Online throughput-competitive algorithm for multicast routing and admission control

We present the first polylog-competitive online algorithm for the general multicast problem in the throughput model. The ratio of the number of requests accepted by the optimum offline algorithm to the expected number of requests accepted by our algorithm is O((log n+log log M)(log n+log M)log n), where M is the number of multicast groups and n is the number of nodes in the graph. We show that this is close to optimum by presenting an _O_(log n log M) lower bound on this ratio for any randomized online algorithm against an oblivious adversary, when M is much larger than the link capacities. Our lower bound applies even in the restricted cause where the link capacities are much larger than bandwidth requested by a single multicast. We also present a simple proof showing that it is impossible to be competitive against an adaptive online adversary. As in the previous online routing algorithms, our algorithm uses edge-costs when deciding on which is the best path to use. In contrast to the previous competitive algorithms in the throughput model, our cost is not a direct function of the edge load. The new cost definition allows us to decouple the effects of routing and admission decisions of different multicast groups

Scheduling data transfers in a network and the set scheduling problem

In this paper we consider the online ftp problem. The goal is to service a sequence of file transfer requests given bandwidth constraints of the underlying communication network. The main result of the paper is a technique that leads to algorithms that optimize several natural metrics, such as max-stretch, total flow time, max flow time, and total completion time. In particular, we show how to achieve optimum total flow time and optimum max-stretch if we increase the capacity of the underlying network by a logarithmic factor. We show that the resource augmentation is necessary by proving polynomial lower bounds on the max-stretch and total flow time for the case where online and offline algorithms are using same-capacity edges. Moreover, we also give poly-logarithmic lower bounds on the resource augmentation factor necessary in order to keep the total flow time and max-stretch within a constant factor of optimum

Improved Bounds on the Max-Flow Min-Cut Ratio for Multicommodity Flows

In this paper we consider the worst case ratio between the capacity of min-cuts and the value of max-flow for multicommodity flow problems. We improve the best known bounds for the mincut max-flow ratio for multicommodity flows in undirected graphs, by replacing the O(log D) in the bound by O(log k), where D denotes the sum of all demands, and k denotes the number of commodities. In essence we prove that up to constant factors the worst min-cut max-flow ratios appear in problems where demands are integral and polynomial in the number of commodities. Klein, Rao, Agrawal, and Ravi have previously proved that if the demands and the capacities are integral, then the min-cut max-flow ratio in general undirected graphs is bounded by O(logC log D), where C denotes the sum of all the capacities. Tragoudas has improved this bound to O(logn log D), where n is the number of nodes in the network. Garg, Vazirani and Yannakakis further improved this to O(log k log D). Klein, Plotkin and Rao have pr..

Parallel Symmetry-Breaking in Sparse Graphs

We describe efficient deterministic techniques for breaking symmetry in parallel. These techniques work well on rooted trees and graphs of constant degree or genus. Our primary technique allows us to 3-color a rooted tree in O(lg n) time on an EREW PRAM using a linear number of processors. We use these techniques to construct fast linear processor algorithms for several problems, including (\Delta + 1)-coloring constantdegree graphs and 5-coloring planar graphs. We also prove lower bounds for 2-coloring directed lists and for finding maximal independent sets in arbitrary graphs. 1 Introduction Some problems for which trivial sequential algorithms exist appear to be much harder to solve in a parallel framework. When converting a sequential algorithm to a parallel one, at each step of the parallel algorithm we have to choose a set of operations which may be executed in parallel. Often, we have to choose these operations from a large set A preliminary version of this paper appear..

Adding Multiple Cost Constraints to Combinatorial Optimization Problems, with Applications to Multicommodity Flows

Minimum cost multicommodity flow is an instance of a simpler problem (multicommodity flow) to which a cost constraint has been added. In this paper we present a general scheme for solving a large class of such "cost-added" problems---even if more than one cost is added. One of the main applications of this method is a new deterministic algorithm for approximately solving the minimumcost multicommodity flow problem. Our algorithm finds a (1 + ffl) approximation to the minimum cost flow in ~ O(ffl \Gamma3 kmn) time, where k is the number of commodities, m is the number of edges, and n is the number vertices in the input problem. This improves the previous best deterministic bounds of O(ffl \Gamma4 kmn 2 ) [9] and ~ O(ffl \Gamma2 k 2 m 2 ) [15] by factors of n=ffl and fflkm=n respectively. In fact, it even dominates the best randomized bound of ~ O(ffl \Gamma2 km 2 ) [15]. The algorithm presented in this paper efficiently solves several other interesting generali..

Random Sampling in Graph Optimization Problems

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Topics in Analysis of Algorithms

Contents 1 Preliminaries 4 1.1 The Fundamental Theorem of Linear Inequalities : : : : : : : : : : : : : : : : : : 4 1.1.1 Definitions : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 1.1.2 The Fundamental Theorem : : : : : : : : : : : : : : : : : : : : : : : : : : 4 1.1.3 Applications : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6 1.2 Farkas' Lemma : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 8 1.3 LP Duality : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 10 1.3.1 Standard form of Duality Theorem : : : : : : : : : : : : : : : : : : : : : : 10 1.3.2 Alternate Forms of LP Duality : : : : : : : : : : : : : : : : : : : : : : : : 11 1.3.3 Complementary Slackness : : : : : : : : : : : : : : : : : : : : : : : : : : : 13 1.3.4 Intuition o