Asaf Levin

Approximations for minimum and min-max vehicle routing problems

We consider a variety of vehicle routing problems. The input to a problem consists of a graph G = (N, E) and edge lengths l(e), e ∈ E. Customers located at the vertices have to be visited by a set of vehicles. Two important parameters are k the number of vehicles, and λ the longest distance traveled by a vehicle. We consider two types of problems. (1) Given a bound λ on the length of each path, find a minimum sized collection of paths that cover all the vertices of the graph, or all the edges from a given subset of edges of the input graph. We also consider a variation where it is desired to cover N by a minimum number of stars of length bounded by λ. (2) Given a number k find a collection of k paths that cover either the vertex set of the graph or a given subset of edges. The goal here is to minimize λ, the maximum travel distance. For all these problems we provide constant ratio approximation algorithms and prove their NP-hardness.

Approximation and heuristic algorithms for minimum-delay application-layer multicast trees

In this paper we investigate the problem of finding minimum-delay application-layer multicast trees, such as the trees constructed in overlay networks. It is accepted that shortest path trees are not a good solution for the problem since such trees can have nodes with very large degree, termed high-load nodes. The load on these nodes makes them a bottleneck in the distribution tree, due to computation load and access link bandwidth constraints. Many previous solutions limited the maximum degree of the nodes by introducing arbitrary constraints. In this work, we show how to directly map the node load to the delay penalty at the application host, and create a new model that captures the trade offs between the desire to select shortest path trees and the need to constrain the load on the hosts. In this model the problem is shown to be NP-hard. We therefore present an approximation algorithm and an alternative heuristic algorithm. Our heuristic algorithm is shown by simulations to be scalable for large group sizes, and produces results that are very close to optimal.We investigate the problem of finding the minimum delay application-layer multicast trees, such as the trees constructed in overlay networks. It is accepted that shortest path trees are not a good solution for the problem since such trees can have nodes with very large degree, termed high load nodes. The load on these nodes makes them a bottleneck in the distribution tree, due to computation load and access link bandwidth constrains. Many previous solutions limited the maximal degree of the nodes by introducing arbitrary constraints. In this work, we show how to directly map the node load to the delay penalty at the application host, and create a new model that captures the trade offs between the desire to select shortest path trees and the need to constrain the load on the hosts. In this model the problem is shown to be NP-hard. Therefore, we present a logarithmic approximation algorithm and an alternative heuristic solution. Our heuristic algorithm is shown by simulations to be scalable for large group sizes, and produces results that are very close to optimal.

Improved Approximation Guarantees for Weighted Matching in the Semi-streaming Model

We study the maximum weight matching problem in the semi-streaming model, and improve on the currently best one-pass algorithm due to Zelke [Proceedings of the 25th Annual Symposium on Theoretical Aspects of Computer Science, 2008, pp. 669–680] by devising a deterministic approach whose performance guarantee is 4.91+e. In addition, we study preemptive online algorithms, a class of algorithms related to one-pass semi-streaming algorithms, where we are allowed to maintain only a feasible matching in memory at any point in time. We provide a lower bound of 4.967 on the competitive ratio of any such deterministic algorithm, and hence show that future improvements will have to store in memory a set of edges that is not necessarily a feasible matching. We conclude by presenting an empirical study, conducted in order to compare the practical performance of our approach to that of previously suggested algorithms.

The SONET edge‐partition problem

Motivated by a problem arising in the design of telecommunications networks using the SONET standard, we consider the problem of covering all edges of a graph using subgraphs that contain at most k edges with the objective of minimizing the total number of vertices in the subgraphs. We show that the problem is -hard when k ≥ 3 and present a linear-time -approximation algorithm. For even k values, we present an approximation scheme with a reduced ratio but with increased complexity.

A Faster, Better Approximation Algorithm for the Minimum Latency Problem

We give a 7.18-approximation algorithm for the minimum latency problem that uses only

An Efficient Polynomial Time Approximation Scheme for the Constrained Minimum Spanning Tree Problem Using Matroid Intersection

O(n \log n)

Better bounds for minimizing SONET ADMs

calls to the prize-collecting Steiner tree (PCST) subroutine of Goemans and Williamson. This improves the previous best algorithms in both performance guarantee and running time. A previous algorithm of Goemans and Kleinberg for the minimum latency problem requires an approximation algorithm for the

On Bin Packing with Conflicts

k

A Better-Than-Greedy Approximation Algorithm for the Minimum Set Cover Problem

-minimum spanning tree (

Approximating the Degree-Bounded Minimum Diameter Spanning Tree Problem

k