Approximation Algorithms for NETWORK DESIGN AND ORIENTEERING
Year: 2010
Type: article
Abstract: This thesis presents approximation algorithms for some N P-Hard combinatorial optimization problems on graphs and networks; in particular, we study problems related to Network Design.Under the widely-believed complexity-theoretic assumption that P = N P, there are no efficient (i.e., polynomial-time) algorithms that solve these problems exactly.Hence, if one desires efficient algorithms for such problems, it is necessary to consider approximate solutions: An approximation algorithm for an N P-Hard problem is a polynomial time algorithm which, for any instance of the problem, finds a solution whose value is guaranteed to be within a multiplicative factor ρ of the value of an optimal solution to that instance.We attempt to design algorithms for which this factor ρ, referred to as the approximation ratio of the algorithm, is as small as possible.The field of Network Design comprises a large class of problems that deal with constructing networks of low cost and/or high capacity, routing data through existing networks, and many related issues.In this thesis, we focus chiefly on designing fault-tolerant networks.Two vertices u, v in a network are said to be k-edge-connected if deleting any set of k -1 edges leaves u and v connected; similarly, they are k-vertex connected if deleting any set of k -1 other vertices or edges leaves u and v connected.We focus on building networks that are highly connected, meaning that even if a small number of edges and nodes fail, the remaining nodes will still be able to communicate.A brief description of some of our results is given below.We study the problem of building 2-vertex-connected networks that are large and have low cost.Given an n-node graph with costs on its edges and any integer k, we give an O(log n log k) approximation for the problem of finding a minimum-cost 2-vertex-connected subgraph containing at least k nodes.We also give an algorithm of similar approximation ratio for maximizing the number of nodes in a 2-vertex-connected subgraph subject to a budget constraint on the total List of SymbolsTypically the number of vertices and edges in a graph, respectively.That is, given a graph G(V, E), we use n to denote |V | and m to denote |E|.OPT The value of an optimal solution to the problem instance being considered.λ G (u, v) The edge-connectivity between vertices u and v in a given graph G.κ G (u, v) The vertex-connectivity between vertices u and v in a given graph G.κ G (u, v) The element-connectivity between two terminal vertices u and v in a given graph G.xii
Author: Nitish John Korula
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7017356743