24 research outputs found
Spanning spiders and light-splitting switches
AbstractMotivated by a problem in the design of optical networks, we ask when a graph has a spanning spider (subdivision of a star), or, more generally, a spanning tree with a bounded number of branch vertices. We investigate the existence of these spanning subgraphs in analogy to classical studies of Hamiltonicity
A Streaming Algorithm for the Undirected Longest Path Problem
We present the first streaming algorithm for the longest path problem in undirected graphs. The input graph is given as a stream of edges and RAM is limited to only a linear number of edges at a time (linear in the number of vertices n). We prove a per-edge processing time of O(n), where a naive solution would have required Omega(n^2). Moreover, we give a concrete linear upper bound on the number of bits of RAM that are required.
On a set of graphs with various structure, we experimentally compare our algorithm with three leading RAM algorithms: Warnsdorf (1823), Pohl-Warnsdorf (1967), and Pongrasz (2012). Although conducting only a small constant number of passes over the input, our algorithm delivers competitive results: with the exception of preferential attachment graphs, we deliver at least 71% of the solution of the best RAM algorithm. The same minimum relative performance of 71% is observed over all graph classes after removing the 10% worst cases. This comparison has strong meaning, since for each instance class there is one algorithm that on average delivers at least 84% of a Hamilton path. In some cases we deliver even better results than any of the RAM algorithms
Circumference of 3-connected claw-free graphs and large Eulerian subgraphs of 3-edge-connected graphs
AbstractThe circumference of a graph is the length of its longest cycles. Results of Jackson, and Jackson and Wormald, imply that the circumference of a 3-connected cubic n-vertex graph is Ω(n0.694), and the circumference of a 3-connected claw-free graph is Ω(n0.121). We generalize and improve the first result by showing that every 3-edge-connected graph with m edges has an Eulerian subgraph with Ω(m0.753) edges. We use this result together with the RyjĂĄÄek closure operation to improve the lower bound on the circumference of a 3-connected claw-free graph to Ω(n0.753). Our proofs imply polynomial time algorithms for finding large Eulerian subgraphs of 3-edge-connected graphs and long cycles in 3-connected claw-free graphs
Optimal Multi-TDMA Scheduling in Ring Topology Networks
A scheduling algorithm will be proposed for wireless ring topology networks, utilizing time division multiple access (TDMA) with possible simultaneous operation of nodes. The proposed algorithm finds the optimal schedule to minimize the turnaround time for messages in the network. The properties of the algorithm are mathematically analyzed and proven, and practical test results are also provided
On the Complexity of Alternative Solutions
Diese Dissertation untersucht die KomplexitĂ€t alternativer Lösungen. Das heiĂt, wir betrachten die Frage, ob eine oder mehrere gegebene Lösungen eines Problems, das Finden weiterer Lösungen vereinfacht. In der Praxis relevant ist diese Fragestellung zum Beispiel, wenn sich eine mit groĂem Aufwand berechnete Lösung eines schwierigen Problems im Nachhinein als unzureichend erweist. In diesem Falle ist es notwendig nach alternativen Lösungen zu suchen, wobei nun die bereits gefundene Lösung als Ausgangspunkt der Berechnung genutzt werden kann. DarĂŒber hinaus hat die untersuchte Aufgabenstellung eine Bedeutung in der Erstellung von (auch hier immer beliebteren) japanischen RĂ€tseln wie Sudoku, Kakkuro oder Nurikabe. Beispielsweise werden im Fall von Sudoku, ausgehend von einem vollstĂ€ndig ausgefĂŒllten Gitter (Startlösung), Ziffern so gestrichen dass die Startlösung die eindeutige Lösung des RĂ€tsels bleibt. Dazu muss wĂ€hrend des Streichprozesses wiederholt geprĂŒft werden, ob es neben der Startlösung alternative Lösungen gibt.
Im ersten Teil der Arbeit (Kapitel 3 und 4) betrachten wir die Klasse der NP-vollstĂ€ndigen Probleme. Wir formalisieren den Begriff der Lösung mittels sogenannter Verifier und das Problem alternativer Lösungen fĂŒr NP-Sprachen. Indem wir die HĂ€rte des Problems alternativer Lösungen fĂŒr einige Probleme zeigen, motivieren wir die Vermutung, dass eine gegebene Lösung das Finden alternativer Lösungen nicht vereinfacht. Wir entwickeln den Begriff des universellen Verifiers, der es ermöglicht, einen geeigneten Lösungsbegriff fĂŒr ein Problem formal zu charakterisieren. DarĂŒber hinaus zeigen wir, dass es möglich ist, mit einer einzigen sogenannten -Reduzierung einen Lösungsbegriff fĂŒr ein Problem als geeignet zu identifizieren sowie die HĂ€rte des Problems alternativer Lösungen fĂŒr jede Anzahl gegebener Lösungen zu zeigen. Unter Benutzung dieser Reduzierung, erhĂ€rten wir die obige Vermutung, indem wir fĂŒr eine groĂe Zahl NP-vollstĂ€ndiger Probleme wie zum Beispiel 0/1-Integer Programming, 3Dimensional Matching, Minimum Edge Cost Flow und Vertex Cover zeigen, dass bezĂŒglich eines geeigneten Lösungsbegriffes alternative Lösungen nicht leicht zu berechnen sind.
DarĂŒber hinaus ĂŒbertragen wir die Theorie fĂŒr NP-Probleme auch auf die Klasse RE der aufzĂ€hlbaren Sprachen (Kapitel 5) und die Klassen der Polynomialzeithierarchie (Kapitel 6). FĂŒr RE zeigen wir damit, dass das Problem alternativer Lösungen fĂŒr RE wenig sinnvoll ist, da wir fĂŒr jedes RE-Problem einen geeigneten Lösungsbegriff finden, der höchstens eine Lösung zulĂ€sst. Die Situation in der Polynomialzeithierarchie ist vermutlich Ă€hnlich zum NP-Fall. FĂŒr können wir die HĂ€rte des Problems alternativer Lösungen fĂŒr einige typische Probleme zeigen, z.B. fĂŒr Generalized Subset Sum und Strongest Implicant Core. Deshalb und wegen der starken strukturellen Ăhnlichkeit zu NP (Die Polynomialzeithierarchie ist eine Verallgemeinerung von NP, insbesondere gilt = NP) vermuten wir auch hier, dass alternative Lösungen im Allgemeinen genauso schwer zu finden sind, wie eine erste Lösung
Approximability of the upper chromatic number of hypergraphs
A C-coloring of a hypergraph H = (X, E) is a vertex coloring Ï : X â N such that each edge E â E has at least two vertices with a common color. The related parameter over(Ï, -) (H), called the upper chromatic number of H, is the maximum number of colors in a C-coloring of H. A hypertree is a hypergraph which has a host tree T such that each edge E â E induces a connected subgraph in T. Notations n and m stand for the number of vertices and edges, respectively, in a generic input hypergraph. We establish guaranteed polynomial-time approximation ratios for the difference n - over(Ï, -) (H), which is 2 + 2 ln (2 m) on hypergraphs in general, and 1 + ln m on hypertrees. The latter ratio is essentially tight as we show that n - over(Ï, -) (H) cannot be approximated within (1 - Δ{lunate}) ln m on hypertrees (unless NPâ DTIME(nO (log log n)) ). Furthermore, over(Ï, -) (H) does not have O (n1 - Δ{lunate})-approximation and cannot be approximated within additive error o (n) on the class of hypertrees (unless P = NP). © 2014 Elsevier B.V. All rights reserved
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
Symmetries in Quantum Mechanics
Symmetry and quantum mechanics are two of the most fundamental probes we have of nature. This collection of eleven papers discusses new quantum phenomena in atoms, galaxies, and people (quantum cognition), which is a testimonial to the breadth of the influence of symmetry and quantum mechanics. The book represents an international effort of researchers from educational and research institutions in nine countries, including India, Finland, France, Mexico, Norway, Russia, Spain, Turkey, and the United States. The papers can be divided into four broad categories: Fundamentals of quantum systems, including a new derivation of the uncertainty principle from optimal stochastic control theory, a new model of energy transfer between atoms with no wave function collapse, a new asymmetric optical micro-device with the remarkable property of showing a current with no applied voltage, and a model of quantum cognition to predict the effect of irrelevant information on decision making. 2. Algebraic methods in quantum mechanics, describing an elegant derivation of hydrogen atom Stark effect matrix elements, and a new group theoretical method for the computation of radiative shifts. Teleportation and scattering, including a method to improve the information transfer in teleportation, and the use of permutation symmetry to compute scattering cross sections. Cosmology, including scalar-tensor theory applied to inflation, the characterization of new Levi-Cevita space-times, and a comprehensive analysis of gravitational dispersion forces