Magnús M. Halldórsson

Approximating maximum independent sets by excluding subgraphs

An approximation algorithm for the maximum independent set problem is given, improving the best performance guarantee known toO(n/(logn)2). We also obtain the same performance guarantee for graph coloring. The results can be combined into a surprisingly strongsimultaneous performance guarantee for the clique and coloring problems.The framework ofsubgraph-excluding algorithms is presented. We survey the known approximation algorithms for the independent set (clique), coloring, and vertex cover problems and show how almost all fit into that framework. We show that among subgraph-excluding algorithms, the ones presented achieve the optimal asymptotic performance guarantees.

Capacity of Arbitrary Wireless Networks

In this work we study the problem of determining the throughput capacity of a wireless network. We propose a scheduling algorithm to achieve this capacity within an approximation factor. Our analysis is performed in the physical interference model, where nodes are arbitrarily distributed in Euclidean space. We consider the problem separately from the routing problem and the power control problem, i.e., all requests are single-hop, and all nodes transmit at a fixed power level. The existing solutions to this problem have either concentrated on special-case topologies, or presented optimality guarantees which become arbitrarily bad (linear in the number of nodes) depending on the networks topology. We propose the first scheduling algorithm with approximation guarantee independent of the topology of the network. The algorithm has a constant approximation guarantee for the problem of maximizing the number of links scheduled in one time-slot. Furthermore, we obtain a O(log n) approximation for the problem of minimizing the number of time slots needed to schedule a given set of requests. Simulation results indicate that our algorithm does not only have an exponentially better approximation ratio in theory, but also achieves superior performance in various practical network scenarios. Furthermore, we prove that the analysis of the algorithm is extendable to higher-dimensional Euclidean spaces, and to more realistic bounded-distortion spaces, induced by non-isotropic signal distortions. Finally, we show that it is NP-hard to approximate the scheduling problem to within n 1-epsiv factor, for any constant epsiv > 0, in the non-geometric SINR model, in which path-loss is independent of the Euclidean coordinates of the nodes.

Greed is good: Approximating independent sets in sparse and bounded-degree graphs

AbstractTheminimum-degree greedy algorithm, or Greedy for short, is a simple and well-studied method for finding independent sets in graphs. We show that it achieves a performance ratio of (Δ+2)/3 for approximating independent sets in graphs with degree bounded by Δ. The analysis yields a precise characterization of the size of the independent sets found by the algorithm as a function of the independence number, as well as a generalization of Turán’s bound. We also analyze the algorithm when run in combination with a known preprocessing technique, and obtain an improved

Greed is good: approximating independent sets in sparse and bounded-degree graphs

(2\bar d + 3)/5

Coloring Powers of Planar Graphs

performance ratio on graphs with average degree

Wireless scheduling with power control

\bar d