23,404 research outputs found
Dynamic Algorithms for Graph Coloring
We design fast dynamic algorithms for proper vertex and edge colorings in a
graph undergoing edge insertions and deletions. In the static setting, there
are simple linear time algorithms for - vertex coloring and
-edge coloring in a graph with maximum degree . It is
natural to ask if we can efficiently maintain such colorings in the dynamic
setting as well. We get the following three results. (1) We present a
randomized algorithm which maintains a -vertex coloring with
expected amortized update time. (2) We present a deterministic
algorithm which maintains a -vertex coloring with
amortized update time. (3) We present a simple,
deterministic algorithm which maintains a -edge coloring with
worst-case update time. This improves the recent
-edge coloring algorithm with worst-case
update time by Barenboim and Maimon.Comment: To appear in SODA 201
Distributed Deterministic Edge Coloring using Bounded Neighborhood Independence
We study the {edge-coloring} problem in the message-passing model of
distributed computing. This is one of the most fundamental and well-studied
problems in this area. Currently, the best-known deterministic algorithms for
(2Delta -1)-edge-coloring requires O(Delta) + log-star n time \cite{PR01},
where Delta is the maximum degree of the input graph. Also, recent results of
\cite{BE10} for vertex-coloring imply that one can get an
O(Delta)-edge-coloring in O(Delta^{epsilon} \cdot \log n) time, and an
O(Delta^{1 + epsilon})-edge-coloring in O(log Delta log n) time, for an
arbitrarily small constant epsilon > 0.
In this paper we devise a drastically faster deterministic edge-coloring
algorithm. Specifically, our algorithm computes an O(Delta)-edge-coloring in
O(Delta^{epsilon}) + log-star n time, and an O(Delta^{1 +
epsilon})-edge-coloring in O(log Delta) + log-star n time. This result improves
the previous state-of-the-art {exponentially} in a wide range of Delta,
specifically, for 2^{Omega(\log-star n)} \leq Delta \leq polylog(n). In
addition, for small values of Delta our deterministic algorithm outperforms all
the existing {randomized} algorithms for this problem.
On our way to these results we study the {vertex-coloring} problem on the
family of graphs with bounded {neighborhood independence}. This is a large
family, which strictly includes line graphs of r-hypergraphs for any r = O(1),
and graphs of bounded growth. We devise a very fast deterministic algorithm for
vertex-coloring graphs with bounded neighborhood independence. This algorithm
directly gives rise to our edge-coloring algorithms, which apply to {general}
graphs.
Our main technical contribution is a subroutine that computes an
O(Delta/p)-defective p-vertex coloring of graphs with bounded neighborhood
independence in O(p^2) + \log-star n time, for a parameter p, 1 \leq p \leq
Delta
Heuristic algorithms for the min-max edge 2-coloring problem
In multi-channel Wireless Mesh Networks (WMN), each node is able to use
multiple non-overlapping frequency channels. Raniwala et al. (MC2R 2004,
INFOCOM 2005) propose and study several such architectures in which a computer
can have multiple network interface cards. These architectures are modeled as a
graph problem named \emph{maximum edge -coloring} and studied in several
papers by Feng et. al (TAMC 2007), Adamaszek and Popa (ISAAC 2010, JDA 2016).
Later on Larjomaa and Popa (IWOCA 2014, JGAA 2015) define and study an
alternative variant, named the \emph{min-max edge -coloring}.
The above mentioned graph problems, namely the maximum edge -coloring and
the min-max edge -coloring are studied mainly from the theoretical
perspective. In this paper, we study the min-max edge 2-coloring problem from a
practical perspective. More precisely, we introduce, implement and test four
heuristic approximation algorithms for the min-max edge -coloring problem.
These algorithms are based on a \emph{Breadth First Search} (BFS)-based
heuristic and on \emph{local search} methods like basic \emph{hill climbing},
\emph{simulated annealing} and \emph{tabu search} techniques, respectively.
Although several algorithms for particular graph classes were proposed by
Larjomaa and Popa (e.g., trees, planar graphs, cliques, bi-cliques,
hypergraphs), we design the first algorithms for general graphs.
We study and compare the running data for all algorithms on Unit Disk Graphs,
as well as some graphs from the DIMACS vertex coloring benchmark dataset.Comment: This is a post-peer-review, pre-copyedit version of an article
published in International Computing and Combinatorics Conference
(COCOON'18). The final authenticated version is available online at:
http://www.doi.org/10.1007/978-3-319-94776-1_5
Minimum Sum Edge Colorings of Multicycles
In the minimum sum edge coloring problem, we aim to assign natural numbers to
edges of a graph, so that adjacent edges receive different numbers, and the sum
of the numbers assigned to the edges is minimum. The {\em chromatic edge
strength} of a graph is the minimum number of colors required in a minimum sum
edge coloring of this graph. We study the case of multicycles, defined as
cycles with parallel edges, and give a closed-form expression for the chromatic
edge strength of a multicycle, thereby extending a theorem due to Berge. It is
shown that the minimum sum can be achieved with a number of colors equal to the
chromatic index. We also propose simple algorithms for finding a minimum sum
edge coloring of a multicycle. Finally, these results are generalized to a
large family of minimum cost coloring problems
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