14 research outputs found
Fast approximation of centrality and distances in hyperbolic graphs
We show that the eccentricities (and thus the centrality indices) of all
vertices of a -hyperbolic graph can be computed in linear
time with an additive one-sided error of at most , i.e., after a
linear time preprocessing, for every vertex of one can compute in
time an estimate of its eccentricity such that
for a small constant . We
prove that every -hyperbolic graph has a shortest path tree,
constructible in linear time, such that for every vertex of ,
. These results are based on an
interesting monotonicity property of the eccentricity function of hyperbolic
graphs: the closer a vertex is to the center of , the smaller its
eccentricity is. We also show that the distance matrix of with an additive
one-sided error of at most can be computed in
time, where is a small constant. Recent empirical studies show that
many real-world graphs (including Internet application networks, web networks,
collaboration networks, social networks, biological networks, and others) have
small hyperbolicity. So, we analyze the performance of our algorithms for
approximating centrality and distance matrix on a number of real-world
networks. Our experimental results show that the obtained estimates are even
better than the theoretical bounds.Comment: arXiv admin note: text overlap with arXiv:1506.01799 by other author
A spanning tree approach to the absolute p-center problem
Cataloged from PDF version of article.We consider the absolute p-center problem on a general network and propose a spanning tree approach which is motivated by the fact that the problem is NP-hard on general networks but solvable in polynomial time on trees. We first prove that every connected network possesses a spanning tree whose p-center solution is also a solution for the network under consideration. Then we propose two classes of spanning trees that are shortest path trees rooted at certain points of the network. We give an experimental study, based on 1440 instances, to test how often these classes of trees include an optimizing tree. We report our computational results on the performance of both types of trees. © 1999 Elsevier Science Ltd. All rights reserved
Approximation Algorithms for Min-Distance Problems
We study fundamental graph parameters such as the Diameter and Radius in directed graphs, when distances are measured using a somewhat unorthodox but natural measure: the distance between u and v is the minimum of the shortest path distances from u to v and from v to u. The center node in a graph under this measure can for instance represent the optimal location for a hospital to ensure the fastest medical care for everyone, as one can either go to the hospital, or a doctor can be sent to help.
By computing All-Pairs Shortest Paths, all pairwise distances and thus the parameters we study can be computed exactly in O~(mn) time for directed graphs on n vertices, m edges and nonnegative edge weights. Furthermore, this time bound is tight under the Strong Exponential Time Hypothesis [Roditty-Vassilevska W. STOC 2013] so it is natural to study how well these parameters can be approximated in O(mn^{1-epsilon}) time for constant epsilon>0. Abboud, Vassilevska Williams, and Wang [SODA 2016] gave a polynomial factor approximation for Diameter and Radius, as well as a constant factor approximation for both problems in the special case where the graph is a DAG. We greatly improve upon these bounds by providing the first constant factor approximations for Diameter, Radius and the related Eccentricities problem in general graphs. Additionally, we provide a hierarchy of algorithms for Diameter that gives a time/accuracy trade-off
Approximation Algorithms for Min-Distance Problems in DAGs
Graph parameters such as the diameter, radius, and vertex eccentricities are not defined in a useful way in Directed Acyclic Graphs (DAGs) using the standard measure of distance, since for any two nodes, there is no path between them in one of the two directions. So it is natural to consider the distance between two nodes as the length of the shortest path in the direction in which this path exists, motivating the definition of the min-distance. The min-distance between two nodes u and v is the minimum of the shortest path distances from u to v and from v to u.
As with the standard distance problems, the Strong Exponential Time Hypothesis [Impagliazzo-Paturi-Zane 2001, Calabro-Impagliazzo-Paturi 2009] leaves little hope for computing min-distance problems faster than computing All Pairs Shortest Paths, which can be solved in O?(mn) time. So it is natural to resort to approximation algorithms in O?(mn^{1-?}) time for some positive ?. Abboud, Vassilevska W., and Wang [SODA 2016] first studied min-distance problems achieving constant factor approximation algorithms on DAGs, and Dalirrooyfard et al [ICALP 2019] gave the first constant factor approximation algorithms on general graphs for min-diameter, min-radius and min-eccentricities. Abboud et al obtained a 3-approximation algorithm for min-radius on DAGs which works in O?(m?n) time, and showed that any (2-?)-approximation requires n^{2-o(1)} time for any ? > 0, under the Hitting Set Conjecture. We close the gap, obtaining a 2-approximation algorithm which runs in O?(m?n) time. As the lower bound of Abboud et al only works for sparse DAGs, we further show that our algorithm is conditionally tight for dense DAGs using a reduction from Boolean matrix multiplication. Moreover, Abboud et al obtained a linear time 2-approximation algorithm for min-diameter along with a lower bound stating that any (3/2-?)-approximation algorithm for sparse DAGs requires n^{2-o(1)} time under SETH. We close this gap for dense DAGs by obtaining a 3/2-approximation algorithm which works in O(n^{2.350}) time and showing that the approximation factor is unlikely to be improved within O(n^{? - o(1)}) time under the high dimensional Orthogonal Vectors Conjecture, where ? is the matrix multiplication exponent
Implementation of new and classical set covering based algorithms for solving the absolute p-center problem
Ankara : The Department of Industrial Engineering and the Institute of Engineering and Science of Bilkent University, 2011.Thesis (Master's) -- Bilkent University, 2011.Includes bibliographical references leaves 80-82.The p-center problem is a model of locating p facilities on a network in order to
minimize the maximum coverage distance between each vertex and its closest facility. The
main application areas of p-center problem are emergency service locations such as fire and
police stations, hospitals and ambulance services. If the p facilities can be located anywhere
on a network including vertices and interior points of edges, the resulting problem is referred
to as the absolute p-center problem and if they are restricted to vertex locations, it is referred
to as the vertex-restricted problem. The absolute p-center problem is considerably more
complicated to solve than the vertex-restricted version. In the literature, most of the
computational analysis and new algorithm developments are performed through the vertex
restricted case of the p-center problem. The absolute p-center problem has received much less
attention in the literature. In this thesis, our focus is on the absolute p-center problem based on
an algorithm for the p-center problem proposed by Tansel (2009). Our work is the first one to
solve large instances up to 900 vertices on the absolute p-center problem. The algorithm
focuses on solving the p-center problem with a finite series of minimum set covering
problems, but the set covering problems used in the algorithm are constructed differently
compared to the ones traditionally used in the literature. The proposed algorithm is applicable
for both absolute and vertex-restricted p-center problems with weighted and unweighted
cases.Saç, YiğitM.S
A spanning tree approach to the absolute p-center problem
We consider the absolute p-center problem on a general network and propose a spanning tree approach which is motivated by the fact that the problem is NP-hard on general networks but solvable in polynomial time on trees. We first prove that every connected network possesses a spanning tree whose p-center solution is also a solution for the network under consideration. Then we propose two classes of spanning trees that are shortest path trees rooted at certain points of the network. We give an experimental study, based on 1440 instances, to test how often these classes of trees include an optimizing tree. We report our computational results on the performance of both types of trees. © 1999 Elsevier Science Ltd. All rights reserved
Localización simple de servicios deseados y no deseados en redes con múltiples criterios
Análisis y desarrollo de varios modelos de localización de servicios deseados y no deseados en redes con múltiples criterios. Asimismo, se han propuesto algunas mejoras en modelos de localización de servicios no deseados en redes con un solo criterio. Por consiguiente, con respecto a la localización de servicios deseados sobre redes, se propone un algoritmo polinomial para solucionar el problema del cent-dian biobjetivo. También se ha estudiado la localización de un servicio en una red con múltiples objetivos tipo mediana. Asimismo, se ha desarrollado un algoritmo polinomial para solucionar el problema cent-dian multicriterio en redes con múltiples pesos por nodo y múltiples longitudes por arista. Con respecto a los problemas de localización de servicios no deseados, primero tratamos el problema de localización del 1-centro no deseado en redes. Demostramos que las cotas superiores ya propuestas en trabajos anteriores pueden ser ajustadas. Por medio de una formulación más adecuada del problema, se ha desarrollado un nuevo algoritmo polinomial el cual es más sencillo y computacionalmente más rápido que los ya divulgados en la literatura. También se ha analizado el problema de localizar una mediana no deseada en una red, obteniendo una nueva y mejor cota superior. Se presenta un nuevo algoritmo para solucionar este problema. Por otra parte, siguiendo la resolución del problema maxian, también se ha propuesto un nuevo algoritmo para solucionar el problema del anti-cent-dian en redes. Finalmente, se han estudiado los problemas del centro no deseado y de la mediana no deseada en redes multicriterio, estableciendo nuevas propiedades y reglas para eliminar aristas ineficientes. También se presenta el modelo anti-cent-dian como combinación convexa de los dos últimos problemas. Se propone una regla eficaz para quitar aristas que contienen puntos ineficientes, así como un algoritmo polinomial. Además, este modelo se puede modificar ligeramente para generalizar otros modelos presentados en la literatura
A spanning tree approach to solving the absolute p-center problem
Ankara : The Department of Industrial Engineering and the Institute of Engineering and Sciences of Bilkent Univ., 1995.Thesis (Master's) -- Bilkent University, 1995.Includes bibliographical references leaves 78-81The p-center problem on a network is a model to locate p new facilities that will
serve n existing demand points on that network. The objective is to minimize
the maximum of the weighted distances between each demand point and its
nearest new facility. This type of problem usually arises in the location of
emergency facilities like hospitals, police and fire stations. The problem is
known to be VP-Hard on a cyclic network, but polynomial-time solvable on a
tree network. In this study, a spanning tree approach to solving the problem on
a cyclic network is discussed. First, the existence of an optimal spanning tree
that gives the network optimal solution, is proved. Then, two specific types of
spanning trees are introduced and experimentally tested whether they contain
the optimal tree or not. Also, some properties of such an optimal tree are
discussed and some special cases for which the optimal tree can be determined
in polynomial time, are identified.Bozkaya, BurçinM.S
Exact solution methodologies for the p-center problem under single and multiple allocation strategies
Ankara : The Department of Industrial Engineering and the Graduate School of Engineering and Science of Bilkent Univ., 2013.Thesis (Ph. D.) -- Bilkent University, 2013.Includes bibliographical references leaves 89-95.The p-center problem is a relatively well known facility location problem that
involves locating p identical facilities on a network to minimize the maximum
distance between demand nodes and their closest facilities. The focus of the
problem is on the minimization of the worst case service time. This sort of
objective is more meaningful than total cost objectives for problems with a time
sensitive service structure. A majority of applications arises in emergency service
locations such as determining optimal locations of ambulances, fire stations and
police stations where the human life is at stake. There is also an increased
interest in p-center location and related location covering problems in the contexts
of terror fighting, natural disasters and human-caused disasters. The p-center
problem is NP-hard even if the network is planar with unit vertex weights, unit
edge lengths and with the maximum vertex degree of 3. If the locations of the
facilities are restricted to the vertices of the network, the problem is called the
vertex restricted p-center problem; if the facilities can be placed anywhere on the
network, it is called the absolute p-center problem. The p-center problem with
capacity restrictions on the facilities is referred to as the capacitated p-center
problem and in this problem, the demand nodes can be assigned to facilities with
single or multiple allocation strategies. In this thesis, the capacitated p-center
problem under the multiple allocation strategy is studied for the first time in the
literature.
The main focus of this thesis is a modelling and algorithmic perspective in
the exact solution of absolute, vertex restricted and capacitated p-center problems.
The existing literature is enhanced by the development of mathematical
formulations that can solve typical dimensions through the use of off the-shelf commercial solvers. By using the structural properties of the proposed formulations,
exact algorithms are developed. In order to increase the efficiency of the
proposed formulations and algorithms in solving higher dimensional problems,
new lower and upper bounds are provided and these bounds are utilized during
the experimental studies. The dimensions of problems solved in this thesis are
the highest reported in the literature.Çalık, HaticePh.D