27,325 research outputs found
A Deterministic Algorithm for the Vertex Connectivity Survivable Network Design Problem
In the vertex connectivity survivable network design problem we are given an
undirected graph G = (V,E) and connectivity requirement r(u,v) for each pair of
vertices u,v. We are also given a cost function on the set of edges. Our goal
is to find the minimum cost subset of edges such that for every pair (u,v) of
vertices we have r(u,v) vertex disjoint paths in the graph induced by the
chosen edges. Recently, Chuzhoy and Khanna presented a randomized algorithm
that achieves a factor of O(k^3 log n) for this problem where k is the maximum
connectivity requirement. In this paper we derandomize their algorithm to get a
deterministic O(k^3 log n) factor algorithm. Another problem of interest is the
single source version of the problem, where there is a special vertex s and all
non-zero connectivity requirements must involve s. We also give a deterministic
O(k^2 log n) algorithm for this problem
Distance and the pattern of intra-European trade
Given an undirected graph G = (V, E) and subset of terminals T ⊆ V, the element-connectivity κ ′ G (u, v) of two terminals u, v ∈ T is the maximum number of u-v paths that are pairwise disjoint in both edges and non-terminals V \ T (the paths need not be disjoint in terminals). Element-connectivity is more general than edge-connectivity and less general than vertex-connectivity. Hind and Oellermann [21] gave a graph reduction step that preserves the global element-connectivity of the graph. We show that this step also preserves local connectivity, that is, all the pairwise element-connectivities of the terminals. We give two applications of this reduction step to connectivity and network design problems. • Given a graph G and disjoint terminal sets T1, T2,..., Tm, we seek a maximum number of elementdisjoint Steiner forests where each forest connects each Ti. We prove that if each Ti is k element k connected then there exist Ω( log hlog m) element-disjoint Steiner forests, where h = | i Ti|. If G is planar (or more generally, has fixed genus), we show that there exist Ω(k) Steiner forests. Our proofs are constructive, giving poly-time algorithms to find these forests; these are the first non-trivial algorithms for packing element-disjoint Steiner Forests. • We give a very short and intuitive proof of a spider-decomposition theorem of Chuzhoy and Khanna [12] in the context of the single-sink k-vertex-connectivity problem; this yields a simple and alternative analysis of an O(k log n) approximation. Our results highlight the effectiveness of the element-connectivity reduction step; we believe it will find more applications in the future
Single-Sink Network Design with Vertex Connectivity Requirements
We study single-sink network design problems in undirected graphs
with vertex connectivity requirements. The input to these problems
is an edge-weighted undirected graph , a sink/root vertex
, a set of terminals , and integer . The goal is
to connect each terminal to via emph{vertex-disjoint}
paths. In the {em connectivity} problem, the objective is to find a
min-cost subgraph of that contains the desired paths. There is a
-approximation for this problem when cite{FleischerJW}
but for , the first non-trivial approximation was obtained
in the recent work of Chakraborty, Chuzhoy and Khanna
cite{ChakCK08}; they describe and analyze an algorithm with an
approximation ratio of where .
In this paper, inspired by the results and ideas in cite{ChakCK08},
we show an -approximation bound for a simple
greedy algorithm. Our analysis is based on the dual of a natural
linear program and is of independent technical interest. We use the
insights from this analysis to obtain an -approximation for the more general single-sink {em
rent-or-buy} network design problem with vertex connectivity
requirements. We further extend the ideas to obtain a
poly-logarithmic approximation for the single-sink {em buy-at-bulk}
problem when and the number of cable-types is a fixed
constant; we believe that this should extend to any fixed . We
also show that for the non-uniform buy-at-bulk problem, for each
fixed , a small variant of a simple algorithm suggested by
Charikar and Kargiazova cite{CharikarK05} for the case of
gives an approximation for larger .
These results show that for each of these problems, simple and
natural algorithms that have been developed for have good
performance for small
Graph measures and network robustness
Network robustness research aims at finding a measure to quantify network
robustness. Once such a measure has been established, we will be able to
compare networks, to improve existing networks and to design new networks that
are able to continue to perform well when it is subject to failures or attacks.
In this paper we survey a large amount of robustness measures on simple,
undirected and unweighted graphs, in order to offer a tool for network
administrators to evaluate and improve the robustness of their network. The
measures discussed in this paper are based on the concepts of connectivity
(including reliability polynomials), distance, betweenness and clustering. Some
other measures are notions from spectral graph theory, more precisely, they are
functions of the Laplacian eigenvalues. In addition to surveying these graph
measures, the paper also contains a discussion of their functionality as a
measure for topological network robustness
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