565 research outputs found
The complexity of 3-colouring H-colourable graphs.
We study the complexity of approximation on satisfiable instances for graph
homomorphism problems. For a fixed graph , the -colouring problem is to
decide whether a given graph has a homomorphism to . By a result of Hell and
Ne\v{s}et\v{r}il, this problem is NP-hard for any non-bipartite graph . In
the context of promise constraint satisfaction problems, Brakensiek and
Guruswami conjectured that this hardness result extends to promise graph
homomorphism as follows: fix any non-bipartite graph and another graph
with a homomorphism from to , it is NP-hard to find a homomorphism to
from a given -colourable graph. Arguably, the two most important special
cases of this conjecture are when is fixed to be the complete graph on 3
vertices (and is any graph with a triangle) and when is the complete
graph on 3 vertices (and is any 3-colourable graph). The former case is
equivalent to the notoriously difficult approximate graph colouring problem. In
this paper, we confirm the Brakensiek-Guruswami conjecture for the latter case.
Our proofs rely on a novel combination of the universal-algebraic approach to
promise constraint satisfaction, that was recently developed by Barto, Bul\'in
and the authors, with some ideas from algebraic topology.Comment: To appear in FOCS 201
The dynamics of proving uncolourability of large random graphs I. Symmetric Colouring Heuristic
We study the dynamics of a backtracking procedure capable of proving
uncolourability of graphs, and calculate its average running time T for sparse
random graphs, as a function of the average degree c and the number of vertices
N. The analysis is carried out by mapping the history of the search process
onto an out-of-equilibrium (multi-dimensional) surface growth problem. The
growth exponent of the average running time is quantitatively predicted, in
agreement with simulations.Comment: 5 figure
On the Complexity of Digraph Colourings and Vertex Arboricity
It has been shown by Bokal et al. that deciding 2-colourability of digraphs
is an NP-complete problem. This result was later on extended by Feder et al. to
prove that deciding whether a digraph has a circular -colouring is
NP-complete for all rational . In this paper, we consider the complexity
of corresponding decision problems for related notions of fractional colourings
for digraphs and graphs, including the star dichromatic number, the fractional
dichromatic number and the circular vertex arboricity. We prove the following
results:
Deciding if the star dichromatic number of a digraph is at most is
NP-complete for every rational .
Deciding if the fractional dichromatic number of a digraph is at most is
NP-complete for every .
Deciding if the circular vertex arboricity of a graph is at most is
NP-complete for every rational .
To show these results, different techniques are required in each case. In
order to prove the first result, we relate the star dichromatic number to a new
notion of homomorphisms between digraphs, called circular homomorphisms, which
might be of independent interest. We provide a classification of the
computational complexities of the corresponding homomorphism colouring problems
similar to the one derived by Feder et al. for acyclic homomorphisms.Comment: 21 pages, 1 figur
On the Complexity of Role Colouring Planar Graphs, Trees and Cographs
We prove several results about the complexity of the role colouring problem.
A role colouring of a graph is an assignment of colours to the vertices of
such that two vertices of the same colour have identical sets of colours in
their neighbourhoods. We show that the problem of finding a role colouring with
colours is NP-hard for planar graphs. We show that restricting the
problem to trees yields a polynomially solvable case, as long as is either
constant or has a constant difference with , the number of vertices in the
tree. Finally, we prove that cographs are always -role-colourable for
and construct such a colouring in polynomial time
- …