347 research outputs found
Treewidth and related graph parameters
For modeling some practical problems, graphs play very important roles.
Since many modeled problems can be NP-hard in general, some restrictions
for inputs are required. Bounding a graph parameter of the inputs is one of
the successful approaches. We study this approach in this thesis. More precisely,
we study two graph parameters, spanning tree congestion and security
number, that are related to treewidth.
Let G be a connected graph and T be a spanning tree of G. For e ∈ E(T),
the congestion of e is the number of edges in G connecting two components
of T − e. The edge congestion of G in T is the maximum congestion over all
edges in T. The spanning tree congestion of G is the minimum congestion
of G in its spanning trees. In this thesis, we show the spanning tree congestion
for the complete k-partite graphs, the two-dimensional tori, and the twodimensional
Hamming graphs. We also address lower bounds of spanning
tree congestion for the multi-dimensional hypercubes, the multi-dimensional
grids, and the multi-dimensional Hamming graphs.
The security number of a graph is the cardinality of a smallest vertex subset
of the graph such that any “attack” on the subset is “defendable.” In this thesis,
we determine the security number of two-dimensional cylinders and tori.
This result settles a conjecture of Brigham, Dutton and Hedetniemi [Discrete
Appl. Math. 155 (2007) 1708–1714]. We also show that every outerplanar
graph has security number at most three. Additionally, we present lower and
upper bounds for some classes of graphs.学位記番号:工博甲39
Better Hardness Results for the Minimum Spanning Tree Congestion Problem
In the spanning tree congestion problem, given a connected graph , the
objective is to compute a spanning tree in for which the maximum edge
congestion is minimized, where the congestion of an edge of is the
number of vertex pairs adjacent in for which the path connecting them in
traverses . The problem is known to be NP-hard, but its approximability
is still poorly understood, and it is not even known whether the optimum can be
efficiently approximated with ratio . In the decision version of this
problem, denoted STC-, we need to determine if has a spanning tree with
congestion at most . It is known that STC- is NP-complete for ,
and this implies a lower bound of on the approximation ratio of
minimizing congestion. On the other hand, -STC can be solved in polynomial
time, with the complexity status of this problem for
remaining an open problem. We substantially improve the earlier hardness result
by proving that STC- is NP-complete for . This leaves only the case
open, and improves the lower bound on the approximation ratio to
Small-world interconnection networks for large parallel computer systems
The use of small-world graphs as interconnection networks of multicomputers is proposed and analysed in this work. Small-world interconnection networks are constructed by adding (or modifying) edges to an underlying local graph. Graphs with a rich local structure but with a large diameter are shown to be the most suitable candidates for the underlying graph. Generation models based on random and deterministic wiring processes are proposed and analysed. For the random case basic properties such as degree, diameter, average length and bisection width are analysed, and the results show that a fast transition from a large diameter to a small diameter is experienced when the number of new edges introduced is increased. Random traffic analysis on these networks is undertaken, and it is shown that although the average latency experiences a similar reduction, networks with a small number of shortcuts have a tendency to saturate as most of the traffic flows through a small number of links. An analysis of the congestion of the networks corroborates this result and provides away of estimating the minimum number of shortcuts required to avoid saturation. To overcome these problems deterministic wiring is proposed and analysed. A Linear Feedback Shift Register is used to introduce shortcuts in the LFSR graphs. A simple routing algorithm has been constructed for the LFSR and extended with a greedy local optimisation technique. It has been shown that a small search depth gives good results and is less costly to implement than a full shortest path algorithm. The Hilbert graph on the other hand provides some additional characteristics, such as support for incremental expansion, efficient layout in two dimensional space (using two layers), and a small fixed degree of four. Small-world hypergraphs have also been studied. In particular incomplete hypermeshes have been introduced and analysed and it has been shown that they outperform the complete traditional implementations under a constant pinout argument. Since it has been shown that complete hypermeshes outperform the mesh, the torus, low dimensional m-ary d-cubes (with and without bypass channels), and multi-stage interconnection networks (when realistic decision times are accounted for and with a constant pinout), it follows that incomplete hypermeshes outperform them as well
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