1 research outputs found
Conflict-Free Colouring using Maximum Independent Set and Minimum Colouring
Given a hypergraph , the conflict-free colouring problem is to colour
vertices of using minimum colours so that each hyperedge in sees a
unique colour. We present a polynomial time reduction from the conflict-free
colouring problem in hypergraphs to the maximum independent set problem in a
class of simple graphs, which we refer to as \textit{conflict graphs}. We also
present another characterization of the conflict-free colouring number in terms
of the chromatic number of graphs in an associated family of simple graphs,
which we refer to as \textit{co-occurrence graphs}. We present perfectness
results for co-occurrence graphs and a special case of conflict graphs. Based
on these results and a linear program that returns an integer solution in
polynomial time, we obtain a polynomial time algorithm to compute a minimum
conflict-free colouring of interval hypergraphs, thus solving an open problem
due to Cheilaris et al.\cite{CPLGARSS2014}. Finally, we use the co-occurrence
graph characterization to prove that for an interval hypergraph, the
conflict-free colouring number is the minimum partition of its intervals into
sets such that each set has an exact hitting set (a hitting set in which each
interval is hit exactly once).Comment: arXiv admin note: text overlap with arXiv:1707.0507