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

We consider the scheduling of biprocessor jobs under sum objective (BPSMSM). Given a collection of unit-length jobs where each job requires the use of two processors, find a schedule such that no two jobs involving the same processor run concurrently. The objective is to minimize the sum of the completion times of the jobs. Equivalently, we would like to find a sum edge coloring of a given multigraph, that is, a partition of its edge set into matchings M1,…,Mt minimizing Σi=1ti|Mi|. This problem is APX-hard, even in the case of bipartite graphs [Marx 2009]. This special case is closely related to the classic open shop scheduling problem. We give a 1.8298-approximation algorithm for BPSMSM improving the previously best ratio known of 2 [Bar-Noy et al. 1998]. The algorithm combines a configuration LP with greedy methods, using nonstandard randomized rounding on the LP fractions. We also give an efficient combinatorial 1.8886-approximation algorithm for the case of simple graphs, which gives an improved 1.79568 + O(log d¯/d¯)-approximation in graphs of large average degree d¯.

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Sum edge coloring of multigraphs via configuration LP

We study streaming algorithms for the interval selection problem: finding a maximum cardinality subset of disjoint intervals on the line. A deterministic 2-approximation streaming algorithm for this problem is developed, together with an algorithm for the special case of proper intervals, achieving improved approximation ratio of 3/2. We complement these upper bounds by proving that they are essentially the best possible in the streaming setting: It is shown that an approximation ratio of 2 − e (or 3/2 − e for proper intervals) cannot be achieved unless the space is linear in the input size. In passing, we also answer an open question of Adler and Azar (J. Scheduling 2003) regarding the space complexity of constant-competitive randomized preemptive online algorithms for the same problem.

Space-Constrained Interval Selection

SDP-based algorithms for maximum independent set problems on hypergraphs

The physical (or SINR) model of wireless communication is more intricate than radio networks and still not well understood. If two neighbors of a node are transmitting, the node may be able to decode one of the transmissions, depending on the relative nearness of the transmitters. Thus, even the lack of proper reception carries indirect information. We explore here the power of such indirect signaling to infer the approximate topology of the network. In particular, we obtain a polylogarithmic time algorithm to compute a backbone: a set of nodes of constant density that dominates every e-neighborhood. A backbone has wide utility for information dissemination, functioning as a sparse spanner. It also leads to fast broadcast, running in O(Diameter) time after a polylogarithmic precomputation, which previously was only known when additional features such as carrier sense, collision detection, geometric coordinates, or power control were available.

Leveraging Indirect Signaling for Topology Inference and Fast Broadcast

We follow the travails of Enzo the baker, Orsino the oven man, and Beppe the planner. Their situation have a common theme: They know the input, in the form of a sequence of items, and they are not computationally constrained. Their issue is that they don't know in advance the time of reckoning, i.e. when their boss might show up, when they will be measured in terms of their progress on the prefix of the input sequence seen so far. Their goal is therefore to find a particular solution whose size on any prefix of the known input sequence is within best possible performance guarantees.

Magnús M. Halldórsson

Papers

Sum edge coloring of multigraphs via configuration LP

Space-Constrained Interval Selection

SDP-based algorithms for maximum independent set problems on hypergraphs

Leveraging Indirect Signaling for Topology Inference and Fast Broadcast

Return of the boss problem: competing online against a non-adaptive adversary