flow with congestion at most 1 p ln k in g if g admits a splittable flow with congestion at most 1 we complement this result in two directions first we present a graph g that admits a splittable flow with congestion at most 1 yet no confluent flow with congestion smaller than hk the kth harmonic number thus establishing tight upper and lower bounds to within an additive constant less than 1 second we show that it is np hard to approximate the congestion of an optimal confluent flow to within a factor of log2k 2 thus resolving the polynomial time approximability to within a multiplicative constant we also consider a demand maximization version of the problem we show that if g admits a splittable flow of congestion at most 1 then a variant of the congestion minimization algorithm yields a confluent flow in g with congestion at most 1 that satisfies 1 3 fraction of total demand we show that the gap between confluent flows and splittable flows is much smaller if the underlying graph is k connected in particular we prove that k connected graphs with k sinks admit confluent flows of congestion less than c p dmax where c is the congestion of the best splittable flow and dmax is the maximum demand of any node in g the proof of this existence theorem is non constructive and relies on topological techniques introduced by lovasz

a new approximation algorithm for the unbalanced min s t cut problem we give a new approximation algorithm for the min k size s t cut problem the algorithm is very simple and has only three lines to state we analyze the integrality gap of a natural lp for the problem being the unbalanced version of the famous min s t cut problem the min k size s t cut problem asks to find a k size s t cut with the minimum capacity where a k size s t cut means an s t cut with its s side having size at most k this problem is fundamental and has extensive applications especially in community identification in social and information networks it is known that the min k size s t cut problem is np hard and can be approximated within o log n where n is the number of vertices in the input graph in this paper we give a new approximation algorithm for the min k size s t cut problem based on the parametric flow technique the algorithm is very simple and has only three lines to state its approximation ratio is k 1 k 1 k where k is the size of the s side of an optimal solution

a structure theory for the parametric submodular intersection problem a linearly parameterized polymatroid intersection problem appears in the context of principal partitions we consider a submodular intersection problem on a pair of strong map sequences of submodular functions which is an extension of the linearly parameterized polymatroid intersection problem to a nonlinearly parameterized one we introduce the concept of a basis frame on a finite nonempty set of cardinality n that gives a mapping from the set of all base polyhedra in n dimensional space into n dimensional vectors such that each base polyhedron is mapped to one of its bases we show the existence of a simple universal representation of all optimal solutions of the parameterized submodular intersection problem by means of basis frames

decompositions network flows and a precedence constrained single machine scheduling problem we present an in depth theoretical algorithmic and computational study of a linear programming lp relaxation to the precedence constrained single machine scheduling problem 1 prec s jw jc jto minimize a weighted sum of job completion times on the theoretical side we study the structure of tight parallel inequalities in thelp relaxation and show thatevery permutation schedule that is consistent with sidney s decomposition has total cost no more than twice the optimum on the algorithmic side we provide a parametric extension to sidney s decomposition and show that a finest decomposition can be obtained by essentially solving a parametric minimum cut problem finally we report results obtained by an algorithm based on these developments on randomly generated instances with up to 2 000 jobs