Given a collection ℱ of subsets of S = {1,…,n}, set cover is the problem of selecting as few as possible subsets from ℱ such that their union covers S,, and max k-cover is the problem of selecting k subsets from ℱ such that their union has maximum cardinality. Both these problems are NP-hard. We prove that (1 - o(1)) ln n is a threshold below which set cover cannot be approximated efficiently, unless NP has slightly superpolynomial time algorithms. This closes the gap (up to low-order terms) between the ratio of approximation achievable by the greedy alogorithm (which is (1 - o(1)) ln n), and provious results of Lund and Yanakakis, that showed hardness of approximation within a ratio of (log2n) / 2 ≃0.72 ln n. For max k-cover, we show an approximation threshold of (1 - 1/e)(up to low-order terms), under assumption that P ≠ NP.

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A threshold of ln n for approximating set cover

Compounding the confusion is the use of the broad term cognitive radio as a synonym for dynamic spectrum access. As an initial attempt at unifying the terminology, the taxonomy of dynamic spectrum access is provided. In this article, an overview of challenges and recent developments in both technological and regulatory aspects of opportunistic spectrum access (OSA). The three basic components of OSA are discussed. Spectrum opportunity identification is crucial to OSA in order to achieve nonintrusive communication. The basic functions of the opportunity identification module are identified

A Survey of Dynamic Spectrum Access

In this provocative book, Barwise and Perry tackle the slippery subject of \"meaning, \" a subject that has long vexed linguists, language philosophers, and logicians.

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Situations and Attitudes.

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Clique is hard to approximate within n1−ε

LEARNING The objectives of BIOS 781 are to present: OBJECTIVES: 1. basic population and quantitative genetic principles, including classical genetics, chromosomal theory of inheritance, and meiotic recombination 2. an exposure to QTL mapping methods of complex quantitative traits and linkage methods to detect co-segregation with disease 3. methods for assessing marker-disease linkage disequilibrium, including case-control approaches 4. methods for genome-wide association and stratification control.

Genetic Data Analysis

The data migration problem is to compute an efficient plan for moving data stored on devices in a network from one configuration to another. We consider this problem with the objective of minimizing the sum of completion times of all storage devices. It is modeled by a transfer graph, where vertices represent the storage devices, and the edges indicate the data transfers required between pairs of devices. Each vertex has a nonnegative weight, and each edge has a release time and a processing time. A vertex completes when all the edges incident on it complete; the constraint is that two edges incident on the same vertex cannot be processed simultaneously. The objective is to minimize the sum of weighted completion times of all vertices. Kim (Journal of Algorithms, 55:42--57, 2005) gave a 9-approximation algorithm for the problem when edges have arbitrary processing times and are released at time zero. We improve Kim's result by giving a 5.06-approximation algorithm. We also address the open shop scheduling problem, Ovrjv ∑wjCj, and show that it is a special case of the data migration problem. Queyranne and Sviridenko (Journal of Scheduling, 5:287-305, 2002) gave a 5.83-approximation algorithm for the nonpreemptive version of the open shop problem. They state as an obvious open question whether there exists an algorithm for open shop scheduling that gives a performance guarantee better than 5.83. Our 5.06 algorithm for data migration proves the existence of such an algorithm. Crucial to our improved result is a property of the linear programming relaxation for the problem. Similar linear programs have been used for various other scheduling problems. Our technique may be useful in obtaining improved results for these problems as well.

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Improved results for data migration and open shop scheduling

We consider graph coloring problems where the cost of a coloring is the sum of the costs of the colors, and the cost of a color is a monotone concave function of the total weight of the class. This models resource allocation problems where the cost of a resource depends on the use of the resource. The specific case of interval graphs is of special interest as multi-criteria interval scheduling. We give an algorithm for all perfect graphs that yields a robust coloring: a particular solution that simultaneously approximates all concave functions. For graphs with uniform weights, we show how to modify the solution to approximate any monotone cost function. We complement these results with a number of hardness results and some exact algorithms on restricted classes of graphs.

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Robust cost colorings

Independent sets in bounded-degree hypergraphs

This paper considers the approximability of the largest common subtree and the largest common subgraph problems, which have applications in molecular biology. It is shown that approximating the problems within a factor of ne is NP-complete, while a general search algorithm which approximates both problems within a factor of O(n/log n) is presented. Moreover, several variants of the largest common subtree problem are studied.

On the Approximation of Largest Common Subtrees and Largest Common Point Sets

Scheduling dependent jobs on multiple machines is modeled by the graph multi-coloring problem. In this paperweconsider the problem of minimizing the average completion time of all jobs. This is formalized as the sum multicoloring (SMC) problem: Given a graph and the number of colors required by each vertex, find a multi-coloring which minimizes the sum of the largest colors assigned to the vertices. It reduces to the known sum coloring (SC) problem in the special case of unit execution times.This paper reports a comprehensive study of the SMC problem, treating three models: with and without preemption allowed, as well as co-scheduling where tasks cannot start while others are running. We establish a linear relation between the approximability of the maximum independent set (IS) and SMC in all three models, via a link to the SC problem. Thus, for classes of graphs where IS is ?-approximable, we obtain O(?)-approximations for preemptive and coscheduling SMC, and O(? ? log n) for non-preemptive SMC. In addition, we give constant-approximation algorithms for SMC under different models, on a number of fundamental classes of graphs, including bipartite, line, bounded degree, and planar graphs.

Magnús M. Halldórsson

Papers

Improved results for data migration and open shop scheduling

Robust cost colorings

Independent sets in bounded-degree hypergraphs

On the Approximation of Largest Common Subtrees and Largest Common Point Sets

Sum Multi-coloring of Graphs