31,872 research outputs found
Optimal Collision/Conflict-free Distance-2 Coloring in Synchronous Broadcast/Receive Tree Networks
This article is on message-passing systems where communication is (a)
synchronous and (b) based on the "broadcast/receive" pair of communication
operations. "Synchronous" means that time is discrete and appears as a sequence
of time slots (or rounds) such that each message is received in the very same
round in which it is sent. "Broadcast/receive" means that during a round a
process can either broadcast a message to its neighbors or receive a message
from one of them. In such a communication model, no two neighbors of the same
process, nor a process and any of its neighbors, must be allowed to broadcast
during the same time slot (thereby preventing message collisions in the first
case, and message conflicts in the second case). From a graph theory point of
view, the allocation of slots to processes is know as the distance-2 coloring
problem: a color must be associated with each process (defining the time slots
in which it will be allowed to broadcast) in such a way that any two processes
at distance at most 2 obtain different colors, while the total number of colors
is "as small as possible". The paper presents a parallel message-passing
distance-2 coloring algorithm suited to trees, whose roots are dynamically
defined. This algorithm, which is itself collision-free and conflict-free, uses
colors where is the maximal degree of the graph (hence
the algorithm is color-optimal). It does not require all processes to have
different initial identities, and its time complexity is , where d
is the depth of the tree. As far as we know, this is the first distributed
distance-2 coloring algorithm designed for the broadcast/receive round-based
communication model, which owns all the previous properties.Comment: 19 pages including one appendix. One Figur
The complexity of the T-coloring problem for graphs with small degree
AbstractIn the paper we consider a generalized vertex coloring model, namely T-coloring. For a given finite set T of nonnegative integers including 0, a proper vertex coloring is called a T-coloring if the distance of the colors of adjacent vertices is not an element of T. This problem is a generalization of the classic vertex coloring and appeared as a model of the frequency assignment problem. We present new results concerning the complexity of T-coloring with the smallest span on graphs with small degree Î. We distinguish between the cases that appear to be polynomial or NP-complete. More specifically, we show that our problem is polynomial on graphs with Îâ©œ2 and in the case of k-regular graphs it becomes NP-hard even for every fixed T and every k>3. Also, the case of graphs with Î=3 is under consideration. Our results are based on the complexity properties of the homomorphism of graphs
Digraph Coloring and Distance to Acyclicity
In -Digraph Coloring we are given a digraph and are asked to partition its
vertices into at most sets, so that each set induces a DAG. This well-known
problem is NP-hard, as it generalizes (undirected) -Coloring, but becomes
trivial if the input digraph is acyclic. This poses the natural parameterized
complexity question what happens when the input is "almost" acyclic. In this
paper we study this question using parameters that measure the input's distance
to acyclicity in either the directed or the undirected sense.
It is already known that, for all , -Digraph Coloring is NP-hard
on digraphs of DFVS at most . We strengthen this result to show that, for
all , -Digraph Coloring is NP-hard for DFVS . Refining our
reduction we obtain two further consequences: (i) for all , -Digraph
Coloring is NP-hard for graphs of feedback arc set (FAS) at most ;
interestingly, this leads to a dichotomy, as we show that the problem is FPT by
if FAS is at most ; (ii) -Digraph Coloring is NP-hard for graphs
of DFVS , even if the maximum degree is at most ; we show
that this is also almost tight, as the problem becomes FPT for DFVS and
.
We then consider parameters that measure the distance from acyclicity of the
underlying graph. We show that -Digraph Coloring admits an FPT algorithm
parameterized by treewidth, whose parameter dependence is . Then,
we pose the question of whether the factor can be eliminated. Our main
contribution in this part is to settle this question in the negative and show
that our algorithm is essentially optimal, even for the much more restricted
parameter treedepth and for . Specifically, we show that an FPT algorithm
solving -Digraph Coloring with dependence would contradict the
ETH
Average Sensitivity of Graph Algorithms
In modern applications of graphs algorithms, where the graphs of interest are
large and dynamic, it is unrealistic to assume that an input representation
contains the full information of a graph being studied. Hence, it is desirable
to use algorithms that, even when only a (large) subgraph is available, output
solutions that are close to the solutions output when the whole graph is
available. We formalize this idea by introducing the notion of average
sensitivity of graph algorithms, which is the average earth mover's distance
between the output distributions of an algorithm on a graph and its subgraph
obtained by removing an edge, where the average is over the edges removed and
the distance between two outputs is the Hamming distance.
In this work, we initiate a systematic study of average sensitivity. After
deriving basic properties of average sensitivity such as composition, we
provide efficient approximation algorithms with low average sensitivities for
concrete graph problems, including the minimum spanning forest problem, the
global minimum cut problem, the minimum - cut problem, and the maximum
matching problem. In addition, we prove that the average sensitivity of our
global minimum cut algorithm is almost optimal, by showing a nearly matching
lower bound. We also show that every algorithm for the 2-coloring problem has
average sensitivity linear in the number of vertices. One of the main ideas
involved in designing our algorithms with low average sensitivity is the
following fact; if the presence of a vertex or an edge in the solution output
by an algorithm can be decided locally, then the algorithm has a low average
sensitivity, allowing us to reuse the analyses of known sublinear-time
algorithms and local computation algorithms (LCAs). Using this connection, we
show that every LCA for 2-coloring has linear query complexity, thereby
answering an open question.Comment: 39 pages, 1 figur
Computing Square Colorings on Bounded-Treewidth and Planar Graphs
A square coloring of a graph is a coloring of the square of ,
that is, a coloring of the vertices of such that any two vertices that are
at distance at most in receive different colors. We investigate the
complexity of finding a square coloring with a given number of colors. We
show that the problem is polynomial-time solvable on graphs of bounded
treewidth by presenting an algorithm with running time for graphs of treewidth at most . The somewhat
unusual exponent in the running time is essentially
optimal: we show that for any , there is no algorithm with running
time unless the
Exponential-Time Hypothesis (ETH) fails.
We also show that the square coloring problem is NP-hard on planar graphs for
any fixed number of colors. Our main algorithmic result is showing
that the problem (when the number of colors is part of the input) can be
solved in subexponential time on planar graphs. The
result follows from the combination of two algorithms. If the number of
colors is small (), then we can exploit a treewidth bound on the
square of the graph to solve the problem in time . If
the number of colors is large (), then an algorithm based on
protrusion decompositions and building on our result for the bounded-treewidth
case solves the problem in time .Comment: 72 pages, 15 figures, full version of a paper accepted at SODA 202
A general framework for coloring problems: old results, new results, and open problems
In this survey paper we present a general framework for coloring problems that was introduced in a joint paper which the author presented at WG2003. We show how a number of different types of coloring problems, most of which have been motivated from frequency assignment, fit into this framework. We give a survey of the existing results, mainly based on and strongly biased by joint work of the author with several different groups of coauthors, include some new results, and discuss several open problems for each of the variants
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