Online coloring of hypergraphs

We give a tight bound on randomized online coloring of hypergraphs. The bound holds even if the algorithm knows the hypergraph in advance (but not the ordering in which it is presented). More specifically, we show that for any n and k, there is a 2colorable k-uniform hypergraph on n vertices for which any randomized online coloring uses Ω(n/k) colors in expectation. 1 Online Hypergraph Coloring A hypergraph H = (V,E) is formed by a set V of vertices and a collection E of subsets of V . The hypergraph is k-uniform if each of the hyperedges (the elements of E) are of cardinality k. A set S ⊂ V is an independent set if no edge in E is a subset of S. A coloring of H is a partition of V into independent sets. In the online hypergraph coloring problem, the algorithm receives in each round i, i = 1, 2, . . . , n, a vertex vi and the edges induced by (i.e., contained in) Vi = {v1, v2, . . . , vi}, and it must assign the vertex irrevocably a valid color. The objective is to minimize the number of colors. Nagy-György and Imreh [9] gave tight bounds for deterministic online hypergraph coloring. They showed that First-Fit uses at most dn/(k− 1)e colors on k-uniform hypergraphs, while no algorithm can color every such 2-colorable hypergraph with fewer colors. It is easy to see that basically no reasonable algorithm can do much worse. Randomized algorithms tend to attain better performance for many online problems. An oblivious adversary first chooses a graph and its ordering, and the algorithm can then use its random bits to thwart some of the worst-case nature of the instance. For ordinary graph coloring, the best performance ratio known by a randomized algorithm is O(n/ log n) [5], which is considerably better than the best deterministic ratio of O(n log log log n/ log log n) by a deterministic algorithm, due to Kierstead [8]. On the other hand, the best lower bound known for online graph coloring of Ω(n/ log n) [7], due to Halldórsson and Szegedy, holds both for deterministic and randomized algorithms. One feature of the construction of [7] is that it holds also in a transparent model : immediately after the algorithm makes its assignment, the adversary reveals its intended color of the node. It also holds under other relaxations, i.e., when the algorithm is allowed logarithmic lookahead, buffering, or a constant fraction of recolorings. ∗School of Computer Science, Reykjavik University, 101 Reykjavik, Iceland.