A polynomial method for the pos/neg weighted 3-median problem on a tree

Let a connected undirected graph G  =  (V, E) be given. In the classical p-median problem we want to find a set X containing p points in G such that the sum of weighted distances from X to all vertices in V is minimized. We consider the semi-obnoxious case where every vertex has either a positive or negative weight. In this case we have two different objective functions: the sum of the minimum weighted distances from X to all vertices and the sum of the weighted minimum distances. In this paper we show that for the case p = 3 an optimal solution for the second model in a tree can be found in O(n ⁵) time. If the 3-median is restricted to vertices or if the tree is a path then the complexity can be reduced to O(n ³). This research has partially been supported by the Spezialforschungsbereich F 003 “Optimierung und Kontrolle”, Projektbereich Diskrete Optimierung.     

Inverse <Emphasis Type="Italic">p</Emphasis>-median problems with variable edge lengths

The inverse p-median problem with variable edge lengths on graphs is to modify the edge lengths at minimum total cost with respect to given modification bounds such that a prespecified set of p vertices becomes a p-median with respect to the new edge lengths. The problem is shown to be strongly NP{\mathcal{NP}}-hard on general graphs and weakly NP{\mathcal{NP}}-hard on series-parallel graphs. Therefore, the special case on a tree is considered: It is shown that the inverse 2-median problem with variable edge lengths on trees is solvable in polynomial time. For the special case of a star graph we suggest a linear time algorithm.     

A Hybrid Heuristic for the p-Median Problem

Given n customers and a set F of m potential facilities, the p-median problem consists in finding a subset of F with p facilities such that the cost of serving all customers is minimized. This is a well-known NP-complete problem with important applications in location science and classification (clustering). We present a multistart hybrid heuristic that combines elements of several traditional metaheuristics to find near-optimal solutions to this problem. Empirical results on instances from the literature attest the robustness of the algorithm, which performs at least as well as other methods, and often better in terms of both running time and solution quality. In all cases the solutions obtained by our method were within 0.1% of the best known upper bounds.     

Discrete stochastic location models

In this paper, we study how the two classical location models, the simple plant location problem and thep-median problem, are transformed in a two-stage stochastic program with recourse when uncertainty on demands, variable production and transportation costs, and selling prices is introduced. We also discuss the relation between the stochastic version of the SPLP and the stochastic version of thep-median.     

Analysis of Decomposition Algorithms with Benders Cuts for p-Median Problem

In this paper the algorithms for solving the p-median problem based on the Benders decomposition are investigated. A family of problems hard for solving with such algorithms is constructed and then generalized to a special NP-hard case of the p-median problem. It is shown that the effectiveness of the considered algorithms depends on the choice of the optimal values of the dual variables used in Benders cuts. In particular, the depth of the cuts can be equal to one.     

A Markov-Nikolskii type inequality for absolutely monotone polynomials of order <Emphasis Type="Italic">K</Emphasis>

A function Q is called absolutely monotone of order k on an interval I if Q(x) ≥ 0, Q′(x) ≥ 0, …, Q(k)(x) ≥ 0, for all x ε I. An essentially sharp (up to a multiplicative absolute constant) Markov inequality for absolutely monotone polynomials of order k in L _p [−1, 1], p > 0, is established. One may guess that the right Markov factor is cn ²/k, and this indeed turns out to be the case. Similarly sharp results hold in the case of higher derivatives and Markov-Nikolskii type inequalities. There is also a remarkable connection between the right Markov inequality for absolutely monotone polynomials of order k in the supremum norm and essentially sharp bounds for the largest and smallest zeros of Jacobi polynomials. This is discussed in the last section of the paper.     

Paranorm <Emphasis Type="Italic">I</Emphasis>-convergent sequence spaces

In this article we introduced the sequence spaces c ^I (p), c ₀ ^I (p), m ^I (p) and m ₀ ^I (p) for p = (p _k ), a sequence of positive real numbers. We study some algebraic and topological properties of these spaces. We prove the decomposition theorem and obtain some inclusion relations.      

On the probabilistic analysis of an approximation algorithm for solving the <Emphasis Type="Italic">p</Emphasis>-median problem

In order to solve the location problem in the p-median form we present an approximation algorithm with time complexity O(n ₂) and the results of its probabilistic analysis. The input data are defined on a complete graph with distances between the vertices expressed by the independent random variables with the same uniform distribution. The value of the objective function produced by the algorithm amounts to a certain sum of the random variables that we analyze basing on an estimate for the probabilities of large deviations of these sums. We use a limit theorem in the form of the Petrov inequalities, taking into account the dependence of the random variables in the sum. The probabilistic analysis yields some estimates for the relative error and the failure probability of our algorithm, as well as conditions for its asymptotic exactness.     

A Cut and Branch Approach for the Capacitated <Emphasis Type="Italic">p</Emphasis>-Median Problem Based on Fenchel Cutting Planes

The capacitated p-median problem (CPMP) consists of finding p nodes (the median nodes) minimizing the total distance to the other nodes of the graph, with the constraint that the total demand of the nodes assigned to each median does not exceed its given capacity. In this paper we propose a cutting plane algorithm, based on Fenchel cuts, which allows us to considerably reduce the integrality gap of hard CPMP instances. The formulation strengthened with Fenchel cuts is solved by a commercial MIP solver. Computational results show that this approach is effective in solving hard instances or considerably reducing their integrality gap.      

Heuristic Procedures for Solving the Discrete Ordered Median Problem

We present two heuristic methods for solving the Discrete Ordered Median Problem (DOMP), for which no such approaches have been developed so far. The DOMP generalizes classical discrete facility location problems, such as the p-median and p-center. The first procedure proposed in this paper is based on a genetic algorithm developed by Moreno Vega (1996) for p-median and p-center problems. Additionally, a second heuristic approach based on the Variable Neighborhood Search metaheuristic (VNS) proposed by Hansen and Mladenović (1997) for the p-median problem is described. An extensive numerical study is presented to show the efficiency of both heuristics and compare them.     

Upper quasi continuous maps and quasi continuous selections

The paper deals with the existence of a quasi continuous selection of a multifunction for which upper inverse image of any open set with compact complement contains a set of the form (G \ I) ∪ J, where G is open and I, J are from a given ideal. The methods are based on the properties of a minimal multifunction which is generated by a cluster process with respect to a system of subsets of the form (G \ I) ∪ J.     

Quotients of Incidence Algebras and the Euler Characteristic

In this note, we show that, if A ? kQ _A /I _A is a schurian strongly simply connected algebra given by its normed presentation, and Σ is the unique poset whose Hasse quiver coincides with Q _A , then A ? kΣ if and only if I _A has a generating set consisting of exactly χ(Q _A ) elements, where χ(Q _A ) is the Euler characteristic of Q _A . We also prove that a quotient of an incidence algebra A = kΣ/J is strongly simply connected if and only if A is simply connected and kΣ is strongly simply connected.     

The poset scheduling problem

Let P and Q be two finite posets and for each pP and qQ let c(p, q) be a specified (real-valued) cost. The poset scheduling problem is to find a function s: PQ such that _pP c(p,s(p)) is minimized, subject to the constraints that p in P implies s(p) in Q. We prove that the poset scheduling problem is NP-hard. This problem with a totally ordered poset Q is proved to be transformable to the closed set problem or the minimum cut problem in a network.This work was done in the fall semester of 1982 when the second author was visiting Cornell University. The first author was his research associate during that period.The first author was supported by National Science Council of Republic of China under Grant NSC73-0208-M008-08 when he wrote the final version of this paper.     

<Emphasis Type="Italic">I</Emphasis>-projection onto isotonic cones and its applications to maximum likelihood estimation for log-linear models

A frequently occurring problem is to find a probability vector,p∈D, which minimizes theI-divergence between it and a given probability vector π. This is referred to as theI-projection of π ontoD. Darroch and Ratcliff (1972,Ann. Math. Statist.,43, 1470–1480) gave an algorithm whenD is defined by some linear equalities and in this paper, for simplicity of exposition, we propose an iterative procedure whenD is defined by some linear inequalities. We also discuss the relationship betweenI-projection and the maximum likelihood estimation for multinomial distribution. All of the results can be applied to isotonic cone.     

Computing the nearest stable matrix pairs

In this paper, we study the nearest stable matrix pair problem: given a square matrix pair (E,A), minimize the Frobenius norm of (Δ_E,Δ_A) such that (E+Δ_E,A+Δ_A) is a stable matrix pair. We propose a reformulation of the problem with a simpler feasible set by introducing dissipative Hamiltonian matrix pairs: A matrix pair (E,A) is dissipative Hamiltonian if A=(J−R)Q with skew‐symmetric J, positive semidefinite R, and an invertible Q such that Q^TE is positive semidefinite. This reformulation has a convex feasible domain onto which it is easy to project. This allows us to employ a fast gradient method to obtain a nearby stable approximation of a given matrix pair.     

The linking set problem: a polynomial special case of the multiple-choice knapsack problem

We consider the linking set problem, which can be seen as a particular case of the multiple-choice knapsack problem. This problem occurs as a subproblem in a decomposition procedure for solving large-scale p-median problems such as the optimal diversity management problem. We show that if a non-increasing diference property of the costs in the linking set problem holds, then the problem can be solved by a greedy algorithm and the corresponding linear gap is null.