Proof Without Words: The Pigeonhole Principle

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Approximation Algorithms: Good Solutions to Hard Problems

Consider a computer network represented by an undirected graph where the vertices represent computer nodes and the edges represent links between the nodes. Since some of the links in the network may become faulty, link testing devices are placed at some of the nodes. A tester at a particular node can test all links incident to that node. Since the testers are expensive, however, we wish to deploy the minimum number of these devices such that every link is incidient to at least one node containing a tester. In graph theoretic terms, a vertex cover is a subset of the vertices such that every edge is incident to at least one vertex in this set. Our objective then is to find a minimum vertex cover. This is known as the vertex cover problem

Figs, Wasps, Gophers, and Lice: A Computational Exploration of Coevolution

This chapter explores the topic of coevolution: the genetic change in one species in response to the change in another. For example, in some cases, a parasite species might evolve to specialize with its host species. In other cases, the relationship between two species may be mutually beneficial and coevolution may serve to strengthen the benefits of that relationship

Sorting in Parallel

In 1842, L.F. Menabrea anticipated the benefits of parallel computing in an article that appeared in the Swiss Journal Bibliotheque universelle de Geneve: When a long series of identical computations is to be performed, such as those required for the formation of numerical tables, the machine can be brought into play so as to give several results at the same time, which will greatly abridge the whole amount of the processes. Although more than a century passed before Menabrea\u27s vision became a reality, today parallel computers with hundreds and even thousands of processors are used in a broad range of applications

On the Computational Complexity of the Reticulate Cophylogeny Reconstruction Problem

The cophylogeny reconstruction problem is that of finding minimal cost explanations of differences between evolutionary histories of ecologically linked groups of biological organisms. We present a proof that shows that the general problem of reconciling evolutionary histories is NP-complete and provide a sharp boundary where this intractability begins. We also show that a related problem, that of finding Pareto optimal solutions, is NP-hard. As a byproduct of our results, we give a framework by which meta-heuristics can be applied to find good solutions to this problem

The Computational Complexity of Motion Planning

In this paper we show that a generalization of a popular motion planning puzzle called Lunar Lockout is computationally intractable. In particular, we show that the problem is PSPACE-complete. We begin with a review of NP-completeness and polynomial-time reductions, introduce the class PSPACE, and motivate the significance of PSPACE-complete problems. Afterwards, we prove that determining whether a given instance of a generalized Lunar Lockout puzzle is solvable is PSPACE-complete

Efficient Multicast in Heterogeneous Networks of Workstations

This paper studies the problem of efficient multicast in heterogeneous networks of workstations (HNOWs) using a parameterized communication model [3]. This model associates a sending overhead and a receiving overhead with each node as well as a network latency parameter. The problem of finding optimal multicasts in this model is known to be NP-complete in the strong sense. Nevertheless, we show that for two different properties that arise in typical HNOWs, provably near-optimal and optimal solutions, respectively, can be found in polynomial time. Specifically, we show the following two results: When the ratios of receiving overhead to sending overhead among the nodes is bounded by constants, solutions within a bounded ratio of optimal can be found in time O(n log n). Secondly, if the number of distinct types of workstations is fixed then optimal solutions can be found in polynomial time. These results provide a practical means of finding optimal and provably near-optimal multicast schedules in a large class of frequently occurring heterogeneous networks of workstations

The Most Parsimonious Reconciliation Problem in the Presence of Incomplete Lineage Sorting and Hybridization Is NP-Hard

The maximum parsimony phylogenetic reconciliation problem seeks to explain incongruity between a gene phylogeny and a species phylogeny with respect to a set of evolutionary events. While the reconciliation problem is well-studied for species and gene trees subject to events such as duplication, transfer, loss, and deep coalescence, recent work has examined species phylogenies that incorporate hybridization and are thus represented by networks rather than trees. In this paper, we show that the problem of computing a maximum parsimony reconciliation for a gene tree and species network is NP-hard even when only considering deep coalescence. This result suggests that future work on maximum parsimony reconciliation for species networks should explore approximation algorithms and heuristics

Multicast Communication in Circuit-Switched Optical Networks

In this paper we examine the problem of multicast routing in Wavelength-division multiplexed (WDM) optical networks. In particular, we examine wavelength and routing assignment problems in circuit-switched WDM networks. We show that although the routing and wavelength assignment (RWA) problem is NP-complete in general, the wavelength assignment (WA) problem can be solevd in a polynomial time

Disjoint Covers in Replicated Heterogeneous Arrays

Reconfigurable chips are fabricated with redundant elements that can be used to replace the faulty elements. The fault cover problem consists of finding an assignment of redundant elements to the faulty elements such that all of the faults are repaired. In reconfigurable chips that consist of arrays of elements, redundant elements are configured as spare rows and spare columns. This paper considers the problem in which a chip contains several replicates of a heterogeneous array, one or more sets of spare rows, and one or more sets of spare columns. Each set of spare rows is identical to the set of rows in the array, and each set of spare columns is identical to the set of columns in the array. Specifically, an ith spare row can only be used to replace an ith row of an array, and similarly with spare columns. Repairing the chip reduces to finding a cover for the faults in each of the arrays. These covers must be disjoint; that is, a particular spare row or spare column can be used in the cover of at most one array. Results are presented for three fault cover problems that arise under these conditions