536 research outputs found
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
Distributed coloring in sparse graphs with fewer colors
This paper is concerned with efficiently coloring sparse graphs in the
distributed setting with as few colors as possible. According to the celebrated
Four Color Theorem, planar graphs can be colored with at most 4 colors, and the
proof gives a (sequential) quadratic algorithm finding such a coloring. A
natural problem is to improve this complexity in the distributed setting. Using
the fact that planar graphs contain linearly many vertices of degree at most 6,
Goldberg, Plotkin, and Shannon obtained a deterministic distributed algorithm
coloring -vertex planar graphs with 7 colors in rounds. Here, we
show how to color planar graphs with 6 colors in \mbox{polylog}(n) rounds.
Our algorithm indeed works more generally in the list-coloring setting and for
sparse graphs (for such graphs we improve by at least one the number of colors
resulting from an efficient algorithm of Barenboim and Elkin, at the expense of
a slightly worst complexity). Our bounds on the number of colors turn out to be
quite sharp in general. Among other results, we show that no distributed
algorithm can color every -vertex planar graph with 4 colors in
rounds.Comment: 16 pages, 4 figures - An extended abstract of this work was presented
at PODC'18 (ACM Symposium on Principles of Distributed Computing
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