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

two edge connected spanning subgraphs and polyhedra this paper studies the problem of finding a two edge connected spanning subgraph of minimum weight this problem is closely related to the widely studied traveling salesman problem and has applications to the design of reliable communication and transportation networks we discuss the polytope associated with the solutions to this problem we show that when the graph is series parallel the polytope is completely described by the trivial constraints and the so called cut constraints we also give some classes of facet defining inequalities of this polytope when the graph is general

on the stable set polytope of a series parallel graph we give a short proof of chvatal s conjecture that the nontrivial facets of the stable set polytope of a series parallel graph all come from edges and odd holes

a note on k theory and triangulated categories we provide an example of two closed model categories having equivalent homotopy categories but different waldhausen k theories we also show that there cannot exist a functor from small triangulated categories to spaces which recovers quillen s k theory for exact categories and which satisfies localization

the steiner tree problem i formulations compositions and extension of facets in this paper we give some integer programming formulations for the steiner tree problem on undirected and directed graphs and study the associated polyhedra we give some families of facets for the undirected case along with some compositions and extensions we also give a projection that relates the steiner tree polyhedron on an undirected graph to the polyhedron for the corresponding directed graph this is used to show that the lp relaxation of the directed formulation is superior to the lp relaxation of the undirected one

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

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

postnikov towers k invariants and obstruction theory for dg categories abstract by inspiring ourselves in drinfeld s dg quotient we develop postnikov towers k invariants and an obstruction theory for dg categories as an application we obtain the following rigidification theorem let a be a homologically connective dg category and f 0 b h 0 a a dg functor to its homotopy category if the inductive family n f n n 0 of obstruction classes vanishes then a lift f b a for f 0 exists

convergence of a crystalline approximation for an area preserving motion we consider an approximation of area preserving motion in the plane by a generalized crystalline motion the area preserving motion is described by a parabolic partial differential equation with a nonlocal term while the crystalline motion is governed by a system of ordinary differential equations we show the convergence between these two motions the convergence theorem is proved in two steps first an a priori estimate is established for a solution to the generalized crystalline motion second a discrete w1 p norms of the error is estimated for all 1 p and passing p to infinity a discrete w1 error estimate is obtained we also construct an implicit scheme which enjoys several nice properties such as the area preserving and curve shortening and compare our scheme with a simple scheme

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

polyhedral characterization of discrete dynamic programming many interesting combinatorial problems can be optimized efficiently using recursive computations often termed discrete dynamic programming in this paper we develop a paradigm for a general class of such optimizations that yields a polyhedral description for each model in the class the elementary concept of dynamic programs as shortest path problems in acyclic graphs is generalized to one seeking a least cost solution in a directed hypergraph sufficient conditions are then provided for binary integrality of the associated hyperflow problem given a polynomially solvable dynamic program the result is a linear program in polynomially many variables and constraints having the solution vectors of the dynamic program as its extreme point optima that is the linear program provides a succinct characterization of the solutions to the underlying optimization problem expressed through an appropriate change of variables we also discuss projecting this formulation to recover constraints on the original variables and illustrate how some important dynamic programming solvable models fit easily into our paradigm a classic multiechelon lot sizing problem in production and a host of optimization problems on recursively defined classes of graphs are included

stochastic approximations to curve shortening flows via particle systems abstract curvature driven flows have been extensively considered from a deterministic point of view besides their mathematical interest they have been shown to be useful for a number of applications including crystal growth flame propagation and computer vision in this paper we describe a random particle system evolving on the discretized unit circle whose profile converges toward the gauss minkowsky transformation of solutions of curve shortening flows initiated by convex curves our approach may be considered as a type of stochastic crystalline algorithm our proofs are based on certain techniques from the theory of hydrodynamical limits

manifold evolution with tangential redistribution of points in this paper we propose several techniques for tangential redistribution of points on evolving surfaces this is an important issue in numerical approximation of any lagrangian evolution model since the quality of the mesh has a significant impact on the result of the computation we explain the volume oriented and length oriented tangential redistribution methods in a general setting of an m dimensional manifold evolving in an n dimensional manifold then we apply the proposed techniques to several manifold evolution problems we explain the numerical approximation of the models and present experiments illustrating the performance of our redistribution techniques

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

convergence of a crystalline algorithm for the motion of a closed convex curve by a power of curvature v k alpha a finite difference scheme is constructed for the evolution of a plane curve driven by a power of curvature v k alpha alpha 0 where v and k are the normal velocity and the curvature respectively here we assume that the curve is closed strictly convex and immersed in the plane r2 this curve evolution is discretized using a crystalline approximation which is a kind of finite difference scheme for the nonlinear evolution equation we prove convergence of this scheme and show the rate of convergence moreover some numerical examples are presented

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

the traveling salesman problem on a graph and some related integer polyhedra given a graphg n e and a length functionl e the graphical traveling salesman problem is that of finding a minimum length cycle goingat least once through each node ofg this formulation has advantages over the traditional formulation where each node must be visited exactly once we give some facet inducing inequalities of the convex hull of the solutions to that problem in particular the so called comb inequalities of grotschel and padberg are generalized some related integer polyhedra are also investigated finally an efficient algorithm is given wheng is a series parallel graph

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

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

on the bolw up rate for fast bolw up solutions arising in an anisotropic crystalline motion we consider the asymptotic behavior of motion of polygonal convex curves by crystalline curvature in the plane there appear spontaneously two types of singularity one is single point extinction and the other is degenerate pinching we mainly discuss degenerate pinching singularity and show the exact blow up rate for a fast blow up solution which arises in an equivalent blow up problem

a partition based relaxation for steiner trees the steiner tree problem is a classical np hard optimization problem with a wide range of practical applications in an instance of this problem we are given an undirected graph g v e a set of terminals r subseteq v and non negative costs c e for all edges e in e any tree that contains all terminals is called a steiner tree the goal is to find a minimum cost steiner tree the vertices v backslash r are called steiner vertices the best approximation algorithm known for the steiner tree problem is a greedy algorithm due to robins and zelikovsky siam j discrete math 19 1 122 134 2005 it achieves a performance guarantee of 1 frac ln 3 2 approx 1 55 the best known linear programming lp based algorithm on the other hand is due to goemans and bertsimas math program 60 145 166 1993 and achieves an approximation ratio of 2 2 r in this paper we establish a link between greedy and lp based approaches by showing that robins and zelikovsky s algorithm can be viewed as an iterated primal dual algorithm with respect to a novel lp relaxation the lp used in the first iteration is stronger than the well known bidirected cut relaxation an instance is b quasi bipartite if each connected component of g backslash r has at most b vertices we show that robins and zelikovsky s algorithm has an approximation ratio better than 1 frac ln 3 2 for such instances and we prove that the integrality gap of our lp is between frac 8 7 and frac 2b 1 b 1

a crystalline motion uniqueness and geometric properties the author studies a model of crystalline motion in the plane existence and uniqueness of local in time solutions are shown geometric properties of the flow are studied assuming that the wulff shape is a regular n sided polygon the author shows that a small convex grain of ice immersed in the cold melt shrinks provided that the flow does not overly deform the initial interface the author is able to verify the last condition only if the initial interface is a scaled wulff shape in this case the free boundary will remain a scaled wulff shape at later times this is shown by using an isoperimetric inequality

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

a crystalline motion of spiral shaped curves with symmetry we study the evolution of spiral shaped polygonal curves by crystalline curvature crystalline curvature is known to extend the notion of ordinary curvature to the special class of nonsmooth curves with the assumption of symmetry we show that the motion of our spirals can be analyzed up to the rest state possibly beyond singularities

rigidity and exotic models for v1 local g equivariant stable homotopy we prove that the v 1 local g equivariant stable homotopy category for g a finite group has a unique g equivariant model at p 2 this means that at the prime 2 the homotopy theory of g spectra up to fixed point equivalences on k theory is uniquely determined by its triangulated homotopy category and basic mackey structure the result combines the rigidity result for k local spectra of the second author with the equivariant rigidity result for g spectra of the first author further when the prime p is at least 5 and does not divide the order of g we provide an algebraic exotic model as well as a g equivariant exotic model for the v 1 local g equivariant stable homotopy category showing that for primes p ge 5 equivariant rigidity fails in general

multiphase thermomechanics with interfacial structure 2 evolution of an isothermal interface paper 1 1988 g 1 of this series began an investigation whose goal is a thermomechanics of two phase continua based on gibbs s notion of a sharp phase interface endowed with thermomechanical structure in that paper a balance law balance of capillary forces was introduced and then applied in conjunction with suitable statements of the first two laws of thermodynamics the chief results are thermodynamic restrictions on constitutive equations exact and approximate free boundary conditions at the interface and a hierarchy of free boundary problems the simplest versions of these problems the mullins sekerka problems are essentially the classical stefan problem with the free boundary condition u 0 for the temperature replaced by the condition u h k where k is the curvature of the free boundary and h 0 is a material constant this dependence on curvature renders the problem difficult and apart from numerical studies involving linearization stability there are almost no supporting theoretical results

autour de la cohomologie de maclane des corps finis nous decrivons une nouvelle methode de calcul de la cohomologie de maclane des corps finis cette theorie est intimement reliee aux extensions du groupe additif deja etudiees par l breen et a l homologie de hochschild topologique de m bokstedt et donc a la k theorie stable notre approche utilise de maniere cruciale l annulation de la cohomologie de maclane du corpsf p avec pour coefficients l algebre symetrique ou l on a inverse le frobenius nous recourons alors a l analyse des complexes de koszul et de de rham en caracteristique non nulle

algebras and modules in monoidal model categories in recent years the theory of structured ring spectra formerly known as a infty and e infty ring spectra has been simplified by the discovery of categories of spectra with strictly associative and commutative smash products now a ring spectrum can simply be defined as a monoid with respect to the smash product in one of these new categories of spectra in this paper we provide a general method for constructing model category structures for categories of ring algebra and module spectra this provides the necessary input for obtaining model categories of symmetric ring spectra functors with smash product gamma rings and diagram ring spectra algebraic examples to which our methods apply include the stable module category over the group algebra of a finite group and unbounded chain complexes over a differential graded algebra 1991 mathematics subject classification primary 55u35 secondary 18d10

combinatorial model categories have presentations abstract we show that every combinatorial model category is quillen equivalent to a localization of a diagram category where diagram category means diagrams of simplicial sets this says that every combinatorial model category can be built from a category of generators and a set of relations

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

on a perimeter preserving crystalline flow abstract the behavior of admissible polygons moving under a crystalline curvature flow is discussed it is proved that the perimeter of the evolving polygon is preserved its area is increasing its crystalline curvature is uniformly bounded under this flow and the final shape of the solution polygon is the boundary of a wulff shape

topological equivalences for differential graded algebras we investigate the relationship between differential graded algebras dgas and topological ring spectra every dga c gives rise to an eilenberg mac lane ring spectrum denoted hc if hc and hd are weakly equivalent then we say c and d are topologically equivalent quasi isomorphic dgas are topologically equivalent but we produce explicit counterexamples of the converse we also develop an associated notion of topological morita equivalence using a homotopical version of tilting

the heat equation shrinks embedded plane curves to round points soit c 0 s 1 r 2 une courbe lisse plongee dans le plan alors c s 1 0 t r 2 existe en satisfaisant c t k n ou k est la courbure de c et n est son vecteur unite normal entrant c t est lisse pour tout t il converge vers un point quand t t et sa forme limite quand t t est un cercle rond avec convergence dans norme c

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

convergence of a crystalline algorithm for the heat equation in one dimension and for the motion of a graph by weighted curvature motion by weighted mean curvature is a geometric evolution law for surfaces representing steepest descent with respect to an isotropic surface energy it has been proposed that this motion could be computed by solving the analogous evolution law using a crystalline approximation to the surface energy we present the first convergence analysis for a numerical scheme of this type our treatment is restricted to one dimensional surfaces curves in the plane which are graphs in this context the scheme amounts to a new algorithm for solving quasilinear parabolic equations in one space dimension

the heat equation shrinking convex plane curves soient m et m des varietes de riemann et f m m une application reguliere si m est une courbe convexe plongee dans le plan r 2 l equation de la chaleur contracte m a un point

k theory and derived equivalences we show that if two rings have equivalent derived categories then they have the same algebraic k theory similar results are given for g theory and for a large class of abelian categories

evolution of plane curves driven by a nonlinear function of curvature and anisotropy in this paper we study evolution of plane curves satisfying a geometric equation v beta k nu where v is the normal velocity and k and nu are the curvature and tangential angle of a plane curve gamma we follow the direct approach and we analyze the so called intrinsic heat equation governing the motion of plane curves obeying such a geometric equation the intrinsic heat equation is modified to include an appropriate nontrivial tangential velocity functional alpha we show how the presence of a nontrivial tangential velocity can prevent numerical solutions from forming various instabilities from an analytical point of view we present some new results on short time existence of a regular family of evolving curves in the degenerate case when beta k nu gamma nu k m 0 m le 2 and the governing system of equations includes a nontrivial tangential velocity functional

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

the steiner tree problem ii properties and classes of facets this is the second part of two papers addressing the study of the facial structure of the steiner tree polyhedron in this paper we identify several classes of facet defining inequalities and relate them to special classes of graphs on which the steiner tree problem is known to be np 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