90,217 research outputs found
Sum Coloring : New upper bounds for the chromatic strength
The Minimum Sum Coloring Problem (MSCP) is derived from the Graph Coloring
Problem (GCP) by associating a weight to each color. The aim of MSCP is to find
a coloring solution of a graph such that the sum of color weights is minimum.
MSCP has important applications in fields such as scheduling and VLSI design.
We propose in this paper new upper bounds of the chromatic strength, i.e. the
minimum number of colors in an optimal solution of MSCP, based on an
abstraction of all possible colorings of a graph called motif. Experimental
results on standard benchmarks show that our new bounds are significantly
tighter than the previous bounds in general, allowing to reduce substantially
the search space when solving MSCP .Comment: pre-prin
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
Gr\"obner Bases and Nullstellens\"atze for Graph-Coloring Ideals
We revisit a well-known family of polynomial ideals encoding the problem of
graph--colorability. Our paper describes how the inherent combinatorial
structure of the ideals implies several interesting algebraic properties.
Specifically, we provide lower bounds on the difficulty of computing Gr\"obner
bases and Nullstellensatz certificates for the coloring ideals of general
graphs. For chordal graphs, however, we explicitly describe a Gr\"obner basis
for the coloring ideal, and provide a polynomial-time algorithm.Comment: 16 page
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
Conflict-Free Vertex Coloring Of Planar Graphs
The conflict-free coloring problem is a variation of the vertex coloring problem, a classical NP-hard optimization problem. The conflict-free coloring problem aims to color a possibly proper subset of vertices such that there is a unique color within the closed neighborhood (a vertex and its neighbors) of every vertex. This paper presents recent findings and heuristics to solve the conflict-free coloring problem on both general graphs and planar graphs
An NP-Complete Problem in Grid Coloring
A c-coloring of G(n,m)=n x m is a mapping of G(n,m) into {1,...,c} such that
no four corners forming a rectangle have the same color. In 2009 a challenge
was proposed via the internet to find a 4-coloring of G(17,17). This attracted
considerable attention from the popular mathematics community. A coloring was
produced; however, finding it proved to be difficult. The question arises: is
the problem of grid coloring is difficult in general? We present three results
that support this conjecture, (1) an NP completeness result, (2) a lower bound
on Tree-resolution, (3) a lower bound on Tree-CP proofs. Note that items (2)
and (3) yield statements from Ramsey Theory which are of size polynomial in
their parameters and require exponential size in various proof systems.Comment: 25 page
Reinforcement learning based local search for grouping problems: A case study on graph coloring
Grouping problems aim to partition a set of items into multiple mutually
disjoint subsets according to some specific criterion and constraints. Grouping
problems cover a large class of important combinatorial optimization problems
that are generally computationally difficult. In this paper, we propose a
general solution approach for grouping problems, i.e., reinforcement learning
based local search (RLS), which combines reinforcement learning techniques with
descent-based local search. The viability of the proposed approach is verified
on a well-known representative grouping problem (graph coloring) where a very
simple descent-based coloring algorithm is applied. Experimental studies on
popular DIMACS and COLOR02 benchmark graphs indicate that RLS achieves
competitive performances compared to a number of well-known coloring
algorithms
Coloring Hypergraphs Induced by Dynamic Point Sets and Bottomless Rectangles
We consider a coloring problem on dynamic, one-dimensional point sets: points
appearing and disappearing on a line at given times. We wish to color them with
k colors so that at any time, any sequence of p(k) consecutive points, for some
function p, contains at least one point of each color.
We prove that no such function p(k) exists in general. However, in the
restricted case in which points appear gradually, but never disappear, we give
a coloring algorithm guaranteeing the property at any time with p(k)=3k-2. This
can be interpreted as coloring point sets in R^2 with k colors such that any
bottomless rectangle containing at least 3k-2 points contains at least one
point of each color. Here a bottomless rectangle is an axis-aligned rectangle
whose bottom edge is below the lowest point of the set. For this problem, we
also prove a lower bound p(k)>ck, where c>1.67. Hence for every k there exists
a point set, every k-coloring of which is such that there exists a bottomless
rectangle containing ck points and missing at least one of the k colors.
Chen et al. (2009) proved that no such function exists in the case of
general axis-aligned rectangles. Our result also complements recent results
from Keszegh and Palvolgyi on cover-decomposability of octants (2011, 2012).Comment: A preliminary version was presented by a subset of the authors to the
European Workshop on Computational Geometry, held in Assisi (Italy) on March
19-21, 201
Algorithms for the minimum sum coloring problem: a review
The Minimum Sum Coloring Problem (MSCP) is a variant of the well-known vertex
coloring problem which has a number of AI related applications. Due to its
theoretical and practical relevance, MSCP attracts increasing attention. The
only existing review on the problem dates back to 2004 and mainly covers the
history of MSCP and theoretical developments on specific graphs. In recent
years, the field has witnessed significant progresses on approximation
algorithms and practical solution algorithms. The purpose of this review is to
provide a comprehensive inspection of the most recent and representative MSCP
algorithms. To be informative, we identify the general framework followed by
practical solution algorithms and the key ingredients that make them
successful. By classifying the main search strategies and putting forward the
critical elements of the reviewed methods, we wish to encourage future
development of more powerful methods and motivate new applications
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