316 research outputs found
Graph Treewidth and Geometric Thickness Parameters
Consider a drawing of a graph in the plane such that crossing edges are
coloured differently. The minimum number of colours, taken over all drawings of
, is the classical graph parameter "thickness". By restricting the edges to
be straight, we obtain the "geometric thickness". By further restricting the
vertices to be in convex position, we obtain the "book thickness". This paper
studies the relationship between these parameters and treewidth.
Our first main result states that for graphs of treewidth , the maximum
thickness and the maximum geometric thickness both equal .
This says that the lower bound for thickness can be matched by an upper bound,
even in the more restrictive geometric setting. Our second main result states
that for graphs of treewidth , the maximum book thickness equals if and equals if . This refutes a conjecture of Ganley and
Heath [Discrete Appl. Math. 109(3):215-221, 2001]. Analogous results are proved
for outerthickness, arboricity, and star-arboricity.Comment: A preliminary version of this paper appeared in the "Proceedings of
the 13th International Symposium on Graph Drawing" (GD '05), Lecture Notes in
Computer Science 3843:129-140, Springer, 2006. The full version was published
in Discrete & Computational Geometry 37(4):641-670, 2007. That version
contained a false conjecture, which is corrected on page 26 of this versio
Fully Dynamic Matching in Bipartite Graphs
Maximum cardinality matching in bipartite graphs is an important and
well-studied problem. The fully dynamic version, in which edges are inserted
and deleted over time has also been the subject of much attention. Existing
algorithms for dynamic matching (in general graphs) seem to fall into two
groups: there are fast (mostly randomized) algorithms that do not achieve a
better than 2-approximation, and there slow algorithms with \O(\sqrt{m})
update time that achieve a better-than-2 approximation. Thus the obvious
question is whether we can design an algorithm -- deterministic or randomized
-- that achieves a tradeoff between these two: a approximation
and a better-than-2 approximation simultaneously. We answer this question in
the affirmative for bipartite graphs.
Our main result is a fully dynamic algorithm that maintains a 3/2 + \eps
approximation in worst-case update time O(m^{1/4}\eps^{/2.5}). We also give
stronger results for graphs whose arboricity is at most \al, achieving a (1+
\eps) approximation in worst-case time O(\al (\al + \log n)) for constant
\eps. When the arboricity is constant, this bound is and when the
arboricity is polylogarithmic the update time is also polylogarithmic.
The most important technical developement is the use of an intermediate graph
we call an edge degree constrained subgraph (EDCS). This graph places
constraints on the sum of the degrees of the endpoints of each edge: upper
bounds for matched edges and lower bounds for unmatched edges. The main
technical content of our paper involves showing both how to maintain an EDCS
dynamically and that and EDCS always contains a sufficiently large matching. We
also make use of graph orientations to help bound the amount of work done
during each update.Comment: Longer version of paper that appears in ICALP 201
Super-Fast 3-Ruling Sets
A -ruling set of a graph is a vertex-subset
that is independent and satisfies the property that every vertex is
at a distance of at most from some vertex in . A \textit{maximal
independent set (MIS)} is a 1-ruling set. The problem of computing an MIS on a
network is a fundamental problem in distributed algorithms and the fastest
algorithm for this problem is the -round algorithm due to Luby
(SICOMP 1986) and Alon et al. (J. Algorithms 1986) from more than 25 years ago.
Since then the problem has resisted all efforts to yield to a sub-logarithmic
algorithm. There has been recent progress on this problem, most importantly an
-round algorithm on graphs with
vertices and maximum degree , due to Barenboim et al. (Barenboim,
Elkin, Pettie, and Schneider, April 2012, arxiv 1202.1983; to appear FOCS
2012).
We approach the MIS problem from a different angle and ask if O(1)-ruling
sets can be computed much more efficiently than an MIS? As an answer to this
question, we show how to compute a 2-ruling set of an -vertex graph in
rounds. We also show that the above result can be improved
for special classes of graphs such as graphs with high girth, trees, and graphs
of bounded arboricity.
Our main technique involves randomized sparsification that rapidly reduces
the graph degree while ensuring that every deleted vertex is close to some
vertex that remains. This technique may have further applications in other
contexts, e.g., in designing sub-logarithmic distributed approximation
algorithms. Our results raise intriguing questions about how quickly an MIS (or
1-ruling sets) can be computed, given that 2-ruling sets can be computed in
sub-logarithmic rounds
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