500 research outputs found
Folding equilateral plane graphs
22nd International Symposium, ISAAC 2011, Yokohama, Japan, December 5-8, 2011. ProceedingsWe consider two types of folding applied to equilateral plane graph linkages. First, under continuous folding motions, we show how to reconfigure any linear equilateral tree (lying on a line) into a canonical configuration. By contrast, such reconfiguration is known to be impossible for linear (nonequilateral) trees and for (nonlinear) equilateral trees. Second, under instantaneous folding motions, we show that an equilateral plane graph has a noncrossing linear folded state if and only if it is bipartite. Not only is the equilateral constraint necessary for this result, but we show that it is strongly NP-complete to decide whether a (nonequilateral) plane graph has a linear folded state. Equivalently, we show strong NP-completeness of deciding whether an abstract metric polyhedral complex with one central vertex has a noncrossing flat folded state with a specified âoutside regionâ. By contrast, the analogous problem for a polyhedral manifold with one central vertex (single-vertex origami) is only weakly NP-complete
Well-Posedness and Symmetries of Strongly Coupled Network Equations
We consider a diffusion process on the edges of a finite network and allow
for feedback effects between different, possibly non-adjacent edges. This
generalizes the setting that is common in the literature, where the only
considered interactions take place at the boundary, i. e., in the nodes of the
network. We discuss well-posedness of the associated initial value problem as
well as contractivity and positivity properties of its solutions. Finally, we
discuss qualitative properties that can be formulated in terms of invariance of
linear subspaces of the state space, i. e., of symmetries of the associated
physical system. Applications to a neurobiological model as well as to a system
of linear Schroedinger equations on a quantum graph are discussed.Comment: 25 pages. Corrected typos and minor change
Band Connectivity for Topological Quantum Chemistry: Band Structures As A Graph Theory Problem
The conventional theory of solids is well suited to describing band
structures locally near isolated points in momentum space, but struggles to
capture the full, global picture necessary for understanding topological
phenomena. In part of a recent paper [B. Bradlyn et al., Nature 547, 298
(2017)], we have introduced the way to overcome this difficulty by formulating
the problem of sewing together many disconnected local "k-dot-p" band
structures across the Brillouin zone in terms of graph theory. In the current
manuscript we give the details of our full theoretical construction. We show
that crystal symmetries strongly constrain the allowed connectivities of energy
bands, and we employ graph-theoretic techniques such as graph connectivity to
enumerate all the solutions to these constraints. The tools of graph theory
allow us to identify disconnected groups of bands in these solutions, and so
identify topologically distinct insulating phases.Comment: 19 pages. Companion paper to arXiv:1703.02050 and arXiv:1706.08529
v2: Accepted version, minor typos corrected and references added. Now
19+epsilon page
Drawings of Complete Multipartite Graphs up to Triangle Flips
For a drawing of a labeled graph, the rotation of a vertex or crossing is the cyclic order of its incident edges, represented by the labels of their other endpoints. The extended rotation system (ERS) of the drawing is the collection of the rotations of all vertices and crossings. A drawing is simple if each pair of edges has at most one common point. Gioan's Theorem states that for any two simple drawings of the complete graph Kn with the same crossing edge pairs, one drawing can be transformed into the other by a sequence of triangle flips (a.k.a. Reidemeister moves of Type 3). This operation refers to the act of moving one edge of a triangular cell formed by three pairwise crossing edges over the opposite crossing of the cell, via a local transformation. We investigate to what extent Gioan-type theorems can be obtained for wider classes of graphs. A necessary (but in general not sufficient) condition for two drawings of a graph to be transformable into each other by a sequence of triangle flips is that they have the same ERS. As our main result, we show that for the large class of complete multipartite graphs, this necessary condition is in fact also sufficient. We present two different proofs of this result, one of which is shorter, while the other one yields a polynomial time algorithm for which the number of needed triangle flips for graphs on n vertices is bounded by O(n16). The latter proof uses a Carathéodory-type theorem for simple drawings of complete multipartite graphs, which we believe to be of independent interest. Moreover, we show that our Gioan-type theorem for complete multipartite graphs is essentially tight in the following sense: For the complete bipartite graph Km, n minus two edges and Km, n plus one edge for any m, n ℠4, as well as Kn minus a 4-cycle for any n ℠5, there exist two simple drawings with the same ERS that cannot be transformed into each other using triangle flips. So having the same ERS does not remain sufficient when removing or adding very few edges
Operations research: from computational biology to sensor network
In this dissertation we discuss the deployment of combinatorial optimization methods for modeling and solve real life problemS, with a particular emphasis to two biological problems arising from a common scenario: the reconstruction of the three-dimensional shape of a biological molecule from Nuclear Magnetic Resonance (NMR) data.
The fi rst topic is the 3D assignment pathway problem (APP) for a RNA molecule.
We prove that APP is NP-hard, and show a formulation of it based on edge-colored
graphs. Taking into account that interactions between consecutive nuclei in the NMR
spectrum are diff erent according to the type of residue along the RNA chain, each color
in the graph represents a type of interaction. Thus, we can represent the sequence of interactions as the problem of fi nding a longest (hamiltonian) path whose edges follow a given order of colors (i.e., the orderly colored longest path). We introduce three alternative IP formulations of APP obtained with a max flow problem on a directed graph with packing constraints over the partitions, which have been compared among themselves. Since the last two models work on cyclic graphs, for them we proposed an algorithm based on the solution of their relaxation combined with the separation of cycle inequalities in a Branch & Cut scheme.
The second topic is the discretizable distance geometry problem (DDGP), which is
a formulation on discrete search space of the well-known distance geometry problem
(DGP). The DGP consists in seeking the embedding in the space of a undirected graph, given a set of Euclidean distances between certain pairs of vertices. DGP has two important applications: (i) fi nding the three dimensional conformation of a molecule from a subset of interatomic distances, called Molecular Distance Geometry Problem, and (ii) the Sensor Network Localization Problem. We describe a Branch & Prune (BP) algorithm
tailored for this problem, and two versions of it solving the DDGP both in protein
modeling and in sensor networks localization frameworks. BP is an exact and exhaustive
combinatorial algorithm that examines all the valid embeddings of a given weighted
graph G=(V,E,d), under the hypothesis of existence of a given order on V. By
comparing the two version of BP to well-known algorithms we are able to prove the
e fficiency of BP in both contexts, provided that the order imposed on V is maintained
Fine-Grained Complexity Analysis of Two Classic TSP Variants
We analyze two classic variants of the Traveling Salesman Problem using the
toolkit of fine-grained complexity. Our first set of results is motivated by
the Bitonic TSP problem: given a set of points in the plane, compute a
shortest tour consisting of two monotone chains. It is a classic
dynamic-programming exercise to solve this problem in time. While the
near-quadratic dependency of similar dynamic programs for Longest Common
Subsequence and Discrete Frechet Distance has recently been proven to be
essentially optimal under the Strong Exponential Time Hypothesis, we show that
bitonic tours can be found in subquadratic time. More precisely, we present an
algorithm that solves bitonic TSP in time and its bottleneck
version in time. Our second set of results concerns the popular
-OPT heuristic for TSP in the graph setting. More precisely, we study the
-OPT decision problem, which asks whether a given tour can be improved by a
-OPT move that replaces edges in the tour by new edges. A simple
algorithm solves -OPT in time for fixed . For 2-OPT, this is
easily seen to be optimal. For we prove that an algorithm with a runtime
of the form exists if and only if All-Pairs
Shortest Paths in weighted digraphs has such an algorithm. The results for
may suggest that the actual time complexity of -OPT is
. We show that this is not the case, by presenting an algorithm
that finds the best -move in time for
fixed . This implies that 4-OPT can be solved in time,
matching the best-known algorithm for 3-OPT. Finally, we show how to beat the
quadratic barrier for in two important settings, namely for points in the
plane and when we want to solve 2-OPT repeatedly.Comment: Extended abstract appears in the Proceedings of the 43rd
International Colloquium on Automata, Languages, and Programming (ICALP 2016
A Discrete Morse Approach for Computing Homotopy Types: An Exploration of the Morse, Generalized Morse, Matching, and Independence Complexes
In this thesis, we study possible homotopy types of four families of simplicial complexesâthe Morse complex, the generalized Morse complex, the matching complex, and the independence complexâusing discrete Morse theory. Given a simplicial complex, K, we can construct its Morse complex from all possible discrete gradient vector fields on K. A similar construction will allow us to build the generalized Morse complex while considering edges and vertices will allow us to construct the matching complex and independence complex. In Chapter 3, we use the Cluster Lemma and the notion of star clusters to apply matchings to families of Morse, generalized Morse, and matching complexes, computing their homotopy types. Notably, we show that the Morse complex of a subset of extended star graphs is homotopy equivalent to a wedge of spheres and the matching complex of a Dutch windmill graph is homotopy equivalent to a point, sphere, or wedge of spheres. In Chapter 4, we use a degenerate Hasse diagram and strong collapses to compute the homotopy type of many families of Morse complexes. Recognizably, we provide computations showing wedged complexes as suspensions and provide a sufficient condition for strongly collapsible Morse complexes. Lastly, in Chapter 5 we study chord diagramsâa largely unexplored topicâand provide insight into the possible homotopy types of the independence complex of intersection graphs of chord diagrams. We realize spheres and wedges of spheres as possible homotopy types and begin to explore what families of intersection graphs can be represented as a chord diagram. From here, most interesting, we show that ladder graphs can be represented as chord diagrams, and the independence complex of a ladder graph has the homotopy type of a sphere
Topological evolution of networks : case studies in the US airlines and language Wikipedias
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 2009.Cataloged from PDF version of thesis.Includes bibliographical references (p. 195-198).This thesis examines the topology of engineering systems and how that topology changes over time. Topology refers to the relative arrangement and connectivity of the elements of a system. We review network theory relevant to topological evolution and use graph-theoretical methods to analyze real systems, represented as networks. Using existing graph generative models, we develop a profile of canonical graphs and tools to compare a real network to that profile. The developed metrics are used to track topology changes over the history of real networks. This theoretical work is applied to two case studies. The first discusses the US airline industry in terms of routes. We study various airlines and segments of the industry statistically and find commonly occurring patterns. We show that there are topology transitions in the history of airlines in the period 1990-2007. Most airline networks have similar topology and historical patterns, with the exception of Southwest Airlines. We show mathematically that Southwest's topology is different. We propose two heuristic growth models, one featuring hub-seeding derived from the underlying patterns of evolution of JetBlue Airways and one featuring local interconnectedness, derived from the patterns of growth of Southwest. The two models match the topologies of these airlines better than canonical models over time. Results suggest that Southwest is becoming more centralized, closer to the hub-spoke topologies of other airlines. Our second case study discusses the growth of language Wikipedia networks, where nodes are articles and hyperlinks are the connections between them. These knowledge networks are subject to different constraints than air transportation systems. The topology of these networks and their growth principles are completely different. Most Wikipedias studied grow by coalescence, with multiple disconnected thematic clusters of pages growing separately and over time, converging to a giant connected component via weak links. These topologies start out as simple trees, and coalesce into sparse hierarchical structures with random interlinking. One striking exception is the history of the Chinese Wikipedia, which grows fully connected from its inception. We discuss these patterns of growth comparatively across Wikipedias, and in general, compared to airline networks. Our work suggests that complex engineering systems are hybrids of pure canonical forms and that they undergo distinct phase transitions during their evolution. We find commonality among systems and uncover important differences by learning from the exceptions.by Gergana Assenova Bounova.Ph.D
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