6 research outputs found
Locality of not-so-weak coloring
Many graph problems are locally checkable: a solution is globally feasible if
it looks valid in all constant-radius neighborhoods. This idea is formalized in
the concept of locally checkable labelings (LCLs), introduced by Naor and
Stockmeyer (1995). Recently, Chang et al. (2016) showed that in bounded-degree
graphs, every LCL problem belongs to one of the following classes:
- "Easy": solvable in rounds with both deterministic and
randomized distributed algorithms.
- "Hard": requires at least rounds with deterministic and
rounds with randomized distributed algorithms.
Hence for any parameterized LCL problem, when we move from local problems
towards global problems, there is some point at which complexity suddenly jumps
from easy to hard. For example, for vertex coloring in -regular graphs it is
now known that this jump is at precisely colors: coloring with colors
is easy, while coloring with colors is hard.
However, it is currently poorly understood where this jump takes place when
one looks at defective colorings. To study this question, we define -partial
-coloring as follows: nodes are labeled with numbers between and ,
and every node is incident to at least properly colored edges.
It is known that -partial -coloring (a.k.a. weak -coloring) is easy
for any . As our main result, we show that -partial -coloring
becomes hard as soon as , no matter how large a we have.
We also show that this is fundamentally different from -partial
-coloring: no matter which we choose, the problem is always hard
for but it becomes easy when . The same was known previously
for partial -coloring with , but the case of was open
Characterizing Circular Colouring Mixing for
Given a graph , the -mixing problem asks: Can one obtain all
-colourings of , starting from one -colouring , by changing the
colour of only one vertex at a time, while at each step maintaining a
-colouring? More generally, for a graph , the -mixing problem asks:
Can one obtain all homomorphisms , starting from one homomorphism ,
by changing the image of only one vertex at a time, while at each step
maintaining a homomorphism ?
This paper focuses on a generalization of -colourings, namely
-circular colourings. We show that when , a graph
is -mixing if and only if for any -colouring of , and
any cycle of , the wind of the cycle under the colouring equals a
particular value (which intuitively corresponds to having no wind). As a
consequence we show that -mixing is closed under a restricted
homomorphism called a fold. Using this, we deduce that -mixing is
co-NP-complete for all , and by similar ideas we show that if
the circular chromatic number of a connected graph is ,
then folds to . We use the characterization to settle a
conjecture of Brewster and Noel, specifically that the circular mixing number
of bipartite graphs is . Lastly, we give a polynomial time algorithm for
-mixing in planar graphs when .Comment: 21 page
Distributed recoloring
Given two colorings of a graph, we consider the following problem: can we recolor the graph from one coloring to the other through a series of elementary changes, such that the graph is properly colored after each step? We introduce the notion of distributed recoloring: The input graph represents a network of computers that needs to be recolored. Initially, each node is aware of its own input color and target color. The nodes can exchange messages with each other, and eventually each node has to stop and output its own recoloring schedule, indicating when and how the node changes its color. The recoloring schedules have to be globally consistent so that the graph remains properly colored at each point, and we require that adjacent nodes do not change their colors simultaneously. We are interested in the following questions: How many communication rounds are needed (in the deterministic LOCAL model of distributed computing) to find a recoloring schedule? What is the length of the recoloring schedule? And how does the picture change if we can use extra colors to make recoloring easier? The main contributions of this work are related to distributed recoloring with one extra color in the following graph classes: trees, 3-regular graphs, and toroidal grids.Peer reviewe
Distributed Recoloring of Interval and Chordal Graphs
International audienceOne of the fundamental and most-studied algorithmic problems in distributed computing on networks is graph coloring, both in bounded-degree and in general graphs. Recently, the study of this problem has been extended in two directions. First, the problem of recoloring, that is computing an efficient transformation between two given colorings (instead of computing a new coloring), has been considered, both to model radio network updates, and as a useful subroutine for coloring. Second, as it appears that general graphs and bounded-degree graphs do not model real networks very well (with, respectively, pathological worst-case topologies and too strong assumptions), coloring has been studied in more specific graph classes. In this paper, we study the intersection of these two directions: distributed recoloring in two relevant graph classes, interval and chordal graphs. More formally, the question of recoloring a graph is as follows: we are given a network, an input coloring α and a target coloring β, and we want to find a schedule of colorings to reach β starting from α. In a distributed setting, the schedule needs to be found within the LOCAL model, where nodes communicate with their direct neighbors synchronously. The question we want to answer is: how many rounds of communication are needed to produce a schedule, and what is the length of this schedule? In the case of interval and chordal graphs, we prove that, if we have less than 2ω colors, ω being the size of the largest clique, extra colors will be needed in the intermediate colorings. For interval graphs, we produce a schedule after O(poly(∆) log * n) rounds of communication, and for chordal graphs, we need O(ω 2 ∆ 2 log n) rounds to get one. Our techniques also improve classic coloring algorithms. Namely, we get ω + 1-colorings of interval graphs in O(ω log * n) rounds and of chordal graphs in O(ω log n) rounds, which improves on previous known algorithms that use ω + 2 colors for the same running times