Proof Without Words: The Pigeonhole Principle

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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

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

Tree-Based Multicasting in Wormhole-Routed Irregular Topologies

A deadlock-free tree-based multicast routing algorithm is presented for all direct networks, regardless of interconnection topology. The algorithm delivers a message to any number of destinations using only a single startup phase. In contrast to existing tree-based schemes, this algorithm applies to all interconnection topologies, requires only fixed-sized input buffers that are independent of maximum message length, and uses a single asynchronous flit replication mechanism. The theoretical basis of the technique used here is sufficiently general to develop other tree-based multicasting algorithms for regular and irregular topologies. Simulation results demonstrate that this tree-based algorithm provides a very promising means of achieving very low latency multicast

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

Jane: A New Tool for the Cophylogeny Reconstruction Problem

Background This paper describes the theory and implementation of a new software tool, called Jane, for the study of historical associations. This problem arises in parasitology (associations of hosts and parasites), molecular systematics (associations of orderings and genes), and biogeography (associations of regions and orderings). The underlying problem is that of reconciling pairs of trees subject to biologically plausible events and costs associated with these events. Existing software tools for this problem have strengths and limitations, and the new Jane tool described here provides functionality that complements existing tools. Results The Jane software tool uses a polynomial time dynamic programming algorithm in conjunction with a genetic algorithm to find very good, and often optimal, solutions even for relatively large pairs of trees. The tool allows the user to provide rich timing information on both the host and parasite trees. In addition the user can limit host switch distance and specify multiple host switch costs by specifying regions in the host tree and costs for host switches between pairs of regions. Jane also provides a graphical user interface that allows the user to interactively experiment with modifications to the solutions found by the program. Conclusions Jane is shown to be a useful tool for cophylogenetic reconstruction. Its functionality complements existing tools and it is therefore likely to be of use to researchers in the areas of parasitology, molecular systematics, and biogeography

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

The Cophylogeny Reconstruction Problem is NP-Complete

The cophylogeny reconstruction problem arises in the study of host-parasite relationships. Specif- ically, we are given a host tree H, a parasite tree P, and a function \u27 mapping the leaves (extant taxa) of P to the leaves of H. Four biologically plausible operations are considered: cospeciation, duplication, host switching, and loss (Figure 1). A host switch is permitted in conjunction with a duplication event but not with a cospeciation event [1]