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.     

An ant colony algorithm for the pos/neg weighted p-median problem

Let a graph G = (V, E) with vertex set V and edge set E be given. The classical graph version of the p-median problem asks for a subset of cardinality p, so that the (weighted) sum of the minimum distances from X to all other vertices in V is minimized. We consider the semi-obnoxious case, where every vertex has either a positive or a negative weight. This gives rise to two different objective functions, namely the weighted sum of the minimum distances from X to the vertices in V\X and, differently, the sum over the minimum weighted distances from X to V\X. In this paper an Ant Colony algorithm with a tabu restriction is designed for both problems. Computational results show its superiority with respect to a previously investigated variable neighborhood search and a tabu search heuristic.This research has partially been supported by the Spezialforschungsbereich F 003 “Optimierung und Kontrolle”, Projektbereich Diskrete Optimierung.     

Solving the p-median problem with pos/neg weights by variable neighborhood search and some results for special cases

The p-median problem with positive and negative weights has been introduced by Burkard and Krarup [Computing 60 (1998) 193]. In this paper we discuss some special cases of this problem on trees and propose a variable neighborhood search procedure for general networks, which is in fact a modification of the one proposed by Hansen and Mladenovic [Locat. Sci. 5 (1997) 207] for the p-median. We also compare the results with those obtained by a Tabu search procedure.     

On finding the core of a tree with a specified length

A core of a graph G is a path P in G that is central with respect to the property of minizining d(P) = Σ_υεV(G)d(υ, P), where d(υ, P) is the distance from vertex υ to path P. This paper explores some properties of a core of a specified length.     

A linear algorithm for a core of a tree

A core of a graph G is a path P in G that is central with respect to the property of minimizing d(P) = Σ_υ?V(G)d(υ, P), where d(υ, P) is the distance from vertex υ to path P. We present a linear algorithm for finding a core of a tree. Since the core of a graph is not necessarily unique, we also output a list of all the vertices which are in some core.     

Efficient algorithms for a constrained k-tree core problem in a tree network

Let T=(V,E) be a free tree in which each vertex has a weight and each edge has a length. Let n=|V|. Given T and parameters k and l, a (k,l)-tree core is a subtree X of T with diameter l, having k leaves, which minimizes the sum of the weighted distances from all vertices in T to X. In this paper, two efficient algorithms are presented for finding a (k,l)-tree core of T. The first algorithm has O(n²) time complexity for the case that each edge has an arbitrary length. The second algorithm has O(lkn) time complexity for the case that the lengths of all edges are 1. The (k,l)-tree core problem has an application in distributed database systems.     

Finding all systems of weight functions associated with a given extended Chebyshev space

Systems of weight functions and corresponding generalised derivatives are classically used to build extended Chebyshev spaces on a given interval. This is a well-known procedure. Conversely, if the interval is closed and bounded, it is known that a given extended Chebyshev space can always be associated with a system of weight functions via the latter procedure. In the present article we determine all such possibilities, that is, all systems of weight functions which can be used to define a given extended Chebyshev space on a closed bounded interval.     

Finding cliques of maximum weight on a generalization of permutation graphs

We propose a dynamic programming procedure for computing the clique of maximum weight on a class of graphs arising in the solution of train timetabling problems. These graphs generalize, in two ways permutation graphs, defined as the intersection graphs of segments joining two parallel lines. First, two segments are joined by an edge not only if they intersect but also if their endpoints are sufficiently close. Second, two parallel segments may be joined by an edge even if they are arbitrarily far away from each other.     

The pos/neg-weighted 2-medians in balanced trees with subtree-shaped customers

This paper deals with the pos/neg-weighted p-median problem on tree graphs where all customers are modeled as subtrees. We present a polynomial algorithm for the 2-median problem on an arbitrary tree. Then we improve the time complexity to O(n log n) for the problem on a balanced tree, where n is the number of the vertices in the tree.     

The weight of the shortest path tree

The minimal weight of the shortest path tree in a complete graph with independent and exponential (mean 1) random link weights is shown to converge to a Gaussian distribution. We prove a conditional central limit theorem and show that the condition holds with probability converging to 1. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2007     

An optimal time algorithm for finding a maximum weight independent set in a tree

The maximum weight independent set problem for a general graph is NP-hard. But for some special classes of graphs, polynomial time algorithms do exist for solving it. Based on the divide-and-conquer strategy, Pawagi has presented anO(|V|log|V|) time algorithm for solving this problem on a tree. In this paper, we propose anO(|V|) time algorithm to improve Pawagi's result. The proposed algorithm is based on the dynamic programming strategy and is time optimal within a constant factor.     

On the core of cost-revenue games: Minimum cost spanning tree games with revenues

In this paper, we analyze cost sharing problems arising from a general service by explicitly taking into account the generated revenues. To this cost-revenue sharing problem, we associate a cooperative game with transferable utility, called cost-revenue game. By considering cooperation among the agents using the general service, the value of a coalition is defined as the maximum net revenues that the coalition may obtain by means of cooperation. As a result, a coalition may profit from not allowing all its members to get the service that generates the revenues. We focus on the study of the core of cost-revenue games. Under the assumption that cooperation among the members of the grand coalition grants the use of the service under consideration to all its members, it is shown that a cost-revenue game has a nonempty core for any vector of revenues if, and only if, the dual game of the cost game has a large core. Using this result, we investigate minimum cost spanning tree games with revenues. We show that if every connection cost can take only two values (low or high cost), then, the corresponding minimum cost spanning tree game with revenues has a nonempty core. Furthermore, we provide an example of a minimum cost spanning tree game with revenues with an empty core where every connection cost can take only one of three values (low, medium, or high cost).     

The bandwidth of a tree with k leaves is at most

The bandwidth B(G) of a finite simple graph G is the minimum of the quantity max{ƒ(x)-ƒ(y) : xy E(G)} taken over all injective integer labellings ƒ of G. We prove that if a tree T has k leaves then B(T) [k/2]. This improves the previously known upper bound B(T) V(T)/2.     

On the core and nucleolus of minimum cost spanning tree games

We develop two efficient procedures for generating cost allocation vectors in the core of a minimum cost spanning tree (m.c.s.t.) game. The first procedure requires O(n ²) elementary operations to obtain each additional point in the core, wheren is the number of users. The efficiency of the second procedure, which is a natural strengthening of the first procedure, stems from the special structure of minimum excess coalitions in the core of an m.c.s.t. game. This special structure is later used (i) to ease the computational difficulty in computing the nucleolus of an m.c.s.t. game, and (ii) to provide a geometric characterization for the nucleolus of an m.c.s.t. game. This geometric characterization implies that in an m.c.s.t. game the nucleolus is the unique point in the intersection of the core and the kernel. We further develop an efficient procedure for generating fair cost allocations which, in some instances, coincide with the nucleolus. Finally, we show that by employing Sterns' transfer scheme we can generate a sequence of cost vectors which converges to the nucleolus. Part of this research was done while the author was visiting the Department of Operations Research at Stanford University. This research was partially supported by Natural Sciences and Engineering Research Council Canada Grant A-4181.     

Finding groups with a simple genetic algorithm*

A new solution to the old problem of partitioning a matrix of social proximities into groups is proposed. It draws on a heuristic developed in computer science, the simple genetic algorithm. The algorithm is described and its utility is demonstrated with applications to three standard data sets.     

Finding Global Minima with a Computable Filled Function

The Filled Function Method is an approach to finding global minima of multidimensional nonconvex functions. The traditional filled functions have features that may affect the computability when applied to numerical optimization. This paper proposes a new filled function. This function needs only one parameter and does not include exponential terms. Also, the lower bound of weight factor a is usually smaller than that of one previous formulation. Therefore, the proposed new function has better computability than the traditional ones.     

p-adic Gibbs measures for the hard core model with three states on the Cayley tree

We study p-adic hard core models with three states on the Cayley tree. It is known that there are four types of such models. We find conditions that must be imposed on the order k of the Cayley tree and on the prime p for a translation-invariant p-adic Gibbs measure to exist.