Further algorithmic aspects of the local lemma
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
Coloring nonuniform hypergraphs: a new algorithmic approach to the general Lovász local lemma
Proceedings of the ninth international conference on on Random structures and algorithms
New Algorithmic Aspects of the Local Lemma with Applications to Routing and Partitioning
SIAM Journal on Computing
Conflict-Free Coloring of Points and Simple Regions in the Plane
Discrete & Computational Geometry
Conflict-free colorings of shallow discs
Proceedings of the twenty-second annual symposium on Computational geometry
How to play a coloring game against a color-blind adversary
Proceedings of the twenty-second annual symposium on Computational geometry
A bound on the chromatic number of the square of a planar graph
Journal of Combinatorial Theory Series B
Conflict-free coloring for rectangle ranges using O(n.382) colors
Proceedings of the nineteenth annual ACM symposium on Parallel algorithms and architectures
Online Conflict-Free Coloring for Intervals
SIAM Journal on Computing
On The Chromatic Number of Geometric Hypergraphs
SIAM Journal on Discrete Mathematics
Deterministic conflict-free coloring for intervals: From offline to online
ACM Transactions on Algorithms (TALG)
A constructive proof of the Lovász local lemma
Proceedings of the forty-first annual ACM symposium on Theory of computing
Conflict-free coloring
A constructive proof of the general lovász local lemma
Journal of the ACM (JACM)
The potential to improve the choice: list conflict-free coloring for geometric hypergraphs
Proceedings of the twenty-seventh annual symposium on Computational geometry
Unique-maximum and conflict-free coloring for hypergraphs and tree graphs
SOFSEM'12 Proceedings of the 38th international conference on Current Trends in Theory and Practice of Computer Science
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A colouring of the vertices of a hypergraph H is called conflict-free if each hyperedge E of H contains a vertex of ‘unique’ colour that does not get repeated in E. The smallest number of colours required for such a colouring is called the conflict-free chromatic number of H, and is denoted by χCF(H). This parameter was first introduced by Even, Lotker, Ron and Smorodinsky (FOCS 2002) in a geometric setting, in connection with frequency assignment problems for cellular networks. Here we analyse this notion for general hypergraphs. It is shown that $\chi_{\rm CF}(H)\leq 1/2+\sqrt{2m+1/4}$, for every hypergraph with m edges, and that this bound is tight. Better bounds of the order of m1/t log m are proved under the assumption that the size of every edge of H is at least 2t − 1, for some t ≥ 3. Using Lovász's Local Lemma, the same result holds for hypergraphs in which the size of every edge is at least 2t − 1 and every edge intersects at most m others. We give efficient polynomial-time algorithms to obtain such colourings. Our machinery can also be applied to the hypergraphs induced by the neighbourhoods of the vertices of a graph. It turns out that in this case we need far fewer colours. For example, it is shown that the vertices of any graph G with maximum degree Δ can be coloured with log2+ε Δ colours, so that the neighbourhood of every vertex contains a point of ‘unique’ colour. We give an efficient deterministic algorithm to find such a colouring, based on a randomized algorithmic version of the Lovász Local Lemma, suggested by Beck, Molloy and Reed. To achieve this, we need to (1) correct a small error in the Molloy–Reed approach, (2) restate and re-prove their result in a deterministic form.