The continuous and discrete path variance problem on trees

In this paper we consider the problem of locating path-shaped facilities on a tree minimizing the variance objective function. This kind of objective function is generally adopted in location problems which arise in the public sector applications, such as the location of evacuation routes or mass transit routes. We consider the general case in which a positive weight is assigned to each vertex of the tree and positive real lengths are associated to the edges. We study both the case in which the path is continuous, that is, the end points of the optimal path can be either vertices or points along an edge, and the case in which the path is discrete, that is, the end points of the optimal path must lie in some vertex of the tree. Given a tree with n vertices, for both these problems we provide algorithms with O(n2) time complexity and we extend our results also to the case in which the length of the path is bounded above. Even in this case we provide polynomial algorithms with the same O(n2) complexity. In particular, our algorithm for the continuous path-variance problem improves upon a log n term the previous best known algorithm for this problem provided in [T. Cáceres, M.C. López-de-los-Mozos, J.A. Mesa (2004). The path-variance problem on tree networks, Discrete Applied Mathematics, 145, 72-79]. Finally, we show that no nestedness property holds for (discrete and continuous) point-variance problem with respect to the corresponding path-variance.Ministerio de Ciencia y TecnologíaAzioni Integrate Italia-Spagna (Ministero dell'istruzione, dell'università e della ricerca

Extensive facility location problems on networks with equity measures

AbstractThis paper deals with the problem of locating path-shaped facilities of unrestricted length on networks. We consider as objective functions measures conceptually related to the variability of the distribution of the distances from the demand points to a facility. We study the following problems: locating a path which minimizes the range, that is, the difference between the maximum and the minimum distance from the vertices of the network to a facility, and locating a path which minimizes a convex combination of the maximum and the minimum distance from the vertices of the network to a facility, also known in decision theory as the Hurwicz criterion. We show that these problems are NP-hard on general networks. For the discrete versions of these problems on trees, we provide a linear time algorithm for each objective function, and we show how our analysis can be extended also to the continuous case

Centroidal bases in graphs

We introduce the notion of a centroidal locating set of a graph $G$, that is, a set $L$ of vertices such that all vertices in $G$ are uniquely determined by their relative distances to the vertices of $L$. A centroidal locating set of $G$ of minimum size is called a centroidal basis, and its size is the centroidal dimension $CD(G)$. This notion, which is related to previous concepts, gives a new way of identifying the vertices of a graph. The centroidal dimension of a graph $G$ is lower- and upper-bounded by the metric dimension and twice the location-domination number of $G$, respectively. The latter two parameters are standard and well-studied notions in the field of graph identification. We show that for any graph $G$ with $n$ vertices and maximum degree at least~2, $(1+o(1))\frac{\ln n}{\ln\ln n}\leq CD(G) \leq n-1$. We discuss the tightness of these bounds and in particular, we characterize the set of graphs reaching the upper bound. We then show that for graphs in which every pair of vertices is connected via a bounded number of paths, $CD(G)=\Omega\left(\sqrt{|E(G)|}\right)$, the bound being tight for paths and cycles. We finally investigate the computational complexity of determining $CD(G)$ for an input graph $G$, showing that the problem is hard and cannot even be approximated efficiently up to a factor of $o(\log n)$. We also give an $O\left(\sqrt{n\ln n}\right)$-approximation algorithm

The path-variance problem on tree networks

Extensive facility location models on graphs deal with the location of a special type of subgraphs such as paths, trees or cycles and can be considered as extensions of classical point location models. Variance is one of the measures applied in models in which some equality requirement is imposed. In this paper the problem of locating a minimum variance path in a tree network is addressed, and an O(n2 log n) time algorithm is proposed.Ministerio de Ciencia y Tecnología BFM2000-1052-C02-0

Locating and Protecting Facilities Subject to Random Disruptions and Attacks

Recent events such as the 2011 Tohoku earthquake and tsunami in Japan have revealed the vulnerability of networks such as supply chains to disruptive events. In particular, it has become apparent that the failure of a few elements of an infrastructure system can cause a system-wide disruption. Thus, it is important to learn more about which elements of infrastructure systems are most critical and how to protect an infrastructure system from the effects of a disruption. This dissertation seeks to enhance the understanding of how to design and protect networked infrastructure systems from disruptions by developing new mathematical models and solution techniques and using them to help decision-makers by discovering new decision-making insights. Several gaps exist in the body of knowledge concerning how to design and protect networks that are subject to disruptions. First, there is a lack of insights on how to make equitable decisions related to designing networks subject to disruptions. This is important in public-sector decision-making where it is important to generate solutions that are equitable across multiple stakeholders. Second, there is a lack of models that integrate system design and system protection decisions. These models are needed so that we can understand the benefit of integrating design and protection decisions. Finally, most of the literature makes several key assumptions: 1) protection of infrastructure elements is perfect, 2) an element is either fully protected or fully unprotected, and 3) after a disruption facilities are either completely operational or completely failed. While these may be reasonable assumptions in some contexts, there may exist contexts in which these assumptions are limiting. There are several difficulties with filling these gaps in the literature. This dissertation describes the discovery of mathematical formulations needed to fill these gaps as well as the identification of appropriate solution strategies

Center location problems on tree graphs with subtree-shaped customers

We consider the p-center problem on tree graphs where the customers are modeled as continua subtrees. We address unweighted and weighted models as well as distances with and without addends. We prove that a relatively simple modification of Handler’s classical linear time algorithms for unweighted 1- and 2-center problems with respect to point customers, linearly solves the unweighted 1- and 2-center problems with addends of the above subtree customer model. We also develop polynomial time algorithms for the p-center problems based on solving covering problems and searching over special domains

A Positive Theory of Network Connectivity

This paper develops a positive theory of network connectivity, seeking to explain the micro-foundations of alternative network topologies as the result of self-interested actors. By building roads, landowners hope to increase their parcelsÕ accessibility and economic value. A simulation model is performed on a grid-like land use layer with a downtown in the center, whose structure resembles the early form of many Midwest- ern and Western (US) cities. The topological attributes for the networks are evaluated. This research posits that road networks experience an evolutionary process where a tree-like structure first emerges around the centered parcel before the network pushes outward to the periphery. In addition, road network topology undergoes clear phase changes as the economic values of parcels vary. The results demonstrate that even without a centralized authority, road networks have the property of self-organization and evolution, and, that in the absence of intervention, the tree-like or web-like nature of networks is a result of the underlying economics.road network, land parcel, network evolution, network growth, phase change, centrality measures, degree centrality, closeness centrality, betweenness centrality, network structure, treeness, circuitness, topology

Finding the ℓ-core of a tree

AbstractAn ℓ-core of a tree T=(V,E) with |V|=n, is a path P with length at most ℓ that is central with respect to the property of minimizing the sum of the distances from the vertices in P to all the vertices of T not in P. The distance between two vertices is the length of the shortest path joining them. In this paper we present efficient algorithms for finding the ℓ-core of a tree. For unweighted trees we present an O(nℓ) time algorithm, while for weighted trees we give a procedure with time complexity of O(nlog2n). The algorithms use two different types of recursive principle in their operation