the lagrangian relaxation method for solving integer programming problems this article originally appeared in management science january 1981 volume 27 number 1 pp 1 18 published by the institute of management sciences one of the most computationally useful ideas of the 1970s is the observation that many hard integer programming problems can be viewed as easy problems complicated by a relatively small set of side constraints dualizing the side constraints produces a lagrangian problem that is easy to solve and whose optimal value is a lower bound for minimization problems on the optimal value of the original problem the lagrangian problem can thus be used in place of a linear programming relaxation to provide bounds in a branch and bound algorithm this approach has led to dramatically improved algorithms for a number of important problems in the areas of routing location scheduling assignment and set covering this paper is a review of lagrangian relaxation based on what has been learned in the last decade

algorithms for a multi level network optimization problem in this paper we work on a multi level network optimization problem that integrates into the same model important aspects of i discrete facility location ii topological network design and iii network dimensioning potential applications for the model are discussed stressing its growing importance the multi level network optimization problem treated is defined and a mathematical programming formulation is presented we make use of a branch and bound algorithm based on lagrangean relaxation lower bounds to introduce some new powerful auxiliary algorithms to exactly solve the problem we conduct a set of computational experiments that indicate the quality of the proposed approach

dynamic vehicle routing with priority classes of stochastic demands in this paper we introduce a dynamic vehicle routing problem in which there are multiple vehicles and multiple priority classes of service demands service demands of each priority class arrive in the environment randomly over time and require a random amount of on site service that is characteristic of the class to service a demand one of the vehicles must travel to the demand location and remain there for the required on site service time the quality of service provided to each class is given by the expected delay between the arrival of a demand in the class and that demand s service completion the goal is to design a routing policy for the service vehicles which minimizes a convex combination of the delays for each class first we provide a lower bound on the achievable values of the convex combination of delays then we propose a novel routing policy and analyze its performance under heavy load conditions i e when the fraction of time the service vehicles spend performing on site service approaches one the policy performs within a constant factor of the lower bound where the constant depends only on the number of classes and is independent of the number of vehicles the arrival rates of demands the on site service times and the convex combination coefficients

adaptive methods for the computation of pagerank we observe that the convergence patterns of pages in the pagerank algorithm have a nonuniform distribution specifically many pages converge to their true pagerank quickly while relatively few pages take a much longer time to converge furthermore we observe that these slow converging pages are generally those pages with high pagerank we use this observation to devise a simple algorithm to speed up the computation of pagerank in which the pagerank of pages that have converged are not recomputed at each iteration after convergence this algorithm which we call adaptive pagerank speeds up the computation of pagerank by nearly 30

the steiner tree polytope and related polyhedra we consider the vertex weighted version of the undirected steiner tree problem in this problem a cost is incurred both for the vertices and the edges present in the steiner tree we completely describe the associated polytope by linear inequalities when the underlying graph is series parallel for general graphs this formulation can be interpreted as a partial extended formulation for the steiner tree problem by projecting this formulation we obtain some very large classes of facet defining valid inequalities for the steiner tree polytope

updating kernel methods in spectral decomposition by affinity perturbations abstract many machine learning based algorithms contain a training step that is done once the training step is usually computational expensive since it involves processing of huge matrices if the training profile is extracted from an evolving dynamic dataset it has to be updated as some features of the training dataset are changed this paper proposes a solution how to update this profile efficiently therefore we investigate how to update the training profile when the data is constantly evolving we assume that the data is modeled by a kernel method and processed by a spectral decomposition in many algorithms for clustering and classification a low dimensional representation of the affinity kernel graph of the embedded training dataset is computed then it is used for classifying newly arrived data points we present methods for updating such embeddings of the training datasets in an incremental way without the need to perform the entire computation upon the occurrences of changes in a small number of the training samples efficient computation of such an algorithm is critical in many web based applications

a meta heuristic approach for solving the urban network design problem abstract this paper proposes an optimisation model and a meta heuristic algorithm for solving the urban network design problem the problem consists in optimising the layout of an urban road network by designing directions of existing roads and signal settings at intersections a non linear constrained optimisation model for solving this problem is formulated adopting a bi level approach in order to reduce the complexity of solution methods and the computation times a scatter search algorithm based on a random descent method is proposed and tested on a real dimension network initial results show that the proposed approach allows local optimal solutions to be obtained in reasonable computation times

topological design of computer communication networks the overall design problem abstract mathematical formulations of the topological design problem of computer communication networks are developed the topological design of computer networks involves decisions on where to place network control processors ncps selecting the set of backbone links to connect ncps linking end user nodes to the ncps and deciding on the set of routes which support communications between communicating end user node pairs the overall design problem is formulated as a nonlinear combinatorial optimization problem a lagrangean relaxation of the problem is presented and effective solution procedures of the lagrangean problem are developed the lagrangean solutions provide lower bounds on the optimal solutions to the complete problem the lagrangean based solutions are further improved using subgradient optimization procedures three heuristics are developed for generating feasible solutions to the problem the heuristic solutions are demonstrated on problems involving 200 end user nodes and up to 30 ncp locations

a heuristic lagrangean algorithm for the capacitated plant location problem abstract lagrangean techniques have been widely applied to the uncapacitated plant location problem and in some cases they have proven to be successfull even when capacitated problems with additional constraints are taken into account in our paper we study the application of these techniques to the capacitated plant location problem when the model considered is a pure integer one several lagrangean decompositions are considered and for some of them heuristic algorithms have been designed to solve the resulting lagrangean subproblems the heuristics consisting of a two phase procedure the first location phase defines a set of multipliers from the analysis of the dual lp relaxation and makes a choice of the plants considering the resulting subproblems as a particular case of the general assignment problems several heuristics have been studied for this second phase based either on a decomposition of knapsack type subproblems through a definition of a set of penalties or of looking into the duality gap and trying to reduce it computational experience is reported

probabilistic and worst case analyses of classical problems of combinatorial optimization in euclidean space the classical problems reviewed are the traveling salesman problem minimal spanning tree minimal matching greedy matching minimal triangulation and others each optimization problem is considered for finite sets of points in d and the feature of principal interest is the value of the associated objective function special attention is given to the asymptotic behavior of this value under probabilistic assumptions but both probabilistic and worst case analyses are surveyed

a tree search algorithm for the p median problem abstract in this paper we present two lower bounds for the p median problem the problem of locating p facilities medians on a network these bounds are based on two separate lagrangean relaxations of a zero one formulation of the problem with subgradient optimisation being used to maximise these bounds penalty tests based on these lower bounds and a heuristically determined upper bound to the problem are developed and shown to result in a large reduction in problem size the incorporation of the lower bounds and the penalty tests into a tree search procedure is described and computational results are given for problems with an arbitrary number of medians and having up to 200 vertices a comparison is also made between these algorithms and the dual based algorithm of erlenkotter

reducing the hierarchical network design problem abstract the hierarchical network design problem hndp can be seen as a minimum spanning tree problem involving two weight functions it is shown that the hndp is a special case of the directed steiner tree problem theorems are given to obtain optimal arcs that are used in a lagrangean relaxation of the linear programming model reduction tests reducing the size of the problem graph and or eliminating variables from this program are specialized and developed for the hndp the presented techniques are illustrated on example problems from the literature and computational results for the reduction tests are given

derivatives and perturbations of eigenvectors for a matrix bf a z whose entries are complex valued functions of a complex variable z results are presented concerning derivatives of an eigenvector bf x z of bf a z associated with a simple eigenvalue lambda z when bf x z is restricted to satisfy a constraint of the form sigma bf x z 1 where sigma is a rather arbitrary scaling function the differentiation formulas lead to a new approach for analyzing the sensitivity of an eigenvector under small perturbations in the underlying matrix finally an application is given which concerns the analysis of a finite markov chain subject to perturbations in the transition probabilities

natural gas cash out problem bilevel stochastic optimization approach a stochastic formulation of the natural gas cash out problem is given in a form of a bilevel multi stage stochastic programming model with recourse after reducing the original formulation to a bilevel linear problem a stochastic scenario tree is defined by its node events and time series forecasting is used to produce stochastic values for data of natural gas price and demand numerical experiments were run to compare the stochastic solution with the perfect information solution and the expected value solutions

a dual based algorithm for multi level network design given an undirected network with l possible facility types for each edge and a partition of the nodes into l levels or grades the multi level network design mlnd problem seeks a fixed cost minimizing design that spans all the nodes and connects the nodes at each level by facilities of the corresponding or higher grade this problem generalizes the well known steiner network problem and the hierarchical network design problem and has applications in telecommunication transportation and electric power distribution network design in a companion paper we studied alternative model formulations for a two level version of this problem and analyzed the worst case performance of several heuristics based on steiner network and spanning tree solutions this paper develops a dual based algorithm for the mlnd problem the method first performs problem preprocessing to fix certain design variables and then applies a dual ascent procedure to generate upper and lower bounds on the optimal value we report extensive computational results on large random two level test problems containing up to 500 nodes and 5 000 edges with varying cost structures the integer programming formulation of the largest of these problems has 20 000 integer variables and over 5 million constraints our tests indicate that the dual based algorithm is very effective producing solutions that are within 0 9 of optimality

incremental nonlinear dimensionality reduction by manifold learning understanding the structure of multidimensional patterns especially in unsupervised cases is of fundamental importance in data mining pattern recognition and machine learning several algorithms have been proposed to analyze the structure of high dimensional data based on the notion of manifold learning these algorithms have been used to extract the intrinsic characteristics of different types of high dimensional data by performing nonlinear dimensionality reduction most of these algorithms operate in a batch mode and cannot be efficiently applied when data are collected sequentially in this paper we describe an incremental version of isomap one of the key manifold learning algorithms our experiments on synthetic data as well as real world images demonstrate that our modified algorithm can maintain an accurate low dimensional representation of the data in an efficient manner

modeling and heuristic worst case performance analysis of the two level network design problem this paper studies a multi facility network synthesis problem called the two level network design tlnd problem that arises in the topological design of hierarchical communication transportation and electric power distribution networks we are given an undirected network containing two types of nodes primary and secondary and fixed costs for installing either a primary or a secondary facility on each edge primary nodes require higher grade interconnections than secondary nodes using the more expensive primary facilities the tlnd problem seeks a minimum cost connected design that spans all the nodes and connects primary nodes via edges containing primary facilities the design can use either primary or secondary edges to connect the secondary nodes the tlnd problem generalizes the well known steiner network problem and the hierarchical network design problem in this paper we study the relationship between alternative model formulations for this problem e g directed and undirected models an

the hierarchical network design problem abstract in this paper the authors introduce the hierarchical network design problem hndp the object of the hndp is to identify the least cost two level hierarchical network the network must include a primary path from a predetermined starting node to a predetermined terminus node in addition each node not on the primary path must be connected to some node on that path by means of a secondary path the problem is initially formulated as an integer linear program an heuristic is then presented which employs a k shortest path algorithm and a minimum spanning tree algorithm heuristic results of two sample problems are presented and compared to the results obtained by solving the integer lp formulation potential applications of the formulation are also discussed

when trees collide an approximation algorithm for thegeneralized steiner problem on networks we give the first approximation algorithm for the generalized network steiner problem a problem in network design an instance consists of a network with link costs and for each pair i j of nodes an edge connectivity requirement r ij the goal is to find a minimum cost network using the available links and satisfying the requirements our algorithm outputs a solution whose cost is within 2 lceil log 2 r 1 rceil of optimal where r is the highest requirement value in the course of proving the performance guarantee we prove a combinatorial min max approximate equality relating minimum cost networks to maximum packings of certain kinds of cuts as a consequence of the proof of this theorem we obtain an approximation algorithm for optimally packing these cuts we show that this algorithm has application to estimating the reliability of a probabilistic network

heuristics with constant error guarantees for the design of tree networks a tree network is a collection of trees rooted at a common central node several types of network design problems can be viewed as requiring the formation of a spanning tree network of minimum length subject to a bound on the sum of weights on the nodes of any component tree such problems are np complete and experience has shown that only small examples can be solved to optimality this paper describes an efficient heuristic algorithm based on partitioning of a traveling salesman tour when all the nodes have a unit weight and the bound is k the heuristic finds a solution whose cost is at most 3 2 k times the minimum in the general case the error bound is 4

approximation algorithms for orienteering and discounted reward tsp in this paper we give the first constant factor approximation algorithm for the rooted orienteering problem as well as a new problem that we call the discounted reward traveling salesman problem tsp motivated by robot navigation in both problems we are given a graph with lengths on edges and rewards on nodes and a start node s in the orienteering problem the goal is to find a path starting at s that maximizes the reward collected subject to a hard limit on the total length of the path in the discounted reward tsp instead of a length limit we are given a discount factor gamma and the goal is to maximize the total discounted reward collected where the reward for a node reached at time t is discounted by gamma t this problem is motivated by an approximation to a planning problem in the markov decision process mdp framework under the commonly employed infinite horizon discounted reward optimality criterion the approximation arises from a need to deal with exponentially large state spaces that emerge when trying to model one time events and nonrepeatable rewards such as for package deliveries we also consider tree and multiple path variants of these problems and provide approximations for those as well although the unrooted orienteering problem where there is no fixed start node s has been known to be approximable using algorithms for related problems such as k tsp in which the amount of reward to be collected is fixed and the total length is approximately minimized ours is the first to approximate the rooted question solving an open problem in e m arkin j s b mitchell and g narasimhan proceedings of the 14 th acm symposium on computational geometry 1998 pp 307 316 and b awerbuch y azar a blum and s vempala siam j comput 28 1998 pp 254 262 we complement our approximation result for orienteering by showing that the problem is apx hard

a dual ascent approach for steiner tree problems on a directed graph the steiner tree problem on a directed graph stdg is to find a directed subtree that connects a root node to every node in a designated node setv we give a dual ascent procedure for obtaining lower bounds to the optimal solution value the ascent information is also used in a heuristic procedure for obtaining feasible solutions to the stdg computational results indicate that the two procedures are very effective in solving a class of stdg s containing up to 60 nodes and 240 directed 120 undirected arcs