In this paper we explore some implications of view-ing graphs as geometric objects. This approach of-fers a new perspective on a number of graph-theoretic and algorithmic problems. There are several ways to model graphs geometrically and our main concern here is with geometric representations that respect the met-ric of the (possibly weighted) graph. Given a graph G we map its vertices to a normed space in an attempt to (i) Keep down the dimension of the host space and (ii) Guarantee a small distortion, i.e., make sure that distances between vertices in G closely match the dis-tances between their geometric images. In this paper we develop efficient algorithms for em-bedding graphs low-dimensionally with a small distor-tion. Further algorithmic applications include: 0 A simple, unified approach to a number of prob-lems on multicommodity flows, including the Leighton-Rae Theorem [29] and some of its ex-tensions. 0 For graphs embeddable in low-dimensional spaces with a small distortion, we can find low-diameter decompositions (in the sense of [4] and [34]). The parameters of the decomposition depend only on the dimension and the distortion and not on the size of the graph. 0 In graphs embedded this way, small balanced separators can be found efficiently. Faithful low-dimensional representations of statisti-cal data allow for meaningful and efficient cluster-ing, which is one of the most basic tasks in pattern-recognition. For the (mostly heuristic) methods used

We give the first approximation algorithm for the generalized network Steiner problem, a problem in network design. An instance consists of a network with link-costs and, for each pair fi; jg of nodes, an edge-connectivity requirement r ij . The goal is to find a minimum-cost network using the available links and satisfying the requirements. Our algorithm outputs a solution whose cost is within 2dlog 2 (r + 1)e of optimal, where r is the highest requirement value. In the course of proving the performance guarantee, we prove a combinatorial min-max approximate equality relating minimum-cost networks to maximum packings of certain kinds of cuts. As a consequence of the proof of this theorem, we obtain an approximation algorithm for optimally packing these cuts; we show that this algorithm has application to estimating the reliability of a probabilistic network.

Abstract not found

We present a factor 2 approximation algorithm for finding a minimum-cost subgraph having at least a specified number of edges in each cut. This class of problems includes, among others, the generalized Steiner network problem, which is also known as the survivable network design problem. Our algorithm first solves the linear relaxation of this problem, and then iteratively rounds off the solution. The key idea in rounding off is that in a basic solution of the LP relaxation, at least one edge gets included at least to the extent of half. We include this edge into our integral solution and solve the residual problem. 1 Introduction We consider the problem of finding a minimum-cost subgraph of a given graph such that the number of edges crossing each cut is at least a specified requirement. Formally, given an undirected multigraph G = (V; E), a non-negative cost function c : E ! Q+ , and a requirement function f : 2 V ! Z , solve the following integer program (IP): min X e2E c e x...

This paper presents a new algorithm for dynamic routing of bandwidth guaranteed tunnels where tunnel routing requests arrive one-by-one and there is no a priori knowledge regarding future requests. This problem is motivated by service provider needs for fast deployment of bandwidth guaranteed services and the consequent need in backbone networks for fast provisioning of bandwidth guaranteed paths. Offline routing algorithms cannot be used since they require a priori knowledge of all tunnel requests that are to be routed. Instead, on-line algorithms that handle requests arriving one-by-one and that satisfy as many potential future demands as possible are needed. The newly developed algorithm is an on-line algorithm and is based on the idea that a newly routed tunnel must follow a route that does not "interfere too much" with a route that may be critical to satisfy a future demand. We show that this problem is NP-hard. We then develop a path selection heuristic that is based on the idea ...

Consider the multicommodity flow problem in which the object is to maximize the sum of commodities routed. We prove the following approximate max-flow min-multicut theorem: min multicut O(logk) max flow min multicut; where k is the number of commodities. Our proof is constructive; it enables us to find a multicut within O(log k) of the max flow (and hence also the optimal multicut). In addition, the proof technique provides a unified framework in which one can also analyse the case of flows with specified demands, of Leighton-Rao and Klein et.al., and thereby obtain an improved bound for the latter problem. 1 Introduction Much of flow theory, and the theory of cuts in graphs, is built around a single theorem - the celebrated max-flow min-cut theorem of Ford and Fulkerson [FF], and Elias, Feinstein and Shannon [EFS]. The power of this theorem lies in that it relates two fundamental graph-theoretic entities via the potent mechanism of a min-max relation. The importance of this theor...

The primal-dual method is a standard tool in the design of algorithms for combinatorial optimization problems. This chapter shows how the primal-dual method can be modified to provide good approximation algorithms for a wide variety of NP-hard problems. We concentrate on results from recent research applying the primal-dual method to problems in network design.

Abstract. It is shown that the minimum cut ratio is within a factor of O(log k) of the maximum concurrent flow for k-commodity flow instances with arbitrary capacities and demands. This improves upon the previously best-known bound of O(log 2 k) and is existentially tight, up to a constant factor. An algorithm for finding a cut with ratio within a factor of O(log k) of the maximum concurrent flow, and thus of the optimal min-cut ratio, is presented.

We study the approximability of edge-disjoint paths and related problems. In the edge-disjoint paths problem (EDP), we are given a network G with source-sink pairs (si, ti), 1 ≤ i ≤ k, and the goal is to find a largest subset of source-sink pairs that can be simultaneously connected in an edge-disjoint manner. We show that in directed networks, for any ɛ> 0, EDP is NP-hard to approximate within m 1/2−ɛ. We also design simple approximation algorithms that achieve essentially matching approximation guarantees for some generalizations of EDP. Another related class of routing problems that we study concerns EDP with the additional constraint that the routing paths be of bounded length. We show that, for any ɛ> 0, bounded length EDP is hard to approximate within m 1/2−ɛ even in undirected networks, and give an O ( √ m)-approximation algorithm for it. For directed networks, we show that even the single source-sink pair case (i.e. find the maximum number of paths of bounded length between a given sourcesink pair) is hard to approximate within m 1/2−ɛ, for any ɛ> 0.

Abstract not found