14,754 research outputs found
On DP-Coloring of Digraphs
DP-coloring is a relatively new coloring concept by Dvo\v{r}\'ak and Postle
and was introduced as an extension of list-colorings of (undirected) graphs. It
transforms the problem of finding a list-coloring of a given graph with a
list-assignment to finding an independent transversal in an auxiliary graph
with vertex set . In this paper, we
extend the definition of DP-colorings to digraphs using the approach from
Neumann-Lara where a coloring of a digraph is a coloring of the vertices such
that the digraph does not contain any monochromatic directed cycle.
Furthermore, we prove a Brooks' type theorem regarding the DP-chromatic number,
which extends various results on the (list-)chromatic number of digraphs.Comment: 23 pages, 6 figure
An enhanced formulation for solving graph coloring problems with the Douglas–Rachford algorithm
We study the behavior of the Douglas–Rachford algorithm on the graph vertex-coloring problem. Given a graph and a number of colors, the goal is to find a coloring of the vertices so that all adjacent vertex pairs have different colors. In spite of the combinatorial nature of this problem, the Douglas–Rachford algorithm was recently shown to be a successful heuristic for solving a wide variety of graph coloring instances, when the problem was cast as a feasibility problem on binary indicator variables. In this work we consider a different formulation, based on semidefinite programming. The much improved performance of the Douglas–Rachford algorithm, with this new approach, is demonstrated through various numerical experiments.F. J. Aragón Artacho and R. Campoy were partially supported by MICINN of Spain and ERDF of EU, Grants MTM2014-59179-C2-1-P and PGC2018-097960-B-C22. F. J. Aragón Artacho was supported by the Ramón y Cajal program by MINECO of Spain and ERDF of EU (RYC-2013-13327) and R. Campoy was supported by MINECO of Spain and ESF of EU (BES-2015-073360) under the program “Ayudas para contratos predoctorales para la formación de doctores 2015”
Adaptive feasible and infeasible tabu search for weighted vertex coloring
The Weighted Vertex Coloring Problem of a vertex weighted graph is to partition the vertex set into k disjoint independent sets such that the sum of the costs of these sets is minimized, where the cost of each set is given by the maximum weight of a vertex (representative) in that set. To solve this NP-hard problem, we present the adaptive feasible and infeasible search algorithm (AFISA) that relies on a mixed search strategy exploring both feasible and infeasible solutions. From an initial feasible solution, AFISA seeks improved solutions by oscillating between feasible and infeasible regions. To prevent the search from going too far from feasibility boundaries, we introduce a control mechanism that adaptively makes the algorithm to go back and forth between feasible and infeasible solutions. To explore the search space, we use a tabu search optimization procedure to ensure an intensified exploitation of candidate solutions and an adaptive perturbation strategy to escape local optimum traps. We show extensive experimental results on 161 benchmark instances and present new upper bounds that are useful for future studies. We assess the benefit of the key features of the proposed approach. This work demonstrates that examining both feasible and infeasible solutions during the search is a highly effective search strategy for the considered coloring problem and could beneficially be applied to other constrained problems as well
Pliable Index Coding via Conflict-Free Colorings of Hypergraphs
In the pliable index coding (PICOD) problem, a server is to serve multiple
clients, each of which possesses a unique subset of the complete message set as
side information and requests a new message which it does not have. The goal of
the server is to do this using as few transmissions as possible. This work
presents a hypergraph coloring approach to the PICOD problem. A
\textit{conflict-free coloring} of a hypergraph is known from literature as an
assignment of colors to its vertices so that each edge of the graph contains
one uniquely colored vertex. For a given PICOD problem represented by a
hypergraph consisting of messages as vertices and request-sets as edges, we
present achievable PICOD schemes using conflict-free colorings of the PICOD
hypergraph. Various graph theoretic parameters arising out of such colorings
(and some new variants) then give a number of upper bounds on the optimal PICOD
length, which we study in this work. Our achievable schemes based on hypergraph
coloring include scalar as well as vector linear PICOD schemes. For the scalar
case, using the correspondence with conflict-free coloring, we show the
existence of an achievable scheme which has length where
refers to a parameter of the hypergraph that captures the maximum
`incidence' number of other edges on any edge. This result improves upon known
achievability results in PICOD literature, in some parameter regimes.Comment: 21 page
An abstract approach to polychromatic coloring: shallow hitting sets in ABA-free hypergraphs and pseudohalfplanes
The goal of this paper is to give a new, abstract approach to
cover-decomposition and polychromatic colorings using hypergraphs on ordered
vertex sets. We introduce an abstract version of a framework by Smorodinsky and
Yuditsky, used for polychromatic coloring halfplanes, and apply it to so-called
ABA-free hypergraphs, which are a generalization of interval graphs. Using our
methods, we prove that (2k-1)-uniform ABA-free hypergraphs have a polychromatic
k-coloring, a problem posed by the second author. We also prove the same for
hypergraphs defined on a point set by pseudohalfplanes. These results are best
possible. We could only prove slightly weaker results for dual hypergraphs
defined by pseudohalfplanes, and for hypergraphs defined by pseudohemispheres.
We also introduce another new notion that seems to be important for
investigating polychromatic colorings and epsilon-nets, shallow hitting sets.
We show that all the above hypergraphs have shallow hitting sets, if their
hyperedges are containment-free
Sublinear Time and Space Algorithms for Correlation Clustering via Sparse-Dense Decompositions
We present a new approach for solving (minimum disagreement) correlation
clustering that results in sublinear algorithms with highly efficient time and
space complexity for this problem. In particular, we obtain the following
algorithms for -vertex -labeled graphs :
-- A sublinear-time algorithm that with high probability returns a constant
approximation clustering of in time assuming access to the
adjacency list of the -labeled edges of (this is almost quadratically
faster than even reading the input once). Previously, no sublinear-time
algorithm was known for this problem with any multiplicative approximation
guarantee.
-- A semi-streaming algorithm that with high probability returns a constant
approximation clustering of in space and a single pass over
the edges of the graph (this memory is almost quadratically smaller than
input size). Previously, no single-pass algorithm with space was known
for this problem with any approximation guarantee.
The main ingredient of our approach is a novel connection to sparse-dense
graph decompositions that are used extensively in the graph coloring
literature. To our knowledge, this connection is the first application of these
decompositions beyond graph coloring, and in particular for the correlation
clustering problem, and can be of independent interest
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