21 research outputs found
Coloring tournaments with few colors: Algorithms and complexity
A -coloring of a tournament is a partition of its vertices into
acyclic sets. Deciding if a tournament is 2-colorable is NP-hard. A natural
problem, akin to that of coloring a 3-colorable graph with few colors, is to
color a 2-colorable tournament with few colors. This problem does not seem to
have been addressed before, although it is a special case of coloring a
2-colorable 3-uniform hypergraph with few colors, which is a well-studied
problem with super-constant lower bounds.
We present an efficient decomposition lemma for tournaments and show that it
can be used to design polynomial-time algorithms to color various classes of
tournaments with few colors, including an algorithm to color a 2-colorable
tournament with ten colors. For the classes of tournaments considered, we
complement our upper bounds with strengthened lower bounds, painting a
comprehensive picture of the algorithmic and complexity aspects of coloring
tournaments
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
Improved hardness for H-colourings of G-colourable graphs
We present new results on approximate colourings of graphs and, more
generally, approximate H-colourings and promise constraint satisfaction
problems.
First, we show NP-hardness of colouring -colourable graphs with
colours for every . This improves
the result of Bul\'in, Krokhin, and Opr\v{s}al [STOC'19], who gave NP-hardness
of colouring -colourable graphs with colours for , and the
result of Huang [APPROX-RANDOM'13], who gave NP-hardness of colouring
-colourable graphs with colours for sufficiently large .
Thus, for , we improve from known linear/sub-exponential gaps to
exponential gaps.
Second, we show that the topology of the box complex of H alone determines
whether H-colouring of G-colourable graphs is NP-hard for all (non-bipartite,
H-colourable) G. This formalises the topological intuition behind the result of
Krokhin and Opr\v{s}al [FOCS'19] that 3-colouring of G-colourable graphs is
NP-hard for all (3-colourable, non-bipartite) G. We use this technique to
establish NP-hardness of H-colouring of G-colourable graphs for H that include
but go beyond , including square-free graphs and circular cliques (leaving
and larger cliques open).
Underlying all of our proofs is a very general observation that adjoint
functors give reductions between promise constraint satisfaction problems.Comment: Mention improvement in Proposition 2.5. SODA 202
LIPIcs, Volume 244, ESA 2022, Complete Volume
LIPIcs, Volume 244, ESA 2022, Complete Volum
LIPIcs, Volume 274, ESA 2023, Complete Volume
LIPIcs, Volume 274, ESA 2023, Complete Volum