3,003 research outputs found

    On the tractability of some natural packing, covering and partitioning problems

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    In this paper we fix 7 types of undirected graphs: paths, paths with prescribed endvertices, circuits, forests, spanning trees, (not necessarily spanning) trees and cuts. Given an undirected graph G=(V,E)G=(V,E) and two "object types" A\mathrm{A} and B\mathrm{B} chosen from the alternatives above, we consider the following questions. \textbf{Packing problem:} can we find an object of type A\mathrm{A} and one of type B\mathrm{B} in the edge set EE of GG, so that they are edge-disjoint? \textbf{Partitioning problem:} can we partition EE into an object of type A\mathrm{A} and one of type B\mathrm{B}? \textbf{Covering problem:} can we cover EE with an object of type A\mathrm{A}, and an object of type B\mathrm{B}? This framework includes 44 natural graph theoretic questions. Some of these problems were well-known before, for example covering the edge-set of a graph with two spanning trees, or finding an ss-tt path PP and an s′s'-t′t' path P′P' that are edge-disjoint. However, many others were not, for example can we find an ss-tt path P⊆EP\subseteq E and a spanning tree T⊆ET\subseteq E that are edge-disjoint? Most of these previously unknown problems turned out to be NP-complete, many of them even in planar graphs. This paper determines the status of these 44 problems. For the NP-complete problems we also investigate the planar version, for the polynomial problems we consider the matroidal generalization (wherever this makes sense)

    Cycles containing many vertices of large degree

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    AbstractLet G be a 2-connected graph of order n, r a real number and Vr=v ϵ V(G)¦d(v)⩾r. It is shown that G contains a cycle missing at most max {0, n − 2r} vertices of Vr, yielding a common generalization of a result of Dirac and one of Shi Ronghua. A stronger conclusion holds if r⩾13(n+2): G contains a cycle C such that either V(C)⊇Vr or ¦ V(C)¦⩾2r. This theorem extends a result of Häggkvist and Jackson and is proved by first showing that if r⩾13(n+2), then G contains a cycle C for which ¦Vr∩V(C)¦is maximal such that N(x)⊆V(C) whenever x ϵ Vr − V(C) (∗). A result closely related to (∗) is stated and a result of Nash-Williams is extended using (∗)

    Distributed Flow Scheduling in an Unknown Environment

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    Flow scheduling tends to be one of the oldest and most stubborn problems in networking. It becomes more crucial in the next generation network, due to fast changing link states and tremendous cost to explore the global structure. In such situation, distributed algorithms often dominate. In this paper, we design a distributed virtual game to solve the flow scheduling problem and then generalize it to situations of unknown environment, where online learning schemes are utilized. In the virtual game, we use incentives to stimulate selfish users to reach a Nash Equilibrium Point which is valid based on the analysis of the `Price of Anarchy'. In the unknown-environment generalization, our ultimate goal is the minimization of cost in the long run. In order to achieve balance between exploration of routing cost and exploitation based on limited information, we model this problem based on Multi-armed Bandit Scenario and combined newly proposed DSEE with the virtual game design. Armed with these powerful tools, we find a totally distributed algorithm to ensure the logarithmic growing of regret with time, which is optimum in classic Multi-armed Bandit Problem. Theoretical proof and simulation results both affirm this claim. To our knowledge, this is the first research to combine multi-armed bandit with distributed flow scheduling.Comment: 10 pages, 3 figures, conferenc

    Existence of Dλ-cycles and Dλ-paths

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    A cycle of C of a graph G is called a Dλ-cycle if every component of G − V(C) has order less than λ. A Dλ-path is defined analogously. In particular, a D1-cycle is a hamiltonian cycle and a D1-path is a hamiltonian path. Necessary conditions and sufficient conditions are derived for graphs to have a Dλ-cycle or Dλ-path. The results are generalizations of theorems in hamiltonian graph theory. Extensions of notions such as vertex degree and adjacency of vertices to subgraphs of order greater than 1 arise in a natural way

    On the Constructive Content of Proofs

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    This thesis aims at exploring the scopes and limits of techniques for extracting programs from proofs. We focus on constructive theories of inductive definitions and classical systems allowing choice principles. Special emphasis is put on optimizations that allow for the extraction of realistic programs. Our main field of application is infinitary combinatorics. Higman's Lemma, having an elegant non-constructive proof due to Nash-Williams, constitutes an interesting case for the problem of discovering the constructive content behind a classical proof. We give two distinct solutions to this problem. First, we present a proof of Higman's Lemma for an arbitrary alphabet in a theory of inductive definitions. This proof may be considered as a constructive counterpart to Nash-Williams' minimal-bad-sequence proof. Secondly, using a refined AA-translation method, we directly transform the classical proof into a constructive one and extract a program. The crucial point in the latter is that we do not need to avoid the axiom of classical dependent choice but directly assign a realizer to its translation. A generalization of Higman's Lemma is Kruskal's Theorem. We present a constructive proof of Kruskal's Theorem that is completely formalized in a theory of inductive definitions. As a practical part, we show that these methods can be carried out in an interactive theorem prover. Both approaches to Higman's Lemma have been implemented in Minlog.Ziel der vorliegenden Arbeit ist es, die Reichweiten und Grenzen von Techniken zur Extraktion von Programmen aus Beweisen zu erforschen. Wir konzentrieren uns dabei auf konstruktive Theorien Induktiver Definitionen und klassische Systeme mit Auswahlprinzipien. Besonderes Gewicht liegt auf Optimierungen, die zur Extraktion von realisischen Programmen f"uhren. Unser Hauptanwendungsgebiet ist die unendliche Kombinatorik. Higmans Lemma, ein Satz mit einem eleganten klassischen, auf Nash-Williams zur"uckgehenden Beweis, ist ein interessantes Fallbeispiel f"ur die Suche nach dem konstruktiven Gehalt in einem klassischen Beweis. Wir zeigen zwei unterschiedliche L"osungen zu dieser Problemstellung auf. Zun"achst pr"asentieren wir einen induktiven Beweis von Higmans Lemma f"ur ein beliebiges Alphabet, der als konstruktives Pendant zum klassischen Beweis angesehen werden kann. Als zweiten Ansatz verwandeln wir mit Hilfe der verfeinerten AA-"Ubersetzungs-methode den klassischen Beweis in einen konstruktiven und extrahieren ein Programm. Der entscheidende Punkt ist hierbei, dass wir einen direkten Realisierer f"ur das "ubersetzte Auswahlaxiom verwenden. Die Verallgemeinerung von Higmans Lemma f"uhrt zu Kruskals Satz. Wir geben einen konstruktiven Beweis von Kruskals Theorem, der vollst"andig auf den Induktiven Definitionen basiert. Der praktische Teil der Arbeit befasst sich mit der Ausf"uhrbarkeit dieser Methoden und Beweise in dem Beweissystem Minlog
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