No abstract

In this paper we consider the problem of (k, ν)-balanced graph partitioning -dividing the vertices of a graph into k almost equal size components (each of size less than ν · n k ) so that the capacity of edges between different components is minimized. This problem is a natural generalization of several other problems such as minimum bisection, which is the (2, 1)-balanced partitioning problem. We present a bicriteria polynomial time approximation algorithm with an O(log 2 n)-approximation for any constant ν > 1. For ν = 1 we show that no polytime approximation algorithm can guarantee a finite approximation ratio unless P = N P . Previous work has only considered the (k, ν)-balanced partitioning problem for ν ≥ 2.

In this paper we present a polynomial time approximation algorithm for designing a multicast overlay network. The algorithm finds a solution that satisfies capacity and reliability constraints to within a constant factor of optimal, and cost to within a logarithmic factor. The class of networks that our algorithm applies to includes the one used by Akamai Technologies to deliver live media streams over the Internet. In particular, we analyze networks consisting of three stages of nodes. The nodes in the first stage are the sources where live streams originate. A source forwards each of its streams to one or more nodes in the second stage, which are called reflectors. A reflector can split an incoming stream into multiple identical outgoing streams, which are then sent on to nodes in the third and final stage, which are called the sinks.As the packets in a stream travel from one stage to the next, some of them may be lost. The job of a sink is to combine the packets from multiple instances of the same stream (by reordering packets and discarding duplicates) to form a single instance of the stream with minimal loss. We assume that the loss rate between any pair of nodes in the network is known, and that losses between different pairs are independent, but discuss extensions in which some losses may be correlated.

We consider the problem of Simultaneous Source Location -selecting locations for sources in a capacitated graph such that a given set of demands can be satisfied simultaneously, with the goal of minimizing the number of locations chosen. For general directed and undirected graphs we give an O(log D) approximation algorithm, where D is the sum of demands, and prove matching Ω(log D) hardness results assuming P = NP. For undirected trees, we give an exact algorithm and show how this can be combined with a result of Räcke to give a solution that exceeds edge capacities by at most O(log 2 n log log n), where n is the number of nodes. For undirected graphs of bounded treewidth we show that the problem is still NP-Hard, but we are able to give a PTAS with at most (1 + ) violation of the capacities for arbitrarily small , or a (k + 1)-approximation with exact capacities, where k is the treewidth.

The developed methodology for assessing the level of quality of passenger service presented below is relied on existing developments in this area, international standards for determining the quality of service and passengers’ questionnaire. The main goal of the proposed methodology is to determine the necessary social effect of the urban passenger transport, as well as the costs that must be incurred to ensure it and the income that will be received at a given level of service for the population of the city (region). When determining all these indicators, it is necessary to take into account the interests of all participants in the passenger service market (urbanities, owners of private motor transport enterprises and passengers). According to the methodology, it is proposed to determine the social effect of the urban transport through the level of quality of passenger service. This indicator should include both the qualitative characteristics of passenger service and the level of transport tariffs of different carriers, since for many categories of passengers, the latter indicator will play a decisive role in choosing the type of the vehicle and the carrier.