3,003 research outputs found
On the tractability of some natural packing, covering and partitioning problems
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 and two "object
types" and chosen from the alternatives above, we
consider the following questions. \textbf{Packing problem:} can we find an
object of type and one of type in the edge set of
, so that they are edge-disjoint? \textbf{Partitioning problem:} can we
partition into an object of type and one of type ?
\textbf{Covering problem:} can we cover with an object of type
, and an object of type ? 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 - path and an - path that are
edge-disjoint. However, many others were not, for example can we find an
- path and a spanning tree 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
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
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
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
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 -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 -"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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