asynchronous distributed algorithms for solving linear algebraic equations two asynchronous distributed algorithms are presented for solving a linear equation of the form ax b with at least one solution the equation is simultaneously and asynchronously solved by m agents assuming that each agent knows only a subset of the rows of the partitioned matrix a b the estimates of the equation s solution generated by its neighbors and nothing more neighbor relationships are characterized by a time dependent directed graph whose vertices correspond to agents and whose arcs depict neighbor relationships each agent recursively updates its estimate of a solution at its own event times by utilizing estimates generated by its neighbors which are transmitted with delays the event time sequences of different agents are not assumed to be synchronized it is shown that for any matrix vector pair a b for which the equation has a solution and any repeatedly jointly strongly connected sequence of neighbor graphs defined on the merged sequence of all agents event times the algorithms cause all agents estimates to converge exponentially fast to the same solution to ax b the first algorithm requires a specific initialization step at each agent and the second algorithm works for arbitrary initializations explicit expressions for convergence rates are provided and a relation between local initializations and limiting consensus solutions is established which is used to solve the least 2 norm solution

exponential convergence of a distributed algorithm for solving linear algebraic equations abstract in a recent paper a distributed algorithm was proposed for solving linear algebraic equations of the form a x b assuming that the equation has at least one solution the equation is presumed to be solved by m agents assuming that each agent knows a subset of the rows of the matrix a b the current estimates of the equation s solution generated by each of its neighbors and nothing more neighbor relationships are represented by a time dependent directed graph n t whose vertices correspond to agents and whose arcs characterize neighbor relationships sufficient conditions on n t were derived under which the algorithm can cause all agents estimates to converge exponentially fast to the same solution to a x b these conditions were also shown to be necessary for exponential convergence provided the data about a b available to the agents is non redundant the aim of this paper is to relax this non redundant assumption this is accomplished by establishing exponential convergence under conditions which are the weakest possible for the problem at hand the conditions are based on a new notion of graph connectivity an improved bound on the convergence rate is also derived

distributed computation of linear matrix equations an optimization perspective this paper investigates the distributed computation of the well known linear matrix equation in the form of axb f with the matrices a b x and f of appropriate dimensions over multiagent networks from an optimization perspective in this paper we consider the standard distributed matrix information structures where each agent of the considered multiagent network has access to one of the subblock matrices of a b and f to be specific we first propose different decomposition methods to reformulate the matrix equations in standard structures as distributed constrained optimization problems by introducing substitutional variables we show that the solutions of the reformulated distributed optimization problems are equivalent to least squares solutions to original matrix equations and we design distributed continuous time algorithms for the constrained optimization problems even by using augmented matrices and a derivative feedback technique moreover we prove the exponential convergence of the algorithms to a least squares solution to the matrix equation for any initial condition

the ordered k median problem surrogate models and approximation algorithms in the last two decades a steady stream of research has been devoted to studying various computational aspects of the ordered k median problem which subsumes traditional facility location problems such as median center p centrum etc through a unified modeling approach given a finite metric space the objective is to locate k facilities in order to minimize the ordered median cost function in its general form this function penalizes the coverage distance of each vertex by a multiplicative weight depending on its ranking or percentile in the ordered list of all coverage distances while antecedent literature has focused on mathematical properties of ordered median functions integer programming methods various heuristics and special cases this problem was not studied thus far through the lens of approximation algorithms in particular even on simple network topologies such as trees or line graphs obtaining non trivial approximation guarantees is an open question the main contribution of this paper is to devise the first provably good approximation algorithms for the ordered k median problem we develop a novel approach that relies primarily on a surrogate model where the ordered median function is replaced by a simplified ranking invariant functional form via efficient enumeration surprisingly while this surrogate model is varomega n varomega 1 hard to approximate on general metrics we obtain an o log n approximation for our original problem by employing local search methods on a smooth variant of the surrogate function in addition an improved guarantee of 2 epsilon is obtained on tree metrics by optimally solving the surrogate model through dynamic programming finally we show that the latter optimality gap is tight up to an o epsilon term

the tree of hubs location problem this paper presents the tree of hubs location problem it is a network hub location problem with single assignment where a fixed number of hubs have to be located with the particularity that it is required that the hubs are connected by means of a tree the problem combines several aspects of location network design and routing problems potential applications appear in telecommunications and transportation systems when set up costs for links between hubs are so high that full interconnection between hub nodes is prohibitive we propose an integer programming formulation for the problem furthermore we present some families of valid inequalities that reinforce the formulation and we give an exact separation procedure for them finally we present computational results using the well known ap and cab data sets

o n3 noniterative heuristic algorithm for linear programs with error free implementation we present an o n 3 noniterative heuristic algorithm and describe an implementation of p adic arithmetic to solve the linear programming problems error free an optimality test as well as numerical examples depict the scope of such an algorithm in the few cases where it did not lead to the optimal solution it did provide a nonnegative solution of the system ax b which could be used as the basic feasible solution for any one of the widely used simplex methods say the revised simplex algorithm without the need to include artificial variables in ax b for consistency check thus the algorithm could be at least a pre processor for any simplex algorithm and avoids the enhancement of dimensions of the constraint matrix a and consequent additional space computational complexities and errors

ozsvath szabo invariants and tight contact three manifolds ii let s 3 k be the oriented 3 manifold obtained by rational r surgery on a knot k s 3 using the contact ozsv ath szab o invariants we prove for a class of knots k containing all the algebraic knots that s 3 k carries positive tight contact structures for every r 6 g s k 1 where gs k is the slice genus of k this implies in particular that the brieskorn spheres 2 3 4 and 2 3 3 carry tight positive contact structures as an application of our main result we show that for each m2n there exists a seifert bered rational homology 3 sphere mm carrying at least m pairwise non isomorphic tight nonllable contact structures

reaching a consensus in a dynamically changing environment convergence rates measurement delays and asynchronous events this paper uses recently established properties of compositions of directed graphs together with results from the theory of nonhomogeneous markov chains to derive worst case convergence rates for the headings of a group of mobile autonomous agents which arise in connection with the widely studied vicsek consensus problem the paper also uses graph theoretic constructions to solve modified versions of the vicsek problem in which there are measurement delays asynchronous events or a group leader in all three cases the conditions under which consensus is achieved prove to be almost the same as the conditions under which consensus is achieved in the synchronous delay free leaderless case

a distributed algorithm for solving a linear algebraic equation a distributed algorithm is described for solving a linear algebraic equation of the form ax b assuming the equation has at least one solution the equation is simultaneously solved by m agents assuming each agent knows only a subset of the rows of the partitioned matrix matrix a b the current estimates of the equation s solution generated by its neighbors and nothing more each agent recursively updates its estimate by utilizing the current estimates generated by each of its neighbors neighbor relations are characterized by a time dependent directed graph bbn t whose vertices correspond to agents and whose arcs depict neighbor relations it is shown that for any matrix a for which the equation has a solution and any sequence of repeatedly jointly strongly connected graphs bbn t t 1 2 ldots the algorithm causes all agents estimates to converge exponentially fast to the same solution to ax b it is also shown that under mild assumptions the neighbor graph sequence must actually be repeatedly jointly strongly connected if exponential convergence is to be assured a worst case convergence rate bound is derived for the case when ax b has a unique solution it is demonstrated that with minor modification the algorithm can track the solution to ax b even if a and b are changing with time provided the rates of change of a and b are sufficiently small it is also shown that in the absence of communication delays exponential convergence to a solution occurs even if the times at which each agent updates its estimates are not synchronized with the update times of its neighbors a modification of the algorithm is outlined which enables it to obtain a least squares solution to ax b in a distributed manner even if ax b does not have a solution

a cutting plane algorithm for convex programming that uses analytic centers anoracle for a convex sets n accepts as input any pointz in n and ifz s then it returns yes while ifz s then it returns no along with a separating hyperplane we give a new algorithm that finds a feasible point ins in cases where an oracle is available our algorithm uses the analytic center of a polytope as test point and successively modifies the polytope with the separating hyperplanes returned by the oracle the key to establishing convergence is that hyperplanes judged to be unimportant are pruned from the polytope if a ball of radius 2 l is contained ins ands is contained in a cube of side 2l 1 then we can show our algorithm converges after o nl2 iterations and performs a total of o n4l3 tnl2 arithmetic operations wheret is the number of arithmetic operations required for a call to the oracle the bound is independent of the number of hyperplanes generated in the algorithm an important application in which an oracle is available is minimizing a convex function overs

interior point methods an old and new approach to nonlinear programming in this paper we discuss the main concepts of structural optimization a field of nonlinear programming which was formed by the intensive development of modern interior point schemes

locating tree shaped facilities using the ordered median objective in this paper we consider the location of a tree shaped facility s on a tree network using the ordered median function of the weighted distances to represent the total transportation cost objective this function unifies and generalizes the most common criteria used in location modeling e g median and center if there are n demand points at the nodes of the tree t v e this function is characterized by a sequence of reals 1 n satisfying 1 2 n 0 for a given subtree s let x s x1 xn be the set of weighted distances of the n demand points from s the value of the ordered median objective at s is obtained as follows sort the n elements in x s in nonincreasing order and then compute the scalar product of the sorted list with the sequence two models are discussed in the tactical model there is an explicit bound l on the length of the subtree and the goal is to select a subtree of size l which minimizes the above transportation cost function in the strategic model the length of the subtree is variable and the objective is to minimize the sum of the transportation cost and the length of the subtree we consider both discrete and continuous versions of the tactical and the strategic models we note that the discrete tactical problem is np hard and we solve the continuous tactical problem in polynomial time using a linear programming lp approach we also prove submodularity properties for the strategic problem these properties allow us to solve the discrete strategic version in strongly polynomial time moreover the continuous version is also solved via lp for the special case of the k centrum objective we obtain improved algorithmic results using a dynamic programming dp algorithm and discretization and nestedness results

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

distributed algorithm for solving convex inequalities in this note a distributed subgradient based algorithm is proposed for continuous time multi agent systems to search a feasible solution to convex inequalities the algorithm involves each agent achieving a state constrained by its own inequalities while exchanging local information with other agents under a time varying directed communication graph with the validity of a mild connectivity condition associated with the communication graph it is shown that all agents will reach agreement asymptotically and the consensus state is in the solution set of the inequalities furthermore the method is also extended to solving the distributed optimization problem of minimizing the sum of local objective functions subject to convex inequalities simulation examples are presented to demonstrate the effectiveness of the theoretical results

ordered weighted average optimization in multiobjective spanning tree problem abstract multiobjective spanning tree problems are studied in this paper the ordered median objective function is used as an averaging operator to aggregate the vector of objective values of feasible solutions this leads to the ordered weighted average spanning tree problem a nonlinear combinatorial optimization problem different mixed integer linear programs are proposed based on the most relevant minimum cost spanning tree models in the literature these formulations are analyzed and several enhancements presented their empirical performance is tested over a set of randomly generated benchmark instances the results of the computational experiments show that the choice of an appropriate formulation allows to solve larger instances with more objectives than those previously solved in the literature

revisiting k sum optimization in this paper we address continuous integer and combinatorial k sum optimization problems we analyze different formulations of this problem that allow to solve it through the minimization of a relatively small number of minisum optimization problems this approach provides a general tool for solving a variety of k sum optimization problems and at the same time improves the complexity bounds of many ad hoc algorithms previously reported in the literature for particular versions of this problem moreover the results developed for k sum optimization have been extended to the more general case of the convex ordered median problem improving upon existing solution approaches

distributed parameter estimation in sensor networks nonlinear observation models and imperfect communication the paper studies distributed static parameter vector estimation in sensor networks with nonlinear observation models and noisy intersensor communication it introduces separably estimable observation models that generalize the observability condition in linear centralized estimation to nonlinear distributed estimation it studies two distributed estimation algorithms in separably estimable models the nu with its linear counterpart lu and the nlu their update rule combines a consensus step where each sensor updates the state by weight averaging it with its neighbors states and an innovation step where each sensor processes its local current observation this makes the three algorithms of the consensus innovations type very different from traditional consensus this paper proves consistency all sensors reach consensus almost surely and converge to the true parameter value efficiency and asymptotic unbiasedness for lu and nu it proves asymptotic normality and provides convergence rate guarantees the three algorithms are characterized by appropriately chosen decaying weight sequences algorithms lu and nu are analyzed in the framework of stochastic approximation theory algorithm nlu exhibits mixed time scale behavior and biased perturbations and its analysis requires a different approach that is developed in this paper

karmarkar s linear programming algorithm and newton s method this paper describes a full dimensional version of karmarkar s linear programming algorithm theprojective scaling algorithm which is defined for any linear program in n having a bounded full dimensional polytope of feasible solutions if such a linear program hasm inequality constraints then it is equivalent under an injective affine mappingj n m to karmarkar s original algorithm for a linear program in m havingm n equality constraints andm inequality constraints karmarkar s original algorithm minimizes a potential functiong x and the projective scaling algorithm is equivalent to that version of karmarkar s algorithm whose step size minimizes the potential function in the step direction

network flows that solve linear equations we study distributed network flows as solvers in continuous time for the linear algebraic equation mathbf z mathbf h mathbf y each node i has access to a row mathbf h i mathrm t of the matrix mathbf h and the corresponding entry z i in the vector mathbf z the first consensus projection flow under investigation consists of two terms one from standard consensus dynamics and the other contributing to projection onto each affine subspace specified by the mathbf h i and z i the second projection consensus flow on the other hand simply replaces the relative state feedback in consensus dynamics with projected relative state feedback without dwell time assumption on switching graphs we prove that all node states converge to a common solution of the linear algebraic equation if there is any the convergence is global for the consensus projection flow while local for the projection consensus flow in the sense that the initial values must lie on the affine subspaces if the linear equation has no exact solutions we show that the node states can converge to a ball around the least squares solution whose radius can be made arbitrarily small through selecting a sufficiently large gain for the consensus projection flow for a fixed bidirectional graph semi global convergence to approximate least squares solutions is also demonstrated for switching balanced directed graphs under suitable conditions it is also shown that the projection consensus flow drives the average of the node states to the least squares solution with a complete graph numerical examples are provided as illustrations of the established results

complexity estimates of some cutting plane methods based on the analytic barrier in this paper we establish the efficiency estimates for two cutting plane methods based on the analytic barrier we prove that the rate of convergence of the second method is optimal uniformly in the number of variables we present a modification of the second method in this modified version each test point satisfies an approximate centering condition we also use the standard strategy for updating approximate hessians of the logarithmic barrier function we prove that the rate of convergence of the modified scheme remains optimal and demonstrate that the number of newton steps in the auxiliary minimization processes is bounded by an absolute constant we also show that the approximate hessian strategy significantly improves the total arithmetical complexity of the method

an ol n 3 primal interior point algorithm for convex quadratic programming we present a primal interior point method for convex quadratic programming which is based upon a logarithmic barrier function approach this approach generates a sequence of problems each of which is approximately solved by taking a single newton step it is shown that the method requires o sqrt n l iterations and o n3 5l arithmetic operations by using modified newton steps the number of arithmetic operations required by the algorithm can be reduced to o n3l

aggregation error bounds for a class of location models many location models involve distances and demand points in their objective function in urban contexts there can be millions of demand points this leads to demand point aggregation which produces error we identify a general model structure that includes most such location models and present a means of obtaining error bounds for all models with this structure the error bounds suggest how to do the demand point aggregation so as to keep the error small

a scaling technique for finding the weighted analytical center of a polytope let a bounded full dimensional polytope be defined by the systemax b wherea is anm n matrix letai denote theith row of the matrixa and define theweighted analytic center of the polytope to be the point that minimizes the strictly convex barrier function i 1mwi ln aitx bi the proper selection of weightswi can make any desired point in the interior of the polytope become the weighted analytic center as a result the weighted analytic center has applications in both linear and general convex programming for simplicity we assume that the weights are positive integers if some of thewi s are much larger than others then newton s method for minimizing the resulting barrier function is very unstable and can be very slow previous methods for finding the weighted analytic center relied upon a rather direct application of newton s method potentially resulting in very slow global convergence we present a method for finding the weighted analytic center that is based on the scaling technique of edmonds and karp and is an enhancement of newton s method the scaling algorithm runs in o sqrt m log w iterations wherem is the number of constraints defining the polytope andw is the largest weight given on any constraint each iteration involves taking a step in the newton direction and its complexity is dominated by the time needed to solve a system of linear equations

on vaidya s volumetric cutting plane method for convex programming we describe a simplified and strengthened version of vaidya s volumetric cutting plane method for finding a point in a convex set x1d49e rn at each step the algorithm has a system of linear inequality constraints which defines a polyhedron x1d4ab x1d49e and an interior point x x1d4ab the algorithm then either drops one constraint or calls an oracle to check if x x1d49e and if not obtain a new constraint that separates x from x1d49e following the addition or deletion of a constraint the algorithm takes a small number of newton steps for the volumetric barrier v progress of the algorithm is measured in terms of changes in v the algorithm is terminated when either it is discovered that x x1d49e or v becomes large enough to demonstrate that the volume of x1d49e must be below some prescribed amount the complexity of the algorithm compares favorably with that of the ellipsoid method especially in terms of the number of calls to the separation oracle compared to vaidya s original analysis we decrease the total number

distributed constrained optimization by consensus based primal dual perturbation method various distributed optimization methods have been developed for solving problems which have simple local constraint sets and whose objective function is the sum of local cost functions of distributed agents in a network motivated by emerging applications in smart grid and distributed sparse regression this paper studies distributed optimization methods for solving general problems which have a coupled global cost function and have inequality constraints we consider a network scenario where each agent has no global knowledge and can access only its local mapping and constraint functions to solve this problem in a distributed manner we propose a consensus based distributed primal dual perturbation pdp algorithm in the algorithm agents employ the average consensus technique to estimate the global cost and constraint functions via exchanging messages with neighbors and meanwhile use a local primal dual perturbed subgradient method to approach a global optimum the proposed pdp method not only can handle smooth inequality constraints but also non smooth constraints such as some sparsity promoting constraints arising in sparse optimization we prove that the proposed pdp algorithm converges to an optimal primal dual solution of the original problem under standard problem and network assumptions numerical results illustrating the performance of the proposed algorithm for a distributed demand response control problem in smart grid are also presented

a new algorithm for minimizing convex functions over convex sets let s subseteq mathbb r n be a convex set for which there is an oracle with the following property given any pointz n the oracle returns a yes ifz s whereas ifz s then the oracle returns a no together with a hyperplane that separatesz froms the feasibility problem is the problem of finding a point ins the convex optimization problem is the problem of minimizing a convex function overs we present a new algorithm for the feasibility problem the notion of a volumetric center of a polytope and a related ellipsoid of maximum volume inscribable in the polytope are central to the algorithm our algorithm has a significantly better global convergence rate and time complexity than the ellipsoid algorithm the algorithm for the feasibility problem easily adapts to the convex optimization problem

finding the conditional location of a median path on a tree in this paper we study the problem of locating a median path of limited length on a tree under the condition that some existing facilities are already located the existing facilities may be located at any subset of vertices upper and lower bounds are proposed for both the discrete and continuous models in the discrete model a median path is not allowed to contain partial edges in the continuous model a median path may contain partial edges the proposed upper bounds for these two models are o nlogn and o nlogn a n respectively they improve the previous known bounds from o nlog 2n and o n 2 respectively the proposed lower bounds are both w nlogn

cobordisms of sutured manifolds and the functoriality of link floer homology abstract it has been a central open problem in heegaard floer theory whether cobordisms of links induce homomorphisms on the associated link floer homology groups we provide an affirmative answer by introducing a natural notion of cobordism between sutured manifolds and showing that such a cobordism induces a map on sutured floer homology this map is a common generalization of the hat version of the closed 3 manifold cobordism map in heegaard floer theory and the contact gluing map defined by honda kazez and matic we show that sutured floer homology together with the above cobordism maps forms a type of tqft in the sense of atiyah applied to the sutured manifold cobordism complementary to a decorated link cobordism our theory gives rise to the desired map on link floer homology hence link floer homology is a categorification of the multi variable alexander polynomial we outline an alternative definition of the contact gluing map using only the contact element and handle maps finally we show that a weinstein sutured manifold cobordism preserves the contact element

constrained consensus and optimization in multi agent networks we present distributed algorithms that can be used by multiple agents to align their estimates with a particular value over a network with time varying connectivity our framework is general in that this value can represent a consensus value among multiple agents or an optimal solution of an optimization problem where the global objective function is a combination of local agent objective functions our main focus is on constrained problems where the estimates of each agent are restricted to lie in different convex sets to highlight the effects of constraints we first consider a constrained consensus problem and present a distributed projected consensus algorithm in which agents combine their local averaging operation with projection on their individual constraint sets this algorithm can be viewed as a version of an alternating projection method with weights that are varying over time and across agents we establish convergence and convergence rate results for the projected consensus algorithm we next study a constrained optimization problem for optimizing the sum of local objective functions of the agents subject to the intersection of their local constraint sets we present a distributed projected subgradient algorithm which involves each agent performing a local averaging operation taking a subgradient step to minimize its own objective function and projecting on its constraint set we show that with an appropriately selected stepsize rule the agent estimates generated by this algorithm converge to the same optimal solution for the cases when the weights are constant and equal and when the weights are time varying but all agents have the same constraint set

the nonlinear geometry of linear programming i affine and projective scaling trajectories this series of papers studies a geometric structure underlying karmarkar s projective scaling algorithm for solving linear programming problems a basic feature of the projective scaling algorithm is a vector field depending on the objective function which is defined on the interior of the polytope of feasible solutions of the linear program the geometric structure studied is the set of trajectories obtained by integrating this vector field which we call ptrajectories we also study a related vector field the affine scaling vector field and its associated trajectories called a trajectories the affine scaling vector field is associated to another linear programming algorithm the affine scaling algorithm affine and projective scaling vector fields are each defined for linear programs of a special form called strict standard form and canonical form respectively this paper derives basic properties of p trajectories and a trajectones it reviews the projective and affine scaling algorithms defines the projective and affine scaling vector fields and gives differential equations for p trajectories and a trajectories it shows that projective transformations map p trajectories into p trajectories it presents karmarkar s interpretation of a trajectories as steepest descent paths of the objective function c x with respect to the riemannian geometry ds2 z dx dx x2 restricted to the relative interior of the polytope of feasible solutions p trajectories of a canonical form linear program are radial projections of a trajectories of an associated standard form linear program as a consequence there is a polynomial time linear programming algorithm using the affine scaling vector field of this associated linear program this algorithm is essentially karmarkar s algorithm these trajectories are studied in subsequent papers by two nonlinear changes of variables called legendre transform coordinates and projective legendre transform coordinates respectively it will be shown that p trajectories have an algebraic and a geometric interpretation they are algebraic curves and they are geodesics actually distinguished chords of a geometry isometric to a hilbert geometry on a polytope combinatorially dual to the polytope of feasible solutions the a trajectories of strict standard form linear programs have similar interpretations they are algebraic curves and are geodesics of a geometry isometric to euclidean geometry received by the editors july 28 1986 and in revised form september 28 1987 and march 21 1988 1980 mathematics subject classification 1985 revision primary 90c05 secondary 52a40 34a34 research of the first author partially supported by onr contract n00014 87 k0214 d 1989 american mathematical society 0002 9947 89 1 00 25 per page

on knapsacks partitions and a new dynamic programming technique for trees let g be an acyclic directed graph with weights and values assigned to its vertices in the partially ordered knapsack problem we wish to find a maximum valued subset of vertices whose total weight does not exceed a given knapsack capacity and which contains every predecessor of a vertex if it contains the vertex itself we consider the special case where g is an out tree even though this special case is still np complete we observe how dynamic programming techniques can be used to construct pseudopolynomial time optimization algorithms and fully polynomial time approximation schemes for it in particular we show that a nonstandard approach we call left right dynamic programming is better suited for this problem than the standard bottom up approach and we show how this left right approach can also be adapted to the case of in trees and to a related tree partitioning problem arising in integrated circuit design we conclude by presenting complexity results which indicate that similar success cannot be expected with either problem when the restriction to trees is lifted

minimizing symmetric submodular functions we describe a purely combinatorial algorithm which given a submodular set functionf on a finite setv finds a nontrivial subseta ofv minimizingf a f v a this algorithm an extension of the nagamochi ibaraki minimum cut algorithm as simplified by stoer and wagner m stoer f wagner a simple min cut algorithm proceedings of the european symposium on algorithms esa 94 lncs 855 springer berlin 1994 pp 141 147 and by frank a frank on the edge connectivity algorithm of nagamochi and ibaraki laboratoire artemis imag universite j fourier grenbole 1994 minimizes any symmetric submodular function using o v 3 calls to a function value oracle 1998 the mathematical programming society inc published by elsevier science b v

approximation algorithms for inventory problems with submodular or routing costs we consider the following two deterministic inventory optimization problems with non stationary demands submodular joint replenishment problem this involves multiple item types and a single retailer who faces demands over a finite planning horizon of t periods in each time period any subset of item types can be ordered incurring a joint ordering cost which is submodular moreover items can be held in inventory while incurring a holding cost the objective is to find a sequence of orders that satisfies all demands and minimizes the total ordering and holding costs inventory routing problem this involves a single depot that stocks items and multiple retailer locations facing demands over a finite planning horizon of t periods in each time period any subset of locations can be visited using a vehicle originating from the depot there is also cost incurred for holding items at any retailer the objective here is to satisfy all demands while minimizing the sum of routing and holding costs we present a unified approach that yields mathcal o left frac log t log log t right ologtloglogt factor approximation algorithms for both problems when the holding costs are polynomial functions a special case is the classic linear holding cost model wherein this is the first sub logarithmic approximation ratio for either problem

an algorithm for linear programming which requires o m n n 2 m n 1 5 n l arithmetic operations we present an algorithm for linear programming which requires o m n n 2 m n 1 5 n l arithmetic operations wherem is the number of constraints andn is the number of variables each operation is performed to a precision of o l bits l is bounded by the number of bits in the input the worst case running time of the algorithm is better than that of karmarkar s algorithm by a factor of sqrt m n

the nonlinear geometry of linear programming ii legendre transform coordinates and central trajectories karmarkar s projective scaling algorithm for solving linear programming problems associates to each objective function a vector field defined in the interior of the polytope of feasible solutions of the problem this paper studies the set of trajectories obtained by integrating this vector field called p trajectories as well as a related set of trajectories called a trajectories the a trajectories arise from another linear programming algorithm the affine scaling algorithm the affine and projective scaling vector fields are each defined for linear programs of a special form called standard form and canonical form respectively these trajectories are studied using a nonlinear change of variables called legendre transform coordinates which is a projection of the gradient of a logarithmic barrier function the legendre transform coordinate mapping is given by rational functions and its inverse mapping is algebraic it depends only on the constraints of the linear program and is a one to one mapping for canonical form linear programs when the polytope of feasible solutions is bounded there is a unique point mapping to zero called the center the a trajectories of standard form linear programs are linearized by the legendre transform coordinate mapping when the polytope of feasible solutions is bounded they are the complete set of geodesics of a riemannian geometry isometric to euclidean geometry each a trajectory is part of a real algebraic curve each p trajectory for a canonical form linear program lies in a plane in legendre transform coordinates the p trajectory through 0 in legendre transform coordinates called the central p trajectory is part of a straight line and is contained in the a trajectory through 0 called the central a trajectory each p trajectory is part of a real algebraic curve the central a trajectory is the locus of centers of a family of linear programs obtained by adding an extra equality constraint of the form c x u it is also the set of minima of a parametrized family of logarithmic barrier functions power series expansions are derived for the central a trajectory which is also the central p trajectory these power series have a simple recursive form and are useful in developing higher order analogues of karmarkar s algorithm a trajectories are defined for a general linear program using this definition it is shown that the limit point x 0 of a central a trajectory on the boundary of the feasible solution polytope p is the center of the unique face of p containing x 0 in its relative interior received by the editors october 8 1986 and in revised form june 9 1987 and march 25 1988 1980 mathematics subject classification 1985 revision primary 90c05 secondary 52a40 34a34 the first author was partially supported by onr contract n00014 87 k0214 1989 american mathematical society 0002 9947 89 1 00 25 per page

polymatroid optimization submodularity and joint replenishment games in this paper we consider the problem of maximizing a separable concave function over a polymatroid more specifically we study the submodularity of its optimal objective value in the parameters of the objective function this question is interesting in its own right and is encountered in many applications but our research has been motivated mainly by a cooperative game associated with the well known joint replenishment model by applying our general results on polymatroid optimization we prove that this cooperative game is submodular i e its characteristic cost function is submodular if the joint setup cost is a normalized and nondecreasing submodular function furthermore the same result holds true for a more general one warehouse multiple retailer game which affirmatively answers an open question posed by anily and haviv anily s m haviv 2007 the cost allocation problem for the first order interaction joint replenishment model oper res 55 2 292 302

a distributed algorithm for computing a common fixed point of a finite family of paracontractions a distributed algorithm is described for finding a common fixed point of a family of m 1 nonlinear maps m i mathbb r n rightarrow mathbb r n assuming that each map is a paracontraction and that at least one such common fixed point exists the common fixed point is simultaneously computed by m agents assuming each agent i knows only m i the current estimates of the fixed point generated by its neighbors and nothing more each agent recursively updates its estimate of a fixed point by utilizing the current estimates generated by each of its neighbors neighbor relations are characterized by a time varying directed graph mathbb n t it is shown under suitably general conditions on mathbb n t that the algorithm causes all agent estimates to converge to the same common fixed point of the m nonlinear maps

an improved benders decomposition algorithm for the tree of hubs location problem abstract the tree of hubs location problem is a particularly hard variant of the so called hub location problems when solving this problem by a benders decomposition approach it is necessary to deal with both optimality and feasibility cuts while modern implementations of the benders decomposition method rely on pareto optimal optimality cuts or on rendering feasibility cuts based on minimal infeasible subsystems a new cut selection scheme is devised here under the guiding principle of extracting useful information even when facing infeasible subproblems the proposed algorithm outperforms two other modern variants of the method and it is capable of optimally solving instances five times larger than the ones previously reported on the literature

on the complexity of quantified linear systems in this paper we explore the computational complexity of the conjunctive fragment of the first order theory of linear arithmetic quantified propositional formulas of linear inequalities with k 1 quantifier alternations are log space complete in s k p or p k p depending on the initial quantifier we show that when we restrict ourselves to quantified conjunctions of linear inequalities i e quantified linear systems the complexity classes collapse to polynomial time in other words the presence of universal quantifiers does not alter the complexity of the linear programming problem which is known to be in p our result reinforces the importance of sentence formats from the perspective of computational complexity

the contact invariant in sutured floer homology we describe an invariant of a contact 3 manifold with convex boundary as an element of juhasz s sutured floer homology our invariant generalizes the contact invariant in heegaard floer homology in the closed case due to ozsvath and szabo

long step strategies in interior point primal dual methods in this paper we analyze from a unique point of view the behavior of path following and primal dual potential reduction methods on nonlinear conic problems we demonstrate that most interior point methods with o root n ln 1 epsilon efficiency estimate can be considered as different strategies of minimizing a convex primal dual potential function in an extended primal dual space their efficiency estimate is a direct consequence of large local norm of the gradient of the potential function along a central path it is shown that the neighborhood of this path is a region of the fastest decrease of the potential therefore the long step path following methods are in a sense the best potential reduction strategies we present three examples of such long step strategies we prove also an efficiency estimate for a pure primal dual potential reduction method which can be considered as an implementation of a penalty strategy based on a functional proximity measure using the convex primal dual potential we prove efficiency estimates for karmarkar type and dikin type methods as applied to a homogeneous reformulation of the initial primal dual problem

random walks on polytopes and an affine interior point method for linear programming let k be a polytope in rn defined by m linear inequalities we give a new markov chain algorithm to draw a nearly uniform sample from k the underlying markov chain is the first to have a mixing time that is strongly polynomial when started from a central point we use this result to design an affine interior point algorithm that does a single random walk to solve linear programs approximately

network coding a computational perspective in this work we study the computational perspective of network coding focusing on two issues first we address the computational complexity of finding a network code for acyclic multicast networks second we address the issue of reducing the amount of computation performed by network nodes in particular we consider the problem of finding a network code with the minimum possible number of encoding nodes i e nodes that generate new packets by performing algebraic operations on packets received over incoming links we present a deterministic algorithm that finds a feasible network code for a multicast network over an underlying graph g v e in time 0 e kh v k2h2 h4k3 k h where k is the number of destinations and h is the number of packets our algorithm improves the best known running time for network code construction in addition our algorithm guarantees that the number of encoding nodes in the obtained network code is upper bounded by 0 h3k2 next we address the problem of finding integral and fractional network codes with the minimum number of encoding nodes we prove that in the majority of settings this problem is np hard however we show that if h o 1 k o 1 and the underlying communication graph is acyclic then there exists an algorithm that solves this problem in polynomial time

a potential reduction variant of renegar s short step path following method for linear programming abstract we propose a new polynomial potential reduction method for linear programming which can also be seen as a large step path following method we do an approximate linesearch along the newton direction with respect to renegar s strictly convex potential function if the iterate is far away from the central trajectory if the iterate lies close to the trajectory we update the lower bound for the optimal value dependent on this updating scheme the iteration bound can be proved to be o n l or o nl our method differs from the recently published potential reduction methods in the choice of the potential function and the search direction

holomorphic disks and knot invariants abstract we define a floer homology invariant for knots in an oriented three manifold closely related to the heegaard floer homologies for three manifolds defined in an earlier paper we set up basic properties of these invariants including an euler characteristic calculation and a description of the behavior under connected sums then we establish a relationship with hf for surgeries along the knot applications include calculation of hf of three manifolds obtained by surgeries on some special knots in s 3 and also calculation of hf for certain simple three manifolds which fiber over the circle

reaching a consensus in a dynamically changing environment a graphical approach this paper presents new graph theoretic results appropriate for the analysis of a variety of consensus problems cast in dynamically changing environments the concepts of rooted strongly rooted and neighbor shared are defined and conditions are derived for compositions of sequences of directed graphs to be of these types the graph of a stochastic matrix is defined and it is shown that under certain conditions the graph of a sarymsakov matrix and a rooted graph are one and the same as an illustration of the use of the concepts developed in this paper graph theoretic conditions are obtained which address the convergence question for the leaderless version of the widely studied vicsek consensus problem

an arrow hurwicz uzawa type flow as least squares solver for network linear equations we study the approach to obtaining least squares solutions to systems of linear algebraic equations over networks by using distributed algorithms each node has access to one of the linear equations and holds a dynamic state the aim for the node states is to reach a consensus as a least squares solution of the linear equations by exchanging their states with neighbors over an underlying interaction graph a continuous time distributed least squares solver over networks is developed in the form of the famous arrow hurwicz uzawa flow a necessary and sufficient condition is established on the graph laplacian for the continuous time distributed algorithm to give the least squares solution in the limit with an exponentially fast convergence rate the feasibility of different fundamental graphs is discussed including path graph and random graph moreover a discrete time distributed algorithm is developed by euler s method converging exponentially to the least squares solution at the node states with suitable step size and graph conditions

holomorphic disks and topological invariants for closed three manifolds the aim of this article is to introduce certain topological invariants for closed oriented three manifolds y equipped with a spiny structure given a heegaard splitting of y u 0o u u 1 these theories are variants of the lagrangian floer homology for the g fold symmetric product of relative to certain totally real subspaces associated to u 0 and u 1

distributed algorithms for computing a common fixed point of a group of nonexpansive operators this paper addresses the problem of seeking a common fixed point for a collection of nonexpansive operators over time varying multi agent networks in real hilbert spaces each operator is only privately and approximately known to each individual agent and all agents need to cooperate to solve this problem by local communications over time varying networks to handle this problem inspired by the centralized inexact krasnosel skii mann ikm iteration we propose a distributed algorithm called distributed inexact averaged operator algorithm d iao it is shown that the d iao algorithm can converge weakly to a common fixed point of the family of nonexpansive operators moreover under the linearly regular assumption it is proved that the d iao algorithm converges with a rate o gamma log 16 4k for some constant gamma in 0 1 where k is the iteration number the same converge speed can be ensured by a novel power regularity condition which is strictly weaker than linear regularity to reduce computational complexity another algorithm called d biao is developed and fully analyzed simulation examples are given to support the theoretical res

distributed least squares solver for network linear equations in this paper we study the problem of finding the least square solutions of over determined linear algebraic equations over networks in a distributed manner each node has access to one of the lin

agreeing asynchronously this paper formulates and solves a version of the widely studied vicsek consensus problem in which each member of a group of n 1 agents independently updates its heading at times determined by its own clock it is not assumed that the agents clocks are synchronized or that the ldquoeventrdquo times between which any one agent updates its heading are evenly spaced nor is it assumed that heading updates must occur instantaneously using the concept of ldquoanalytic synchronizationrdquo together with several key results concerned with properties of ldquocompositionsrdquo of directed graphs it is shown that the conditions under which a consensus is achieved are essentially the same as those applicable in the synchronous case provided the notion of an agent s neighbor between its event times is appropriately defined however in sharp contrast with the synchronous case where for analysis an n dimensional state space model is adequate for the asynchronous version of the problem a 2n dimensional state space model is required it is explained how to analyze this model despite the fact that unlike the synchronous case the stochastic matrices involved do not have all positive diagonal entries

convergence rate on periodic gossiping in a sensor network in which each sensor controls a real valued state the goal of a distributed averaging problem is to compute the global average in a decentralized way which is the average of all sensors initial state values across the entire network a t periodic gossiping protocol can solve such a problem which stipulates that each agent must gossip with each of its neighbors exactly once every t time unit the convergence rate of a t periodic gossiping protocol is determined by the magnitude of the second largest eigenvalue of the stochastic matrix associated with the gossip sequence occurring over one period an interesting result is that when the allowable gossip graph is a tree the convergence rate is independent of gossip orders within one period this paper will prove this result by developing several properties of doubly stochastic matrices the properties derived also can be used in analyzing convergence rate problems of other periodic gossip protocols

reprint of a distributed algorithm for efficiently solving linear equations and its applications special issue jcw abstract a distributed algorithm is proposed for solving a linear algebraic equation a x b over a multi agent network where a r n n and the equation has a unique solution x r n each agent knows only a subset of the rows of a b controls a state vector x i t of size smaller than n and is able to receive information from its nearby neighbors neighbor relations are characterized by time dependent directed graphs it is shown that for a large class of time varying networks the proposed algorithm enables each agent to recursively update its own state by only using its neighbors states such that all x i t converge exponentially fast to a specific part of x of interest to agent i applications of the proposed algorithm include solving the least square solution problem and the network localization problem

consensus control for linear systems with optimal energy cost in this paper we design an optimal energy cost controller for linear systems asymptotic consensus given the topology of the graph the controller depends only on relative information of the agents

fast distributed gradient methods we study distributed optimization problems when n nodes minimize the sum of their individual costs subject to a common vector variable the costs are convex have lipschitz continuous gradient with constant l and bounded gradient we propose two fast distributed gradient algorithms based on the centralized nesterov gradient algorithm and establish their convergence rates in terms of the per node communications k and the per node gradient evaluations k our first method distributed nesterov gradient achieves rates o logk k and o logk k our second method distributed nesterov gradient with consensus iterations assumes at all nodes knowledge of l and w the second largest singular value of the n n doubly stochastic weight matrix w it achieves rates o 1 k2 and o 1 k2 0 arbitrarily small further we give for both methods explicit dependence of the convergence constants on n and w simulation examples illustrate our findings

holomorphic triangles and invariants for smooth four manifolds abstract in earlier articles the authors introduced invariants for closed oriented three manifolds defined using a variant of lagrangian floer homology in the symmetric products of riemann surfaces the aim of this article is to introduce invariants of oriented smooth four manifolds built using these floer homology groups this four dimensional theory also endows the corresponding three dimensional theories with additional structure an absolute grading of certain of its floer homology groups

equivalence of heegaard floer homology and embedded contact homology via open book decompositions we sketch the proof of the equivalence between the hat versions of heegaard floer homology and embedded contact homology ech the key point is to express these two floer homology theories in terms of an open book decomposition of the ambient manifold

khovanov homology open books and tight contact structures abstract we define the reduced khovanov homology of an open book s and identify a distinguished contact element in this group which may be used to establish the tightness or non fillability of contact structures compatible with s our construction generalizes the relationship between the reduced khovanov homology of a link and the heegaard floer homology of its branched double cover as an application we give combinatorial proofs of tightness for several contact structures which are not stein fillable lastly we investigate a comultiplication structure on the reduced khovanov homology of an open book which parallels the comultiplication on heegaard floer homology defined in baldwin 2008 4

solving linear programming problems exactly a linear programming problem in an inequality form having a bounded solution is solved error free using an algorithm that sorts the inequalities removes the redundant ones and uses the p adic arithmetic c elsevier science inc 1997

stability of multiagent systems with time dependent communication links we study a simple but compelling model of network of agents interacting via time dependent communication links the model finds application in a variety of fields including synchronization swarming and distributed decision making in the model each agent updates his current state based upon the current information received from neighboring agents necessary and or sufficient conditions for the convergence of the individual agents states to a common value are presented thereby extending recent results reported in the literature the stability analysis is based upon a blend of graph theoretic and system theoretic tools with the notion of convexity playing a central role the analysis is integrated within a formal framework of set valued lyapunov theory which may be of independent interest among others it is observed that more communication does not necessarily lead to faster convergence and may eventually even lead to a loss of convergence even for the simple models discussed in the present paper

computational complexity of some maximum average weight problems with precedence constraints maximum average weight ideal problems in ordered sets arise from modeling variants of the investment problem and in particular learning problems in the context of concepts with tree structured attributes in artificial intelligence similarly trying to construct tests with high reliability leads to a nontrivial maximum average weight ideal problem this paper investigates the computational complexity and shows that the general problem is np complete important special cases e g finding rooted subtrees of maximal average weight however can be handled with efficient algorithms