On the exponential cardinality of FDS for the ordered p-median problem

We study finite dominating sets (FDS) for the ordered median problem. This kind of problems allows to deal simultaneously with a large number of models. We show that there is no valid polynomial size FDS for the general multifacility version of this problem even on path networks.     

Heuristics for the fixed cost median problem

We describe in this paper polynomial heuristics for three important hard problems—the discrete fixed cost median problem (the plant location problem), the continuous fixed cost median problem in a Euclidean space, and the network fixed cost median problem with convex costs. The heuristics for all the three problems guarantee error ratios no worse than the logarithm of the number of customer points. The derivation of the heuristics is based on the presentation of all types of median problems discussed as a set covering problem.     

Extensions to the continuous ordered median problem

Classical location models fix an objective function and then attempt to find optimal points to this objective. In the last years a flexible approach, the ordered median problem, has been introduced. It handles a wide class of objectives, such as the median, the center and the centdian function. In this paper we present new properties of the ordered median problem such as solvability for the situation of attractive and repulsive locations. We also develop a new solution method that even yields local optimal points for non-convex objective functions. Furthermore, we discuss separability of ordered median problems without repulsion and derive a sufficient criterion. Finally, we introduce a useful model extension, the facility class model, which allows to deal with a wider range of real world problems in the ordered median setting.     

Solving the ordered one-median problem in the plane

In this paper, we propose a general approach solution method for the single facility ordered median problem in the plane. All types of weights (non-negative, non-positive, and mixed) are considered. The big triangle small triangle approach is used for the solution. Rigorous and heuristic algorithms are proposed and extensively tested on eight different problems with excellent results.     

Genetic algorithms for solving the discrete ordered median problem

In this paper we present two new heuristic approaches to solve the Discrete Ordered Median Problem (DOMP). Described heuristic methods, named HGA1 and HGA2 are based on a hybrid of genetic algorithms (GA) and a generalization of the well-known Fast Interchange heuristic (GFI). In order to investigate the effect of encoding on GA performance, two different encoding schemes are implemented: binary encoding in HGA1, and integer representation in HGA2. If binary encoding is used (HGA1), new genetic operators that keep the feasibility of individuals are proposed. Integer representation keeps the individuals feasible by default, so HGA2 uses slightly modified standard genetic operators. In both methods, caching GA technique was integrated with the GFI heuristic to improve computational performance. The algorithms are tested on standard ORLIB p-median instances with up to 900 nodes. The obtained results are also compared with the results of existing methods for solving DOMP in order to assess their merits.     

A simulated annealing based solution approach for the two-layered location registration and paging areas partitioning problem in cellular mobile networks

This paper presents a mathematical model and simulated annealing based solution approach for finding optimal location updates and paging area configuration for mobile communication networks. We use a two-layered zone-based location registration and paging scheme in which the costs of location updates and paging signaling traffic are reduced by introducing a two-step paging process. The location updates and paging procedures in a two-layered scheme are first described, and an approximation of the measure required for calculating the paging-related signaling volume is provided based on assumptions of cell shapes and mobile stations’ movement patterns. A simulated annealing (SA)-based solution method is devised along with a greedy heuristic, and computational experiments are conducted to illustrate the superiority of the proposed SA-based method over other solution methods.     

Relationships between total domination,order, size,and maximum degree of graphs

A total dominating set, S, in a graph, G, has the property that every vertex in G is adjacent to a vertex in S. The total dominating number, γ_t(G) of a graph G is the size of a minimum total dominating set in G. Let G be a graph with no component of size one or two and with Δ(G) ≥ 3. In 6 , it was shown that |E(G)| ≤ Δ(G) (|V(G)|–γ_t(G)) and conjectured that |E(G)| ≤ (Δ(G)+3) (|V(G)|–γ_t(G))/2 holds. In this article, we prove that holds and that the above conjecture is false as there for every Δ exist Δ‐regular bipartite graphs G with |E(G)| ≥ (Δ+0.1 ln(Δ)) (|V(G)|–γ_t(G))/2. The last result also disproves a conjecture on domination numbers from 8 . © 2007 Wiley Periodicals, Inc. J Graph Theory 55: 325–337, 2007     

Location-domination in line graphs

A set $D$ of vertices of a graph $G$ is locating if every two distinct vertices outside $D$ have distinct neighbors in $D$; that is, for distinct vertices $u$ and $v$ outside $D$, $N\left(u\right)\cap D\ne N\left(v\right)\cap D$, where $N\left(u\right)$ denotes the open neighborhood of $u$. If $D$ is also a dominating set (total dominating set), it is called a locating-dominating set (respectively, locating-total dominating set) of $G$. A graph $G$ is twin-free if every two distinct vertices of $G$ have distinct open and closed neighborhoods. It is conjectured (Garijo et al., 2014 [15]) and (Foucaud and Henning, 2016 [12]) respectively, that any twin-free graph $G$ without isolated vertices has a locating-dominating set of size at most one-half its order and a locating-total dominating set of size at most two-thirds its order. In this paper, we prove these two conjectures for the class of line graphs. Both bounds are tight for this class, in the sense that there are infinitely many connected line graphs for which equality holds in the bounds.     

A minmax regret median problem on a tree under uncertain locations of the demand points

This paper deals with the location of a new facility on a tree according to the minimization of the weighted distance to the customers. The weights represent demands of the set of nodes. The exact location of each customer will be assumed unknown but close   to its corresponding node. In this context, an algorithm to find a minmax regret median is proposed and its complexity is shown to be O(nlog(n))$O(nlog\left(n\right))$, where n$n$ is the number of nodes of the tree     

The multicriteria p-facility median location problem on networks

In this paper we discuss the multicriteria p-facility median location problem on networks with positive and negative weights. We assume that the demand is located at the nodes and can be different for each criterion under consideration. The goal is to obtain the set of Pareto-optimal locations in the graph and the corresponding set of non-dominated objective values. To that end, we first characterize the linearity domains of the distance functions on the graph and compute the image of each linearity domain in the objective space. The lower envelope of a transformation of all these images then gives us the set of all non-dominated points in the objective space and its preimage corresponds to the set of all Pareto-optimal solutions on the graph. For the bicriteria 2-facility case we present a low order polynomial time algorithm. Also for the general case we propose an efficient algorithm, which is polynomial if the number of facilities and criteria is fixed.     

A modified variable neighborhood search for the discrete ordered median problem

This paper presents a modified Variable Neighborhood Search (VNS) heuristic algorithm for solving the Discrete Ordered Median Problem (DOMP). This heuristic is based on new neighborhoods’ structures that allow an efficient encoding of the solutions of the DOMP avoiding sorting in the evaluation of the objective function at each considered solution. The algorithm is based on a data structure, computed in preprocessing, that organizes the minimal necessary information to update and evaluate solutions in linear time without sorting. In order to investigate the performance, the new algorithm is compared with other heuristic algorithms previously available in the literature for solving DOMP. We report on some computational experiments based on the well-known N-median instances of the ORLIB with up to 900 nodes. The obtained results are comparable or superior to existing algorithms in the literature, both in running times and number of best solutions found.     

Variable neighborhood search for minimum sum-of-squares clustering on networks

Euclidean Minimum Sum-of-Squares Clustering amounts to finding p prototypes by minimizing the sum of the squared Euclidean distances from a set of points to their closest prototype. In recent years related clustering problems have been extensively analyzed under the assumption that the space is a network, and not any more the Euclidean space. This allows one to properly address community detection problems, of significant relevance in diverse phenomena in biological, technological and social systems. However, the problem of minimizing the sum of squared distances on networks have not yet been addressed. Two versions of the problem are possible: either the p prototypes are sought among the set of nodes of the network, or also points along edges are taken into account as possible prototypes. While the first problem is transformed into a classical discrete p-median problem, the latter is new in the literature, and solved in this paper with the Variable Neighborhood Search heuristic. The solutions of the two problems are compared in a series of test examples.     

Comparing different metaheuristic approaches for the median path problem with bounded length

In this paper we consider the Bounded Length Median Path Problem which can be defined as the problem of locating a path-shaped facility that departures from a given origin and arrives at a given destination in a network. The length of the path is assumed to be bounded by a given maximum length. At each vertex of the network (customer-point) the demand for the service is given and the cost to reach the closest service-point is computed. The objective is to minimize the sum of these costs over all the customer-points in the network.     

A note of the knapsack problem with special ordered sets

The knapsack problem with special ordered sets and arbitrarily signed coefficients is shown to be equivalent to a standard problem of the same type but having all coefficients positive. Two propositions are proven which define an algorithm for the linear programming relaxation of the standard problem that is a natural generalization of the Dantzig solution to the problem without special ordered sets/ Several properties of the corvex hull of the associated zero-one polytope are derived.     

Pebbling Graphs of Fixed Diameter

Given a configuration of indistinguishable pebbles on the vertices of a connected graph G on n vertices, a pebbling move is defined as the removal of two pebbles from some vertex, and the placement of one pebble on an adjacent vertex. The m‐pebbling number of a graph G, , is the smallest integer k such that for each vertex v and each configuration of k pebbles on G there is a sequence of pebbling moves that places at least m pebbles on v. When , it is simply called the pebbling number of a graph. We prove that if G is a graph of diameter d and are integers, then , where denotes the size of the smallest distance k dominating set, that is the smallest subset of vertices such that every vertex is at most distance k from the set, and, . This generalizes the work of Chan and Godbole (Discrete Math 208 (2008), 15–23) who proved this formula for . As a corollary, we prove that . Furthermore, we prove that if d is odd, then , which in the case of answers for odd d, up to a constant additive factor, a question of Bukh (J Graph Theory 52 (2006), 353–357) about the best possible bound on the pebbling number of a graph with respect to its diameter.     

Transformational antiatomic characteristic of ordered sets

To any ordered set with a universally maximal element, a semigroup of its transformations with some natural properties that defines the ordered set up to an isomorphism is assigned. The system of such transformation semigroups is proved to be the minimal element in the set of all defining systems of transformation semigroups with respect to the following ordering: one system precedes another if for each ordered set from the class in question, the semigroup of its transformation belonging to the first system is contained in the semigroup of its transformation from the second system. Translated fromMatematicheskie Zametki, Vol. 66, No. 1, pp. 112–119, July, 1999.     

Steiner's problem and fagnano's result on the sphere

This note describes the nature of optimal solutions for the spherical Steiner-Weber location problem for the case of unit weights and either 3 or 4 demand points (requireing 4 demand points to lie in an open hemisphere). Geometrically appealing results which are necessary conditions for optimum solutions and spherical analogs of known planar results are obtained.     

Application of renewal theory to call handover counting and dynamic location management in cellular mobile networks

Mobility management in wireless cellular networks is one of the main issues for resource optimization. It is aimed to keep track of Mobile Stations (MSs) in the different Location Areas (LAs) or Registration Areas (RAs) for an efficient call delivery. The optimization issues of these location strategies look for a minimization of the generated signaling traffic. We describe the three basic strategies for location management: distance-based, time-based and movement-based, and their corresponding optimization cost. We emphasize that counting the number of wireless cell crossings or handovers occurring in the call duration time or during inter-call times is a fundamental issue for mobility management analysis. We present the main approaches in the literature to deal with these problems with a special emphasis to renewal theory to model the probabilistic structure of these optimization problems.