improving upper bounds for the clique number by non valid inequalities the lovasz and lovasz schrijver bounds are well known upper bounds for the clique number of a graph based on the solution of semidefinite programming problems both bounds can be seen as obtained through a relaxation of a completely positive formulation of the maximum clique problem in this paper we propose to improve these bounds by adding inequalities based on independent sets which may be non valid in the sense that they may be violated by optimal solutions of the completely positive formulation some computational experiments have been performed over different classes of graphs and the results are promising

simultaneous column and row generation for large scale linear programs with column dependent rows in this paper we develop a simultaneous column and row generation algorithm that could be applied to a general class of large scale linear programming problems these problems typically arise in the context of linear programming formulations with exponentially many variables the defining property for these formulations is a set of linking constraints which are either too many to be included in the formulation directly or the full set of linking constraints can only be identified if all variables are generated explicitly due to this dependence between columns and rows we refer to this class of linear programs as problems with column dependent rows to solve these problems we need to be able to generate both columns and rows on the fly within an efficient solution approach we emphasize that the generated rows are structural constraints and distinguish our work from the branch and cut and price framework we first characterize the underlying assumptions for the proposed column and row generation algorithm these assumptions are general enough and cover all problems with column dependent rows studied in the literature up until now to the best of our knowledge we then introduce in detail a set of pricing subproblems which are used within the proposed column and row generation algorithm this is followed by a formal discussion on the optimality of the algorithm to illustrate our approach the paper is concluded by applying the proposed framework to the multi stage cutting stock and the quadratic set covering problems

on the 0 1 facets of the set covering polytope in this paper we consider inequalities of the form jxj where j equals 0 or 1 and is a positive integer we give necessary and sufficient conditions for such inequalities to define facets of the set covering polytope associated with a 0 1 constraint matrixa these conditions are in terms of critical edges and critical cutsets defined in the bipartite incidence graph ofa and are in the spirit of the work of balas and zemel on the set packing problem where similar notions were defined in the intersection graph ofa furthermore we give a polynomial characterization of a class of 0 1 facets defined from chorded cycles of the bipartite incidence graph this characterization also yields all the 0 1 liftings of odd hole inequalities for the simple plant location polytope

an exact algorithm for the vehicle routing problem based on the set partitioning formulation with additional cuts this paper presents a new exact algorithm for the capacitated vehicle routing problem cvrp based on the set partitioning formulation with additional cuts that correspond to capacity and clique inequalities the exact algorithm uses a bounding procedure that finds a near optimal dual solution of the lp relaxation of the resulting mathematical formulation by combining three dual ascent heuristics the first dual heuristic is based on the q route relaxation of the set partitioning formulation of the cvrp the second one combines lagrangean relaxation pricing and cut generation the third attempts to close the duality gap left by the first two procedures using a classical pricing and cut generation technique the final dual solution is used to generate a reduced problem containing only the routes whose reduced costs are smaller than the gap between an upper bound and the lower bound achieved the resulting problem is solved by an integer programming solver computational results over the main instances from the literature show the effectiveness of the proposed algorithm

decomposing matrices into blocks in this paper we investigate whether matrices arising from linear or integer programming problems can be decomposed into so called bordered block diagonal form more precisely given some matrix a we try to assign as many rows as possible to some number beta of blocks of size kappa such that no two rows assigned to different blocks intersect in a common column bordered block diagonal form is desirable because it can guide and speed up the solution process for linear and integer programming problems we show that various matrices from the linear programming and mixed integer programming libraries netlib and miplib can indeed be decomposed into this form by computing optimal decompositions or decompositions with proven quality these computations are done with a branch and cut algorithm based on polyhedral investigations of the matrix decomposition problem in practice however one would use heuristics to find a good decomposition we present several heuristic ideas and test their performance finally we investigate the usefulness of optimal matrix decompositions into bordered block diagonal form for integer programming by using such decompositions to guide the branching process in a branch and cut code for general mixed integer programs

some comments on fourier analysis uncertainty and modeling investigation of the problem of simultaneously concentrating a function and its fourier transform has led to some interesting special functions that have widespread applications in engineering they provide a means of proving a rigorous version of an engineering folk theorem called the 2wt theorem many generalizations of these ideas seem to possess a similar elegant mathematical structure a brief descriptive review is given of these developments

enhancing an algorithm for set covering problems abstract in this note we present enhancements to a previously published algorithm for the optimal solution of set covering problems these enhancements relate to the use of a langrangean heuristic feasible solution exclusion constraints gomory f cuts and an improved branching strategy computational results for problems involving up to 400 rows and 4000 columns indicate that the enhanced algorithm gives better computational results than the original algorithm

on the spatial bandwidth of scattered fields it is shown that the scattered fields are almost space bandlimited functions the effective bandwidth w is introduced and evaluated for a very general scattering system as well as the error made using functions bandlimited to w w for representing the scattered field the effective bandwidth is very simply related to the maximum dimension of the scattering system the error drops to negligible values for modest increases of w compared to w in the case of large scatterers important consequences of the above general results are finally stressed

on the set covering polytope i all the facets with coefficients in 0 1 2 while the set packing polytope through its connection with vertex packing has lent itself to fruitful investigations little is known about the set covering polytope we characterize the class of valid inequalities for the set covering polytope with coefficients equal to 0 1 or 2 and give necessary and sufficient conditions for such an inequality to be minimal and to be facet defining we show that all inequalities in the above class are contained in the elementary closure of the constraint set and that 2 is the largest value ofk such that all valid inequalities for the set covering polytope with integer coefficients no greater thank are contained in the elementary closure we point out a connection between minimal inequalities in the class investigated and certain circulant submatrices of the coefficient matrix finally we discuss conditions for an inequality to cut off a fractional solution to the linear programming relaxation of the set covering problem and to improve the lower bound given by a feasible solution to the dual of the linear programming relaxation

a set packing model for the ground holding problem in congested networks abstract air traffic flow management atfm consists of several activities performed by control authorities in order to reduce delays due to traffic congestion ground holding decisions restrict certain flights from tacking off at the scheduled departure time if congestion is expected at the destination airport they are motivated by the fact that it is safer to hold an aircraft on the ground than in the air several integer linear programming models have been proposed to efficiently solve the ground holding problem ghp in this paper we investigate a set packing formulation of the ghp and design a branch and cut algorithm to solve the problem in high congestion scenarios i e when lack of capacity induces flights cancellation the constraint generation is carried out by heuristically solving the separation problem associated with a large class of rank inequalities this procedure exploits the special structure of the ghp s intersection graphs the computational results indicate that the proposed algorithm outperforms other algorithms in which flight cancellation has been allowed

critical cutsets of graphs and canonical facets of set packing polytopes a cutset in a graph g with node set n is the set of edges joining the nodes in a subset of n to those not in the subset we call a cutset critical if its removal produces a graph whose independence number is greater than that of g we use the concept of a critical cutset to give first a necessary then a sufficient condition for an inequality with 0 1 coefficients to be a facet of the set packing polytope defined by g

minimum cost capacity installation for multicommodity network flows consider a directed graphg v a and a set of traffic demands to be shipped between pairs of nodes inv capacity has to be installed on the edges of this graph in integer multiples of a base unit so that traffic can be routed in this paper we consider the problem of minimum cost installation of capacity on the arcs to ensure that the required demands can be shipped simultaneously between node pairs we study two different approaches for solving problems of this type the first one is based on the idea of metric inequalities see onaga and kakusho on feasibility conditions of multicommodity flows in networks ieee transactions on circuit theory ct 18 4 1971 425 429 and uses a formulation with only a variables the second uses an aggregated multicommodity flow formulation and has v a variables we first describe two classes of strong valid inequalities and use them to obtain a complete polyhedral description of the associated polyhedron for the complete graph on three nodes next we explain our solution methods for both of the approaches in detail and present computational results our computational experience shows that the two formulations are comparable and yield effective algorithms for solving real life problems 1998 the mathematical programming society inc published by elsevier science b v

representation of electromagnetic fields over arbitrary surfaces by a finite and nonredundant number of samples it is shown that the electromagnetic em field radiated or scattered by bounded sources can be accurately represented over a substantially arbitrary surface by a finite number of samples even when the observation domain is unbounded the number of required samples is nonredundant and essentially coincident with the number of degrees of freedom of the field this result relies on the extraction of a proper phase factor from the field expression and on the use of appropriate coordinates to parameterize the domain it is demonstrated that the number of degrees of freedom is independent of the observation domain and depends only on the source geometry the case of spheroidal sources and observation domains with rotational symmetry is analyzed in detail and the particular cases of spherical and planar sources are explicitly considered for these geometries precise and fast sampling algorithms of central type are presented which allow an efficient recovery of em fields from a nonredundant finite number of samples such algorithms are stable with respect to random errors affecting the data

the degrees of freedom of wireless networksvia cut set integrals the problem of determining the number of spatial degrees of freedom d o f of the signals carrying information in a wireless network is reduced to the computation of the geometric variation of the environment with respect to the cut through which the information must flow physically this has an appealing interpretation in terms of the diversity induced on the cut by the possible richness of the scattering environment mathematically this variation is expressed as an integral along the cut which we call cut set integral and whose scaling order is evaluated exactly in the case of planar networks embedded in arbitrary three dimensional 3 d environments presented results shed some new light on the problem of computing the capacity of wireless networks showing a fundamental limitation imposed by the size of the cut through which the information must flow in an attempt to remove what may appear as apparent inconsistencies with previous literature we also discuss how our upper bounds relate to corresponding lower bounds obtained using the techniques of multihop hierarchical cooperation and interference alignment

modeling and solving a crew assignment problem in air transportation abstract a typical problem arising in airline crew management consists in optimally assigning the required crew members to each flight segment of a given time period while complying with a variety of work regulations and collective agreements this problem called the crew assignment problem cap is currently decomposed into two independent sub problems which are modeled and solved sequentially a the well known crew pairing problem followed by b the working schedules construction problem in the first sub problem a set of legal minimum cost pairings is constructed covering all the planned flight segments in the second sub problem pairings rest periods training periods annual leaves etc are combined to form working schedules which are then assigned to crew members in this paper we present a new approach to the crew assignment problem arising in the context of airline companies operating short and medium haul flights contrary to most previously published work on the subject our approach is not based on the concept of crew pairings though it is capable of handling many of the constraints present in crew pairing based models moreover contrary to crew pairing based approaches one of its distinctive features is that it formulates and solves the two sub problems a and b simultaneously for the technical crew members pilots and officers with specific constraints we show how this problem can be formulated as a large scale integer linear program with a general structure combining different types of constraints and not exclusively partitioning or covering constraints as usually suggested in previous papers we introduce then a formulation enhancement phase where we replace a large number of binary exclusion constraints by stronger and less numerous ones the clique constraints using data provided by the tunisian airline company tunisair we demonstrate that thanks to this new formulation the crew assignment problem can be solved by currently available integer linear programming technology finally we propose an efficient heuristic method based on a rounding strategy embedded in a partial tree search procedure the implementation of these methods both exact and heuristic ones provides good solutions in reasonable computation times using cplex 6 0 2 guaranteed exact solutions are obtained for 60 of the test instances and solutions within 5 of the lower bound for the others

on the effectiveness of set covering formulations for the vehicle routing problem with time windows the vehicle routing problem with time windows vrptw is one of the most important problems in distribution and transportation a classical and recently popular technique that has proven effective for solving these problems is based on formulating them as a set covering problem the method starts by solving its linear programming relaxation via column generation and then uses a branch and bound strategy to find an integer solution to the set covering problem a solution to the vrptw an empirically observed property is that the optimal solution value of the set covering problem is very close to its linear programming relaxation which makes the branch and bound step extremely efficient in this paper we explain this behavior by demonstrating that for any distribution of service times time windows customer loads and locations the relative gap between fractional and integer solutions of the set covering problem becomes arbitrarily small as the number of customers increases

polyhedral results for two connected networks with bounded rings we study the polyhedron associated with a network design problem which consists in determining at minimum cost a two connected network such that the shortest cycle to which each edge belongs a ring does not exceed a given length k we present here a new formulation of the problem and derive facet results for different classes of valid inequalities we study the separation problems associated to these inequalities and their integration in a branch and cut algorithm and provide extensive computational results

on the facial structure of the set covering polytope dimensional linear programming given a bipartite graphg v u e a cover ofg is a subset d subseteq v with the property that each nodeu u is adjacent to at least one nodev d if a positive weightcv is associated with each nodev v the covering problem cp is to find a cover ofg having minimum total weight

solving airline crew scheduling problems by branch and cut the crew scheduling problem is one that has been studied almost continually for the past 40 years but all prior approaches have always approximated the problem of finding an optimal schedule for even the smallest of an airline s fleets the problem is especially important today since costs for flying personnel of major u s carriers have grown and now often exceed 1 3 billion a year and are the second largest item next to fuel cost of the total operating cost of major u s carriers thus even small percentage savings amount to substantial dollar amounts we present a branch and cut approach to solving to proven optimality large set partitioning problems arising within the airline industry we first provide some background related to this important application and then describe the approach for solving representative problems in this problem class the branch and cut solver generates cutting planes based on the underlying structure of the polytope defined by the convex hull of the feasible integer points and incorporates these cuts into a tree search algorithm that uses automatic reformulation procedures heuristics and linear programming technology to assist in the solution numerical experiments are reported for a sample of 68 large scale real world crew scheduling problems these problems include both pure set partitioning problems and set partitioning problems with side constraints these base constraints represent contractual labor requirements and have heretofore not been represented explicitly in the construction of crew schedules thus making it impossible to provide any measure of how far the obtained solution was from optimality an interesting result of obtaining less costly schedules is that the crews themselves are happier with the schedules because they spend more of their duty time flying than waiting on the ground

multi phase dynamic constraint aggregation for set partitioning type problems dynamic constraint aggregation is an iterative method that was recently introduced to speed up the linear relaxation solution process of set partitioning type problems this speed up is mostly due to the use at each iteration of an aggregated problem defined by aggregating disjoint subsets of constraints from the set partitioning model this aggregation is updated when needed to ensure the exactness of the overall approach in this paper we propose a new version of this method called the multi phase dynamic constraint aggregation method which essentially adds to the original method a partial pricing strategy that involves multiple phases this strategy helps keeping the size of the aggregated problem as small as possible yielding a faster average computation time per iteration and fewer iterations we also establish theoretical results that provide some insights explaining the success of the proposed method tests on the linear relaxation of simultaneous bus and driver scheduling problems involving up to 2 000 set partitioning constraints show that the partial pricing strategy speeds up the original method by an average factor of 4 5

a lagrangian based heuristic for large scale set covering problems we present a new lagrangian based heuristic for solving large scale set covering problems arising from crew scheduling at the italian railways ferrovie dello stato our heuristic obtained impressive results when compared to state of the art codes on a test bed provided by the company which includes instances with sizes ranging from 50 000 variables and 500 constraints to 1 000 000 variables and 5000 constraints 1998 the mathematical programming society inc published by elsevier science b v

octane a new heuristic for pure 0 1 programs we propose a new heuristic for pure 0 1 programs which finds feasible integer points by enumerating extended facets of the octahedron the outer polar of the unit hypercube we give efficient algorithms to carry out the enumeration and we explain how our heuristic can be embedded in a branch and cut framework finally we present computational results on a set of pure 0 1 programs taken from miplib and other sources

subadditive approaches in integer programming linear programming duality is well understood and the reduced cost of a column is frequently used in various algorithms on the other hand for integer programs it is not clear how to define a dual function even though the subadditive dual theory has been developed a long time ago in this work we propose a family of computationally tractable subadditive dual functions for integer programs we develop a solution methodology that computes an optimal primal solution and an optimal subadditive dual function we present computational experiments which show that the new algorithm is tractable

transitive packing a unifying concept in combinatorial optimization this paper attempts to provide a better understanding of the facial structure of polyhedra previously investigated separately it introduces the notion of transitive packing and the transitive packing polytope polytopes that turn out to be special cases of the transitive packing polytope include the node packing acyclic subdigraph bipartite subgraph planar subgraph clique partitioning partition transitive acyclic subdigraph interval order and relatively transitive subgraph polytopes we give cutting plane proofs for several rich classes of valid inequalities of the transitive packing polytope thereby introducing generalized cycle generalized clique generalized antihole generalized antiweb and odd partition inequalities on the one hand these classes subsume several known classes of valid inequalities for several special cases on the other hand they yield many new inequalities for several other special cases for some of the classes we also prove a lower bound on their gomory chvatal rank finally we relate the concept of transitive packing to generalized set packing and covering as well as to balanced and ideal matrices

some facets of the simple plant location polytope we relate the simple plant location problem to the vertex packing problem and derive several classes of facets of their associated integer polytopes

combinatorial optimization current successes and directions for the future our ability to solve large important combinatorial optimization problems has improved dramatically in the past decade the availability of reliable software extremely fast and inexpensive hardware and high level languages that make the modeling of complex problems much faster have led to a much greater demand for optimization tools this paper highlights the major breakthroughs and then describes some very exciting future opportunities previously large research projects required major data collection efforts expensive mainframes and substantial analyst manpower now we can solve much larger problems on personal computers much of the necessary data is routinely collected and tools exist to speed up both the modeling and the post optimality analysis with the information technology revolution taking place currently we now have the opportunity to have our tools embedded into supply chain systems that determine production and distribution schedules process design and location allocation decisions these tools can be used industry wide with only minor modifications being done by each user

facets of the independent set polytope theoretical results pertaining to the independent set polytope p isp conv x 0 1 n ax b are presented a conflict hypergraph is constructed based on the set of dependent sets which facilitates the examination of the facial structure of p isp necessary and sufficient conditions are provided for every nontrivial 0 1 facet defining inequalities of p isp in terms of hypercliques the relationship of hypercliques and some classes of knapsack facet defining inequalities are briefly discussed the notion of lifting is extended to the conflict hypergraph setting to obtain strong valid inequalities and back lifting is introduced to strengthen cut coefficients preliminary computational results are presented to illustrate the usefulness of the theoretical findings

implementing an efficient minimum capacity cut algorithm in this paper we present an efficient implementation of theo mn n2 logn time algorithm originally proposed by nagamochi and ibaraki 1992 for computing the minimum capacity cut of an undirected network to enhance computation various ideas are added so that it can contract as many edges as possible in each iteration to evaluate the performance of the resulting implementation we conducted extensive computational experiments and compared the results with that of padberg and rinaldi s algorithm 1990 which is currently known as one of the practically fastest programs for this problem the results indicate that our program is considerably faster than padberg and rinaldi s program and its running time is not significantly affected by the types of the networks being solved

topics in data assimilation stochastic processes abstract stochastic models with varying degrees of complexity are increasingly widespread in the oceanic and atmospheric sciences one application is data assimilation i e the combination of model output with observations to form the best picture of the system under study for any given quantity to be estimated the relative weights of the model and the data will be adjusted according to estimated model and data error statistics so implementation of any data assimilation scheme will require some assumption about errors which are considered to be random for dynamical models some assumption about the evolution of errors will be needed stochastic models are also applied in studies of predictability the formal theory of stochastic processes was well developed in the last half of the twentieth century one consequence of this theory is that methods of simulation of deterministic processes cannot be applied to random processes without some modification in some cases the rules of ordinary calculus must be modified the formal theory was developed in terms of mathematical formalism that may be unfamiliar to many oceanic and atmospheric scientists the purpose of this article is to provide an informal introduction to the relevant theory and to point out those situations in which that theory must be applied in order to model random processes correctly

dynamical criteria for the evolution of the stochastic dimensionality in flows with uncertainty abstract we estimate and study the evolution of the dominant dimensionality of dynamical systems with uncertainty governed by stochastic partial differential equations within the context of dynamically orthogonal do field equations transient nonlinear dynamics irregular data and non stationary statistics are typical in a large range of applications such as oceanic and atmospheric flow estimation to efficiently quantify uncertainties in such systems it is essential to vary the dimensionality of the stochastic subspace with time an objective here is to provide criteria to do so working directly with the original equations of the dynamical system under study and its do representation we first analyze the scaling of the computational cost of these do equations with the stochastic dimensionality and show that unlike many other stochastic methods the do equations do not suffer from the curse of dimensionality subsequently we present the new adaptive criteria for the variation of the stochastic dimensionality based on instantaneous i stability arguments and ii bayesian data updates we then illustrate the capabilities of the derived criteria to resolve the transient dynamics of two 2d stochastic fluid flows specifically a double gyre wind driven circulation and a lid driven cavity flow in a basin in these two applications we focus on the growth of uncertainty due to internal instabilities in deterministic flows we consider a range of flow conditions described by varied reynolds numbers and we study and compare the evolution of the uncertainty estimates under these varied conditions

a stochastic model of indopacific sea surface temperature anomalies abstract it is often desirable to represent a rapidly varying physical process as stochastic forcing of some slower dynamical system a review of this approximation is presented the white noise limit of rapidly varying processes reduces the dynamical description of the affected slower systems to stochastic differential equations the properties of which are summarized as an application of stochastic differential equations we review the evidence that indopacific sea surface temperature anomalies sstas can be represented as a stable linear process driven by spatially coherent stochastic forcing an inversemodeling approach is used that is the relevant parameters of the best fit stable linear process are obtained from observations and given these parameters the assumptions of stability and linearity are subsequently tested an optimal initial structure for growth predicted by the model is found to occur approximately seven months before most of the major warm and cold el nino la nina events in the data record the optimal structure preferentially occurs in the boreal spring with the mature phase of the extreme event occurring in the subsequent late fall winter the best model to fit the observations including the dependence of the dynamical system on the annual cycle is driven by stochastic forcing with periodic statistics these statistics are inferred using a time dependent fluctuation dissipation relation the variation of the stochastic forcing with the annual cycle is not the same as that of the optimal initial structure yet when a linear numerical model is driven with the derived stochastic forcing the optimal structure including its variation with the annual cycle is recovered this implies that deterministic dynamics play an important role in setting up the optimal structure and therefore a mature phase of el nino may be predicted before the optimal structure actually appears

crew pairing at air france abstract in the airline industry crew schedules consist of a number of pairings these are round trips originating and terminating at the same crew home base composed of legal work days called duties separated by rest periods the purpose of the airline crew pairing problem is to generate a set of minimal cost crew pairings covering all flight legs the set of pairings must satisfy all the rules in the work convention and all the appropriate air traffic regulations the resulting constraints can affect duty construction may restrict each pairing or be imposed on the overall crew schedule the pairing problem is formulated as an integer nonlinear multi commodity network flow problem with additional resource variables nonlinearities occur in the objective function as well as in a large subset of constraints a branch and bound algorithm based on an extension of the dantzig wolfe decomposition principle is used to solve this model the master problem becomes a set partitioning type model as in the classical formulation while pairings are generated using resource constrained shortest path subproblems this primal approach implicitly considers all feasible pairings and also provides the optimality gap value on a feasible solution a nice feature of this decomposition process is that it isolates all nonlinear aspects of the proposed multi commodity model in the subproblems which are solved by means of a specialized dynamic programming algorithm we present the application and implementation of this approach at air france it is one of the first implementations of an optimal approach for a large airline carrier we have chosen a subproblem network representation where the duties rather than the legs are on the arcs this ensures feasibility relative to duty restrictions by definition as opposed to lavoie minoux and odier 1988 the nonlinear cost function is modeled without approximations the computational experiments were conducted using actual air france medium haul data even if the branch and bound trees were not fully explored in all cases the gaps certify that the computed solutions are within a fraction of one percentage point of the optimality our results illustrate that our approach produced substantial improvements over solutions derived by the expert system in use at air france their magnitude led to the eventual implementation of the approach

estimating parameters in stochastic systems a variational bayesian approach this work is concerned with approximate inference in dynamical systems from a variational bayesian perspective when modelling real world dynamical systems stochastic differential equations appear as a natural choice mainly because of their ability to model the noise of the system by adding a variation of some stochastic process to the deterministic dynamics hence inference in such processes has drawn much attention here a new extended framework is derived that is based on a local polynomial approximation of a recently proposed variational bayesian algorithm the paper begins by showing that the new extension of this variational algorithm can be used for state estimation smoothing and converges to the original algorithm however the main focus is on estimating the hyper parameters of these systems i e drift parameters and diffusion coefficients the new approach is validated on a range of different systems which vary in dimensionality and non linearity these are the ornstein uhlenbeck process the exact likelihood of which can be computed analytically the univariate and highly non linear stochastic double well and the multivariate chaotic stochastic lorenz 63 3d model as a special case the algorithm is also applied to the 40 dimensional stochastic lorenz 96 system in our investigation we compare this new approach with a variety of other well known methods such as the hybrid monte carlo dual unscented kalman filter full weak constraint 4d var algorithm and analyse empirically their asymptotic behaviour as a function of observation density or length of time window increases in particular we show that we are able to estimate parameters in both the drift deterministic and the diffusion stochastic part of the model evolution equations using our new methods

capacity scaling of wireless ad hoc networks shannon meets maxwell in this paper we characterize the information theoretic capacity scaling of wireless ad hoc networks with randomly distributed nodes by using an exact channel model from maxwell s equations we successfully resolve the conflict in the literature between the linear capacity scaling by ozgur and the degrees of freedom limit given as the ratio of the network diameter and the wavelength by franceschetti in dense networks where the network area is fixed the capacity scaling is given as the minimum of and the degrees of freedom limit to within an arbitrarily small exponent in extended networks where the network area is linear in the capacity scaling is given as the minimum of and the degrees of freedom limit to within an arbitrarily small exponent hence we recover the linear capacity scaling by ozgur if in dense networks and if in extended networks otherwise the capacity scaling is given as the degrees of freedom limit characterized by franceschetti for achievability a modified hierarchical cooperation is proposed based on a lower bound on the capacity of multiple input multiple output channel between two node clusters using our channel model

a branch and cut algorithm for the resolution of large scale symmetric traveling salesman problems an algorithm is described for solving large scale instances of the symmetric traveling salesman problem stsp to optimality the core of the algorithm is a polyhedral cutting plane procedure that exploits a subset of the system of linear inequalities defining the convex hull of the incidence vectors of the hamiltonian cycles of a complete graph the cuts are generated by several identification procedures that have been described in a companion paper whenever the cutting plane procedure does not terminate with an optimal solution the algorithm uses a tree search strategy that as opposed to branch and bound keeps on producing cuts after branching the algorithm has been implemented in fortran two different linear programming lp packages have been used as the lp solver the implementation of the algorithm and the interface with one of the lp solvers is described in sufficient detail to permit the replication of our experiments computational results are reported with up to 42 stsps with sizes rangin

mixed 0 1 programming by lift and project in a branch and cut framework we investigate the computational issues that need to be addressed when incorporating general cutting planes for mixed 0 1 programs into a branch and cut framework the cuts we use are of the lift and project variety some of the issues addressed have a theoretical answer but others are of an experimental nature and are settled by comparing alternatives on a set of test problems the resulting code is a robust solver for mixed 0 1 programs we compare it with several existing codes on a wide range of test problems it performs as well as or better than some of the best currently available mixed integer programming codes

implementing the dantzig fulkerson johnson algorithm for large traveling salesman problems dantzig fulkerson and johnson 1954 introduced the cutting plane method as a means of attacking the traveling salesman problem this method has been applied to broad classes of problems in combinatorial optimization and integer programming in this paper we discuss an implementation of dantzig et al s method that is suitable for tsp instances having 1 000 000 or more cities our aim is to use the study of the tsp as a step towards understanding the applicability and limits of the general cutting plane method in large scale applications

the mixed vertex packing problem we study a generalization of the vertex packing problem having both binary and bounded continuous variables called the mixed vertex packing problem mvpp the well known vertex packing model arises as a subproblem or relaxation of many 0 1 integer problems whereas the mixed vertex packing model arises as a natural counterpart of vertex packing in the context of mixed 0 1 integer programming we describe strong valid inequalities for the convex hull of solutions to the mvpp and separation algorithms for these inequalities we give a summary of computational results with a branch and cut algorithm for solving the mvpp and using it to solve general mixed integer problems

solving a truck dispatching scheduling problem using branch and cut a branch and cut ip solver is developed for a class of structured 0 1 integer programs arising from a truck dispatching scheduling problem this problem involves a special class of knapsack equality constraints families of facets for the polytopes associated with individual knapsack constraints are identified in addition a notion of conflict graph is utilized to obtain an approximating node packing polytope for the convex hull of all 0 1 solutions the branch and cut solver generates cuts based on both the knapsack equality constraints and the approximating node packing polytope and incorporates these cuts into a tree search algorithm that uses problem reformulation and linear programming based heuristics at each node in the search tree to assist in the solution process numerical experiments are performed on large scale real instances supplied by texaco trading transportation inc the optimal schedules correspond to cost savings for the company and greater job satisfaction for drivers due to more balanced work schedules and income distribution

applying evolutionary programming to selected set partitioning problems evolutionary programming is applied to several instances of the set partitioning problem comparison is made between the distribution of best evolved solutions arising from implementations of the ep with the empirical distribution of a randomly selected trial solution

parallel machine scheduling linear programming and parameter list scheduling heuristics in this paper we consider a class of parallel machine scheduling problems and their associated set partitioning formulations we show that the tightness of the linear programming relaxation of these formulations is directly related to the performance of a class of heuristics called parameter list scheduling heuristics this makes it possible to characterize the worst possible gap between optimal solutions for the scheduling problems and the corresponding linear programming relaxations in the case of the classical parallel machine weighted completion time model we also show that the solution to the linear programming relaxation of the set partitioning formulation is asymptotically optimal under mild assumptions on the distribution of job weights and processing times finally we extend most of the results to the time discretized formulation of machine scheduling problems

computational study of large scale p median problems given a directed graph g v a the p median problem consists of determining p nodes the median nodes minimizing the total distance from the other nodes of the graph we present a branch and cut and price algorithm yielding provably good solutions for instances with v 3795 key ingredients of the algorithm are a delayed column and row generation technique exploiting the special structure of the formulation to solve the lp relaxation and cutting planes to strengthen the formulation and limit the size of the enumeration tree

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

an efficient algorithm for the minimum capacity cut problem given a finite undirected graph with nonnegative edge capacities the minimum capacity cut problem consists of partitioning the graph into two nonempty sets such that the sum of the capacities of edges connecting the two parts is minimum among all possible partitionings the standard algorithm to calculate a minimum capacity cut due to gomory and hu 1961 runs in o n4 time and is difficult to implement we present an alternative algorithm with the same worst case bound which is easier to implement and which was found empirically to be far superior to the standard algorithm we report computational results for graphs with up to 2000 nodes

some thoughts on combinatorial optimisation abstract a group of young researchers from the esi x summer school hec jouy en josas 1994 give their personal views on the current status of and prospects for combinatorial optimisation several issues are considered and discussed with emphasis on a selected number of techniques heuristics and polyhedral approaches and problems knapsack quadratic 0 1 programming machine scheduling routing and network design

worst case analyses linear programming and the bin packing problem in this paper we consider the familiar bin packing problem and its associated set partitioning formulation we show that the optimal solution to the bin packing problem can be no larger than 4 3 zlp wherezlp is the optimal solution value of the linear programming relaxation of the set partitioning formulation an example is provided to show that the bound is tight a by product of our analysis is a new worst case bound on the performance of the well studied first fit decreasing and best fit decreasing heuristics

a tree search algorithm for the crew scheduling problem in this paper we consider the crew scheduling program that is the problem of assigning k crews to n tasks with fixed start and finish times such that each crew does not exceed a limit on the total time it can spend working a zero one integer linear programming formulation of the problem is given which is then relaxed in a lagrangean way to provide a lower bound which is improved by subgradient optimisation finally a tree search procedure incorporating this lower bound is presented to solve the problem to optimality computational results are given for a number of randomly generated test problems involving between 50 and 500 tasks

on the degrees of freedom of scattered fields starting from the observation that fields differing less than a prescribed error cannot be resolved as distinct entities the degrees of freedom of the scattered field are introduced and then computed the degrees of freedom are shown to be practically equal to the nyquist number appropriate to the effective spatial bandwidth of the scattered field and to the extension of the observation domain accordingly a finite number of elements of information can be used to determine the scattered field to a prescribed approximation error it is also shown that the field representation can be made in terms of field values and simple sampling functions provided that a marginal increase in the approximation error is tolerated the results not only completely justify the use of sampling interpolation for representing scattering fields but also demonstrate that such representation is practically an optimal one an algorithm for the reconstruction of scattered fields given the maximum allowed error is then produced

an application of combinatorial optimization to statistical physics and circuit layout design we study the problem of finding ground states of spin glasses with exterior magnetic field and the problem of minimizing the number of vias holes on a printed circuit board or contacts on a chip subject to pin preassignments and layer preferences the former problem comes up in solid state physics and the latter in very large scale integrated vlsi circuit design and in printed circuit board design both problems can be reduced to the max cut problem in graphs based on a partial characterization of the cut polytope we design a cutting plane algorithm and report on computational experience with it our method has been used to solve max cut problems on graphs with up to 1 600 nodes

on cutting plane proofs in combinatorial optimization abstract gomory s cutting plane technique can be viewed as a recursive procedure for proving the validity of linear inequalities over the set of all integer vectors in a prescribed polyhedron the number of rounds of cutting planes needed to obtain all valid linear inequalities is known as the rank of the polyhedron we prove that polyhedra featured in popular formulations of the stable set problem the set covering problem the set partitioning problem the knapsack problem the bipartite subgraph problem the maximum cut problem the acyclic subdigraph problem the asymmetric traveling salesman problem and the traveling salesman problem have arbitrarily high rank in particular we prove conjectures of barahona grotschel and mahjoub chvatal grotschel and pulleyblank and junger

a new approach for crew pairing problems by column generation with an application to air transportation we propose a new approach to crew pairing problems arising in the context of airline companies the problem is first formulated as a large scale set covering problem with many colums each column representing a valid crew pairing we then suggest a solution procedure for the continuous relaxation of this large scale problem based on generalized linear programming in which the column generation subproblem is shown to be equivalent to a shortest path problem in an associated graph computational results obtained on a series of real problems involving up to 329 flight segments are reported confirming both computational efficiency and practical applicability of the new approach indeed not only were the resulting solutions observed to be integral for most test problems but average savings of about 4 to 5 over the best available hand built solutions were shown to be obtained

efficient separation routines for the symmetric traveling salesman problem i general tools and comb separation this is the first of a series of two papers dedicated to efficient separation heuristics for violated inequalities in the context of solving to optimality instances of the traveling salesman problem via branch and cut in this paper we give the basic ideas behind these heuristics and design heuristics for comb separation the second paper will deal with more complex inequalities such as clique tree star or path and ladder inequalities and give computational results

dynamic penalization of fractional directions in the integral simplex using decomposition application to aircrew scheduling abstract to solve integer linear programs primal algorithms follow an augmenting sequence of integer solutions leading to an optimal solution in this work we focus on a particular primal algorithm the integral simplex using decomposition isud to find the next point one solves a linear program to select an augmenting direction for the current point from a cone of feasible directions to ensure that this linear program is bounded a normalization constraint is added and the optimization is performed on a section of the cone the solution of the linear program i e the direction proposed by the algorithm strongly depends on the chosen normalization weights and so does the likelihood that the next solution is integer we modify isud so that the normalization is dynamically updated whenever the direction leads to a fractional solution to penalize that direction we propose update strategies based on new theoretical and experimental results to prove the efficiency of our strategies we show that our version of the algorithm yields better results than the former version and than classical branch and bound techniques on a benchmark of industrial aircrew scheduling instances the benchmark that we propose here is to the best of our knowledge comparable to no other from the literature it provides large scale instances with up to 1700 flights and 115 000 pairings hence as many constraints and variables and the instances are given in a set partitioning form together with initial solutions that accurately mimic those of industrial applications our work shows the strong potential of primal algorithms for the crew scheduling problem which is a key challenge for large airlines

degrees of freedom in multiple antenna channels a signal space approach multiple antenna systems that are limited by the area and geometry of antenna arrays are considered given these physical constraints the limit on the available number of spatial degrees of freedom is derived the commonly used statistical multiple input multiple output mimo model is inadequate antenna theory is applied to take into account the area and geometry constraints and to define the spatial signal space so as to interpret experimental channel measurements in an array independent but manageable description of the physical environment based on these modeling strategies for a spherical array of effective aperture a in a physical environment of angular spread spl omega in solid angle the number of spatial degrees of freedom is shown to be a spl omega for uni polarized antennas and 2a spl omega for tri polarized antennas together with the 2wt degrees of freedom for a system of bandwidth w transmitting in an interval t the total degrees of freedom of a multiple antenna channel is therefore 4wta spl omega

cabob a fast optimal algorithm for winner determination in combinatorial auctions combinatorial auctions where bidders can bid on bundles of items can lead to more economically efficient allocations but determining the winners is complete and inapproximable we present cabob a sophisticated optimal search algorithm for the problem it uses decomposition techniques upper and lower bounding also across components elaborate and dynamically chosen bid ordering heuristics and a host of structural observations cabob attempts to capture structure in any instance without making assumptions about the instance distribution experiments against the fastest prior algorithm cplex 8 0 show that cabob is often faster seldom drastically slower and in many cases drastically faster especially in cases with structure cabob s search runs in linear space and has significantly better anytime performance than cplex we also uncover interesting aspects of the problem itself first problems with short bids which were hard for the first generation of specialized algorithms are easy second almost all of the cats distributions are easy and the run time is virtually unaffected by the number of goods third we test several random restart strategies showing that they do not help on this problem the run time distribution does not have a heavy tail

aircrew schedule generation using repeated matching we consider the aircrew scheduling problem of determining tours of duty tods for aircrews given a set of sectors or flights requiring regular crews a regular crew consists of two crew members but by including supplementary crew a third pilot on some sectors it is possible to extend duty periods to generate more cost efficient tods a related problem is thus to generate tods for these third pilots but the sectors requiring a third pilot are not known in advance to solve these two related problems simultaneously we apply a heuristic procedure that solves a sequence of matching problems i e a repeated matching algorithm numerical results based on the solution of a real problem show that this approach is a valid and efficient method for solving the aircrew scheduling problem especially when there is the option of using supplementary crew to extend duty periods

facets and lifting procedures for the set covering polytope given a family of subsets of an arbitrary groundsete acover of is any setc e having non empty intersection with every subset in in this paper we deal with thecovering polytope i e the convex hull of the incidence vectors of all the covers of in section 2 we review all the known properties of the covering polytope in sections 3 and 4 we introduce two new classes of non boolean facets of such a polytope in sections 5 and 6 we describe some non sequential lifting procedures in section 7 a generalization of the notion ofweb introduced by l e trotter is presented together with the facets of the covering polytope produced by such a structure moreover the strong connections between several combinatorial problems and the covering problem are pointed out and exploiting those connections some examples are presented of new facets for the knapsack and acyclic subdigraph polytopes

column generation in the integral simplex method the integral simplex method for set partitioning problems allows only pivots on one to be made which results in a primal all integer method in this technical note we outline how to tailor the column generation principle to this method because of the restriction to pivots on one only local optimality can be guaranteed and to ensure global optimality we consider the use of implicit enumeration

rank inequalities and separation algorithms for packing designs and sparse triple systems combinatorial designs find numerous applications in computer science and are closely related to problems in coding theory packing designs correspond to codes with constant weight 4 sparse partial steiner triple systems 4 sparse pstss correspond to erasure resilient codes that are useful in handling failures in large disk arrays chee colbourn ling discrete appl math to appear hellerstein gibson karp katz paterson algorithmica 12 1994 182 208 the study of polytopes associated with combinatorial problems has proven to be important for both algorithms and theory but only recently the study of design polytopes has been pursued moura math appl 368 1996 227 254 moura ph d thesis university of toronto 1999 moura proc seventh annu european symp prague 1999 lecture notes in computer science vol 1643 springer berlin 1999 pp 462 475 wengrzik master s thesis universitat berlin 1995 zehendner doctoral thesis universitat augsburg 1986 in this article we study polytopes associated with t v k packing designs and with m sparse pstss subpacking and l sparseness inequalities are introduced and studied they can be regarded as rank inequalities for the independence systems associated with these designs conditions under which subpacking inequalities define facets are derived in particular those which define facets for pstss are determined for m 4 the l sparseness inequalities with 2 l m are proven to induce facets for the m sparse psts polytope this proof uses extremal families of pstss known as erdos configurations separation algorithms for these inequalities are proposed we incorporate some of the sparseness inequalities in a polyhedral algorithm and determine maximal 4 sparse psts v v 16 an upper bound on the size of m sparse pstss is presented

numerical schemes for dynamically orthogonal equations of stochastic fluid and ocean flows the quantification of uncertainties is critical when systems are nonlinear and have uncertain terms in their governing equations or are constrained by limited knowledge of initial and boundary conditions such situations are common in multiscale intermittent and non homogeneous fluid and ocean flows the dynamically orthogonal do field equations provide an adaptive methodology to predict the probability density functions of such flows the present work derives efficient computational schemes for the do methodology applied to unsteady stochastic navier stokes and boussinesq equations and illustrates and studies the numerical aspects of these schemes semi implicit projection methods are developed for the mean and for the do modes and time marching schemes of first to fourth order are used for the stochastic coefficients conservative second order finite volumes are employed in physical space with new advection schemes based on total variation diminishing methods other results include i the definition of pseudo stochastic pressures to obtain a number of pressure equations that is linear in the subspace size instead of quadratic ii symmetric advection schemes for the stochastic velocities iii the use of generalized inversion to deal with singular subspace covariances or deterministic modes and iv schemes to maintain orthonormal modes at the numerical level to verify our implementation and study the properties of our schemes and their variations a set of stochastic flow benchmarks are defined including asymmetric dirac and symmetric lock exchange flows lid driven cavity flows and flows past objects in a confined channel different reynolds number and grashof number regimes are employed to illustrate robustness optimal convergence under both time and space refinements is shown as well as the convergence of the probability density functions with the number of stochastic realizations

a lp based heuristic for a time constrained routing problem abstract in this paper we present a lp based heuristic for the solution of a time constrained routing problem arising from innovative services accessible via world wide web the problem consists of scheduling the visit of a tourist to a given geographical area in order to maximize his satisfaction degree whilst respecting time windows restrictions we refer to this problem as the intelligent tourist problem itp itp is formulated as a set packing problem with side constraints due to the huge number of variables in the formulation the lp relaxation is solved by a column and row generation approach then we run a mip solver over the active columns to get a feasible solution computational experience on real world instances is reported showing the effectiveness of the proposed approach

on the uncapacitated plant location problem ii facets and lifting theorems several lifting procedures for facets of the uncapacitated plant location problem are discussed also necessary and sufficient conditions for nontrivial facets with 0 1 integer coefficients are derived in addition all of the facets for the case of three or more plants and three destinations are identified

the relation of time indexed formulations of single machine scheduling problems to the node packing problem the relation of time indexed formulations of nonpreemptive single machine scheduling problems to the node packing problem is established and then used to provide simple and intuitive alternate proofs of validity and maximality for previously known results on the facial structure of the scheduling problem previous work on the facial structure has focused on describing the convex hull of the set of feasible partial schedules schedules in which not all jobs have to be started the equivalence between the characteristic vectors of this set and those of the set of feasible node packings in a graph whose structure is determined by the parameters of the scheduling problem is established the main contribution of this paper is to show that the facet inducing inequalities for the convex hull of the set of feasible partial schedules that have integral coefficients and right hand side 1 or 2 are the maximal clique inequalities and the maximally and sequentially lifted 5 hole inequalities of the convex hull of the set of feasible node packings in this graph respectively

intrinsic limits of dimensionality and richness in random multipath fields we study the dimensions or degrees of freedom of farfield multipath that is observed in a limited source free region of space the multipath fields are studied as solutions to the wave equation in an infinite dimensional vector space we prove two universal upper bounds on the truncation error of fixed and random multipath fields a direct consequence of the derived bounds is that both fixed and random multipath fields have an effective finite dimension for circular and spherical spatial regions we show that this finite dimension is proportional to the radius and area of the region respectively we use the karhunen loegraveve kl expansion of random multipath fields to quantify the notion of multipath richness the multipath richness is defined as the number of significant eigenvalues in the kl expansion that achieve 99 of the total multipath energy we establish a lower bound on the largest eigenvalue this lower bound quantifies to some extent the well known reduction of multipath richness with reducing the angular power spread of multipath angular power spectrum

on landau s eigenvalue theorem and information cut sets a variation of landau s eigenvalue theorem describing the phase transition of the eigenvalues of a time frequency limiting self adjoint operator is presented the total number of degrees of freedom of square integrable multidimensional bandlimited functions is defined in terms of kolmogorov s n width and computed in some limiting regimes where the original theorem cannot be directly applied results are used to characterize up to order the total amount of information that can be transported in time and space by multiple scattered electromagnetic waves rigorously addressing a question originally posed in the early works of toraldo di francia and gabor applications in the context of wireless communication and electromagnetic sensing are discussed

a generalization of antiwebs to independence systems and their canonical facets we consider independence system polytopes i e polytopes whose extreme points are the incidence vectors of the sets of an independence system we first give a sufficient condition for recognizing boolean facets then the notion of antiweb introduced by trotter for graphs is generalized to independence systems and used for obtaining canonical facets of the associated polytopes we also point out how our results relate with known ones for knapsack set covering and matroid polytopes

on the 0 1 knapsack polytope this paper deals with the 0 1 knapsack polytope in particular we introduce the class ofweight inequalities this class of inequalities is needed to describe the knapsack polyhedron when the weights of the items lie in certain intervals a generalization of weight inequalities yields the so called weight reduction principle and the class of extended weight inequalities the latter class of inequalities includes minimal cover and l k configuration inequalities the properties of lifted minimal cover inequalities are extended to this general class of inequalities

conflict graphs in solving integer programming problems abstract we report on the use of conflict graphs in solving integer programs a conflict graph represents logical relations between binary variables we develop algorithms and data structures that allow the effective and efficient construction management and use of dynamically changing conflict graphs our computational experiments show that the techniques presented work very well

strong lift and project cutting planes for the stable set problem a great deal of research has been focusing since the early seventies on finding strong relaxations for the stable set problem polyhedral combinatorics techniques have been at first developed to strengthen the natural linear formulation afterward strong semidefinite programming relaxations have been deeply investigated nevertheless the resulting integer programming ip algorithms cannot be regarded as being quite successful in practice as most of the relaxations give rise to one out of two extreme situations either provide weak bounds at low computational cost or give good bounds sometimes excellent but too demanding to compute in this paper we present a method to bridge such a gap in particular a new lift and project relaxation is obtained by a problem specific variant of the lifting operator m k k by lovasz and schrijver combined with benders decomposition this yields strong cutting planes generated by solving a cut generating linear program an extensive computational experience shows that embedding these cuts in a branch and cut framework significantly reduces the size of the enumeration trees as well as the cpu times with respect to state of the art ip algorithms

an application of the lovasz schrijver m k k operator to the stable set problem although the lift and project operators of lovasz and schrijver have been the subject of intense study their m k k operator has received little attention we consider an application of this operator to the stable set problem we begin with an initial linear programming lp relaxation consisting of clique and non negativity inequalities and then apply the operator to obtain a stronger extended lp relaxation we discuss theoretical properties of the resulting relaxation describe the issues that must be overcome to obtain an effective practical implementation and give extensive computational results remarkably the upper bounds obtained are sometimes stronger than those obtained with semidefinite programming techniques

on the set covering polytope ii lifting the facets with coefficients in 0 1 2 in an earlier paper mathematical programming 43 1989 57 69 we characterized the class of facets of the set covering polytope defined by inequalities with coefficients equal to 0 1 or 2 in this paper we connect that characterization to the theory of facet lifting in particular we introduce a family of lower dimensional polytopes and associated inequalities having only three nonzero coefficients whose lifting yields all the valid inequalities in the above class with the lifting coefficients given by closed form expressions