104 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
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
Introduction to local certification
A distributed graph algorithm is basically an algorithm where every node of a
graph can look at its neighborhood at some distance in the graph and chose its
output. As distributed environment are subject to faults, an important issue is
to be able to check that the output is correct, or in general that the network
is in proper configuration with respect to some predicate. One would like this
checking to be very local, to avoid using too much resources. Unfortunately
most predicates cannot be checked this way, and that is where certification
comes into play. Local certification (also known as proof-labeling schemes,
locally checkable proofs or distributed verification) consists in assigning
labels to the nodes, that certify that the configuration is correct. There are
several point of view on this topic: it can be seen as a part of
self-stabilizing algorithms, as labeling problem, or as a non-deterministic
distributed decision.
This paper is an introduction to the domain of local certification, giving an
overview of the history, the techniques and the current research directions.Comment: Last update: minor editin
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