104 research outputs found
Spanning Trees with Many Leaves in Graphs without Diamonds and Blossoms
It is known that graphs on n vertices with minimum degree at least 3 have
spanning trees with at least n/4+2 leaves and that this can be improved to
(n+4)/3 for cubic graphs without the diamond K_4-e as a subgraph. We generalize
the second result by proving that every graph with minimum degree at least 3,
without diamonds and certain subgraphs called blossoms, has a spanning tree
with at least (n+4)/3 leaves, and generalize this further by allowing vertices
of lower degree. We show that it is necessary to exclude blossoms in order to
obtain a bound of the form n/3+c.
We use the new bound to obtain a simple FPT algorithm, which decides in
O(m)+O^*(6.75^k) time whether a graph of size m has a spanning tree with at
least k leaves. This improves the best known time complexity for MAX LEAF
SPANNING TREE.Comment: 25 pages, 27 Figure
An FPT Algorithm for Directed Spanning k-Leaf
An out-branching of a directed graph is a rooted spanning tree with all arcs
directed outwards from the root. We consider the problem of deciding whether a
given directed graph D has an out-branching with at least k leaves (Directed
Spanning k-Leaf). We prove that this problem is fixed parameter tractable, when
k is chosen as the parameter. Previously this was only known for restricted
classes of directed graphs.
The main new ingredient in our approach is a lemma that shows that given a
locally optimal out-branching of a directed graph in which every arc is part of
at least one out-branching, either an out-branching with at least k leaves
exists, or a path decomposition with width O(k^3) can be found. This enables a
dynamic programming based algorithm of running time 2^{O(k^3 \log k)} n^{O(1)},
where n=|V(D)|.Comment: 17 pages, 8 figure
Max-Leaves Spanning Tree is APX-hard for Cubic Graphs
We consider the problem of finding a spanning tree with maximum number of
leaves (MaxLeaf). A 2-approximation algorithm is known for this problem, and a
3/2-approximation algorithm when restricted to graphs where every vertex has
degree 3 (cubic graphs). MaxLeaf is known to be APX-hard in general, and
NP-hard for cubic graphs. We show that the problem is also APX-hard for cubic
graphs. The APX-hardness of the related problem Minimum Connected Dominating
Set for cubic graphs follows
The Existence of Spanning Ended System on Claw-Free Graphs
We prove that every connected claw-free graph G contains a spanning k-ended system if and only if cl(G) contains a spanning k-ended system, where cl(G) denotes Ryjáček closure of G
Spartan Daily, February 11, 1963
Volume 50, Issue 66https://scholarworks.sjsu.edu/spartandaily/4404/thumbnail.jp
The construction of a manual for speech correction to be used at St. Mary's clinic, Wilkes-Barre, Pennsylvania
Thesis (Ed.M.)--Boston Universit
Measure-Driven Algorithm Design and Analysis: A New Approach for Solving NP-hard Problems
NP-hard problems have numerous applications in various fields such as networks,
computer systems, circuit design, etc. However, no efficient algorithms have
been found for NP-hard problems. It has been commonly believed that no efficient algorithms
for NP-hard problems exist, i.e., that P6=NP. Recently, it has been observed
that there are parameters much smaller than input sizes in many instances of NP-hard
problems in the real world. In the last twenty years, researchers have been interested
in developing efficient algorithms, i.e., fixed-parameter tractable algorithms, for those
instances with small parameters. Fixed-parameter tractable algorithms can practically
find exact solutions to problem instances with small parameters, though those
problems are considered intractable in traditional computational theory.
In this dissertation, we propose a new approach of algorithm design and analysis:
discovering better measures for problems. In particular we use two measures instead of
the traditional single measure?input size to design algorithms and analyze their time
complexity. For several classical NP-hard problems, we present improved algorithms
designed and analyzed with this new approach,
First we show that the new approach is extremely powerful for designing fixedparameter
tractable algorithms by presenting improved fixed-parameter tractable algorithms
for the 3D-matching and 3D-packing problems, the multiway cut problem, the feedback vertex set problems on both directed and undirected
graph and the max-leaf problems on both directed and undirected graphs. Most of
our algorithms are practical for problem instances with small parameters.
Moreover, we show that this new approach is also good for designing exact algorithms
(with no parameters) for NP-hard problems by presenting an improved exact
algorithm for the well-known satisfiability problem.
Our results demonstrate the power of this new approach to algorithm design and
analysis for NP-hard problems. In the end, we discuss possible future directions on
this new approach and other approaches to algorithm design and analysis
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