facets and algorithms for capacitated lot sizing the dynamic economic lot sizing model which lies at the core of numerous production planning applications is one of the most highly studied models in all of operations research and yet capacitated multi item versions of this problem remain computationally elusive we study the polyhedral structure of an integer programming formulation of a single item capacitated version of this problem and use these results to develop solution methods for multi item applications in particular we introduce a set of valid inequalities for the problem and show that they define facets of the underlying integer programming polyhedron computational results on several single and multiple product examples show that these inequalities can be used quite effectively to develop an efficient cutting plane branch and bound procedure moreover our results show that in many instances adding certain of these inequalities a priori to the problem formulation and avoiding the generation of cutting planes can be equally effective

0 1 2 chva tal gomory cuts given the integer polyhedronp t conv x n ax b wherea m n andb m achvatal gomory cg cut is a valid inequality forp 1 of the type ax b for some such that a n in this paper we study 0 1 2 cg cuts arising for 0 1 2 m we show that the associated separation problem 0 1 2 sep is equivalent to finding a minimum weight member of a binary clutter this implies that 0 1 2 sep is np complete in the general case but polynomially solvable whena is related to the edge path incidence matrix of a tree we show that 0 1 2 sep can be solved in polynomial time for a convenient relaxation of the systemax b this leads to an efficient separation algorithm for a subclass of 0 1 2 cg cuts which often contains wide families of strong inequalities forp 1 applications to the clique partitioning asymmetric traveling salesman plant location acyclic subgraph and linear ordering polytopes are briefly discussed

on the monotonization of polyhedra in polyhedral combinatorics one often has to analyze the facial structure of less than full dimensional polyhedra the presence of implicit or explicit equations in the linear system defining such a polyhedron leads to technical difficulties when analyzing its facial structure it is therefore customary to approach the study of such a polytopep through the study of one of its full dimensional relaxations monotonizations known as the submissive and the dominant ofp finding sufficient conditions for an inequality that induces a facet of the submissive or the dominant of a polyhedron to also induce a facet of the polyhedron itself has been posed in the literature as an important research problem our paper goes a long way towards solving this problem we address the problem in the framework of a generalized monotonization of a polyhedronp g mon p that subsumes both the submissive and the dominant and give a sufficient condition for an inequality that defines a facet of g mon p to define a facet ofp for the important cases of the traveling salesman ts polytope in both its symmetric and asymmetric variants and of the linear ordering polytope we give sufficient conditions trivially easy to verify for a facet of the monotone completion to define a facet of the original polytope itself

on the exact solution of a facility layout problem we consider the layout problem of arranging a number of departments on a line this problem is known as single row facility layout the problem is very difficult to be solved in fact it has as a particular case the linear ordering problem which is strongly np hard in this paper a new mixed integer linear programming model is proposed for the problem theoretical arguments as well as computational results are given which demonstrate the efficiency of the new model relatively to previous mixed integer linear programming models proposed for the problem

general transitivity conditions for fuzzy reciprocal preference matrices a reciprocal fuzzy matrix relation is a non negative matrix q qij such that qij qji 1 for all i j 1 2 n we define general transitivity conditions named fg transitivities for fuzzy reciprocal preference relations and show that they generalize some well known transitivities we also study relationships of these conditions with two models of rational preferences the so called utility model and the multidimensional model

median linear orders heuristics and a branch and bound algorithm abstract after recalling some facts about the median linear orders problem two heuristics are studied to solve it then a branch and bound method is designed which computes all the solutions

clustering of neural code words revealed by a first order phase transition a network of neurons in the central nervous system collectively represents information by its spiking activity states typically observed states i e code words occupy only a limited portion of the state space due to constraints imposed by network interactions geometrical organization of code words in the state space critical for neural information processing is poorly understood due to its high dimensionality here we explore the organization of neural code words using retinal data by computing the entropy of code words as a function of hamming distance from a particular reference codeword specifically we report that the retinal code words in the state space are divided into multiple distinct clusters separated by entropy gaps and that this structure is shared with well known associative memory networks in a recallable phase our analysis also elucidates a special nature of the all silent state the all silent state is surrounded by the densest cluster of code words and located within a reachable distance from most code words this code word space structure quantitatively predicts typical deviation of a state trajectory from its initial state altogether our findings reveal a non trivial heterogeneous structure of the code word space that shapes information representation in a biological network

facets of the linear ordering polytope letd n be the complete digraph onn nodes and letp lo denote the convex hull of all incidence vectors of arc sets of linear orderings of the nodes ofd n i e these are exactly the acyclic tournaments ofd n we show that various classes of inequalities define facets ofp lo e g the 3 dicycle inequalities the simplek fence inequalities and various mobius ladder inequalities and we discuss the use of these inequalities in cutting plane approaches to the triangulation problem of input output matrices i e the solution of permutation resp linear ordering problems

an inexact algorithm for the sequential ordering problem abstract given the directed g n a and the penalty matrix c the sequential ordering problem hereafter sop consists of finding the permutation of the nodes from the set n such that it minimizes a c based function and does not violate the precedence relationships given by the set a strong sufficient conditions for the infeasibility of a sop s instance are embedded in a procedure for the sop s pre processing note that it is one of the key steps in any algorithm that attempts to solve sop by dropping the constraints related to the precedence relationships sop can be converted in the classical asymmetric traveling salesman problem hereafter astp the algorithm obtains hopefully satisfactory solutions by modifying the optimal solution to the related assignment problem hereafter ap if it is not a feasible sequential ordering hereafter fso the new solution patches the subtours if any giving preference to the patches with zero reduced cost in the linking arcs the ap based lower bound on the optimal solution to atsp is tightened by using some of the procedures given in 1 in any case a local search for improving the initial fso is performed it uses 3 and 4 changed based procedures computational results on a broad set of cases are reported

a branch and bound algorithm for the linear ordering problem with cumulative costs abstract the linear ordering problem with cumulative costs is an np hard combinatorial optimization problem arising from an application in umts mobile phone communication systems this paper presents a polynomially computable lower bound that is particularly effective when embedded in a branch and bound algorithm the same idea can be further exploited to sort the children nodes at each node of the search tree in order to find the optimal solution earlier a suitable truncation of the resulting branch and bound algorithm results in a fast constructive heuristic

diffuse scattering on graphs abstract we formulate and analyze difference equations on graphs analogous to time independent diffusion equations arising in the study of diffuse scattering in continuous media moreover we show how to construct solutions in the presence of weak scatterers from the solution to the homogeneous background problem using born series providing necessary conditions for convergence and demonstrating the process through numerous examples in addition we outline a method for finding green s functions for cayley graphs for both abelian and non abelian groups finally we conclude with a discussion of the effects of sparsity on our method and results outlining the simplifications that can be made provided that the scatterers are weak and well separated

real laminations and the topological dynamics of complex polynomials abstract we characterize the laminations associated to complex polynomials with connected julia sets and without irrationally neutral cycles

the linear ordering problem with cumulative costs abstract the optimization problem of finding a permutation of a given set of items that minimizes a certain cost function is naturally modeled by introducing a complete digraph g whose vertices correspond to the items to be sorted depending on the cost function to be used different optimization problems can be defined on g the most familiar one is the min cost hamiltonian path problem or its closed path version the traveling salesman problem arising when the cost of a given permutation only depends on consecutive node pairs a more complex situation arises when a given cost has to be paid whenever an item is ranked before another one in the final permutation in this case a feasible solution is associated with an acyclic tournament the transitive closure of an hamiltonian path and the resulting problem is known as the linear ordering problem lop in this paper we introduce and study a relevant case of lop arising when the overall permutation cost can be expressed as the sum of terms u associated with each item u each defined as a linear combination of the values v of all items v that follow u in the permutation this setting implies a cumulative nonlinear propagation of the value of variables v along the node permutation hence the name linear ordering problem with cumulative costs we illustrate the practical application in wireless telecommunication system that motivated the present study we prove complexity results and propose a mixed integer linear programming model as well as an ad hoc enumerative algorithm for the exact solution of the problem a dynamic programming heuristic is also described extensive computational results on large sets of instances are presented showing that the proposed techniques are capable of solving in reasonable computing times all the instances coming from our application

on the partial order polytope of a digraph we introduce the partial order polytope of a digraphd defined as the convex hull of the incidence vectors of all transitive acyclic arc sets ofd for this polytope we prove some classes of inequalities to be facet defining and show that there is a polynomial separation algorithm for each of these classes the results imply a polynomial separation algorithm for a class of valid inequalities of the clique partitioning polytope that includes the two chorded odd cycle inequalities

noise robust modes of the retinal population code have the geometry of ridges and correspond to neuronal communities an appealing new principle for neural population codes is that correlations among neurons organize neural activity patterns into a discrete set of clusters which can each be viewed as a noise robust population codeword previous studies assumed that these codewords corresponded geometrically with local peaks in the probability landscape of neural population responses here we analyze multiple data sets of the responses of approximately 150 retinal ganglion cells and show that local probability peaks are absent under broad nonrepeated stimulus ensembles which are characteristic of natural behavior however we find that neural activity still forms noise robust clusters in this regime albeit clusters with a different geometry we start by defining a soft local maximum which is a local probability maximum when constrained to a fixed spike count next we show that soft local maxima are robustly present and can moreover be linked across different spike count levels in the probability landscape to form a ridge we found that these ridges comprise combinations of spiking and silence in the neural population such that all of the spiking neurons are members of the same neuronal community a notion from network theory we argue that a neuronal community shares many of the properties of donald hebb s classic cell assembly and show that a simple biologically plausible decoding algorithm can recognize the presence of a specific neuronal community

entropy landscape and non gibbs solutions in constraint satisfaction problems we study the entropy landscape of solutions for the bicoloring problem in random graphs a representative difficult constraint satisfaction problem our goal is to classify which types of clusters of solutions are addressed by different algorithms in the first part of the study we use the cavity method to obtain the number of clusters with a given internal entropy and determine the phase diagram of the problem e g dynamical rigidity and satisfiability unsatisfiability sat unsat transitions in the second part of the paper we analyze different algorithms and locate their behavior in the entropy landscape of the problem for instance we show that a smoothed version of a decimation strategy based on belief propagation is able to find solutions belonging to subdominant clusters even beyond the so called rigidity transition where the thermodynamically relevant clusters become frozen these nonequilibrium solutions belong to the most probable unfrozen clusters

entropy landscape of solutions in the binary perceptron problem the statistical picture of the solution space for a binary perceptron is studied the binary perceptron learns a random classification of input random patterns by a set of binary synaptic weights the learning of this network is difficult especially when the pattern constraint density is close to the capacity which is supposed to be intimately related to the structure of the solution space the geometrical organization is elucidated by the entropy landscape from a reference configuration and of solution pairs separated by a given hamming distance in the solution space we evaluate the entropy at the annealed level as well as replica symmetric level and the mean field result is confirmed by the numerical simulations on single instances using the proposed message passing algorithms from the first landscape a random configuration as a reference we see clearly how the solution space shrinks as more constraints are added from the second landscape of solution pairs we deduce the coexistence of clustering and freezing in the solution space

an exact approach to the one dimensional facility layout problem the one dimensional facility layout problem is concerned with arranging n departments of given lengths on a line while minimizing the weighted sum of the distances between all pairs of departments the problem is np hard because it is a generalization of the minimum linear arrangement problem in this paper a 0 1 quadratic programming model consisting of only o n2 0 1 variables is proposed for the problem subsequently this model is cast as an equivalent mixed integer program and then reduced by preprocessing next additional redundant constraints are introduced and linearized in a higher space to achieve an equivalent mixed 0 1 linear program whose continuous relaxation provides an approximation of the convex hull of solutions to the quadratic program it is shown that the resulting mixed 0 1 linear program is more efficient than previously published mixed integer formulations in the computational results several problem instances taken from the literature were efficiently solved to optimality moreover it is now possible to efficiently solve problems of a larger size

on the acyclic subgraph polytope the acyclic subgraph problem can be formulated as follows given a digraph with arc weights find a set of arcs containing no directed cycle and having maximum total weight we investigate this problem from a polyhedral point of view and determine several classes of facets for the associated acyclic subgraph polytope we also show that the separation problem for the facet defining dicycle inequalities can be solved in polynomial time this implies that the acyclic subgraph problem can be solved in polynomial time for weakly acyclic digraphs this generalizes a result of lucchesi for planar digraphs

inference and learning in sparse systems with multiple states we discuss how inference can be performed when data are sampled from the nonergodic phase of systems with multiple attractors we take as a model system the finite connectivity hopfield model in the memory phase and suggest a cavity method approach to reconstruct the couplings when the data are separately sampled from few attractor states we also show how the inference results can be converted into a learning protocol for neural networks in which patterns are presented through weak external fields the protocol is simple and fully local and is able to store patterns with a finite overlap with the input patterns without ever reaching a spin glass phase where all memories are lost

commutative algebra of statistical ranking abstract a model for statistical ranking is a family of probability distributions whose states are orderings of a fixed finite set of items we represent the orderings as maximal chains in a graded poset the most widely used ranking models are parameterized by rational function in the model parameters so they define algebraic varieties we study these varieties from the perspective of combinatorial commutative algebra one of our models the plackett luce model is non toric five others are toric the birkhoff model the ascending model the csiszar model the inversion model and the bradley terry model for these models we examine the toric algebra its lattice polytope and its markov basis

a cutting plane algorithm for the linear ordering problem the linear ordering problem is an np hard combinatorial optimization problem with a large number of applications including triangulation of input output matrices archaeological senation minimizing total weighted completion time in one machine scheduling and aggregation of individual preferences in a former paper we have investigated the facet structure of the 0 1 polytope associated with the linear ordering problem here we report on a new algorithm that is based on these theoretical results the main part of the algorithm is a cutting plane procedure using facet defining inequalities this procedure is combined with various heuristics and branch and bound techniques our computational results compare favorably with the results of existing codes in particular we could triangulate all input output matrices of size up to 60 60 available to us within acceptable time bounds