Chip games and paintability

We prove that the difference between the paint number and the choice number of a complete bipartite graph $K_{N,N}$ is $\Theta(\log \log N )$. That answers the question of Zhu (2009) whether this difference, for all graphs, can be bounded by a common constant. By a classical correspondence, our result translates to the framework of on-line coloring of uniform hypergraphs. This way we obtain that for every on-line two coloring algorithm there exists a k-uniform hypergraph with $\Theta(2^k )$ edges on which the strategy fails. The results are derived through an analysis of a natural family of chip games

On-line Algorithms for 2-Coloring Hypergraphs via Chip Games

Erdős has shown that for all k-hypergraphs with fewer than 2 k\Gamma1 edges, there exists a 2-coloring of the nodes so that no edge is monochromatic. Erdos has also shown that when the number of edges is greater than k 2 2 k+1 , there exist k-hypergraphs with no such 2-coloring. These bounds are not contructive, however. In this paper, we take an "on-line" look at this problem, showing constructive upper and lower bounds on the number of edges of a hypergraph which allow it to be 2-colored on-line. These bounds become particularly interesting for degree-k k-hypergraphs which always have a good 2-coloring for all k 10 by the Lovász Local Lemma. In this case, our upper bound demonstrates an inherent weakness of on-line strategies by constructing an adversary which defeats any on-line 2-coloring algorithm using degree-k k-hypergraph with (3 + 2 p 2) k edges

On-line Algorithms for 2-Coloring Hypergraphs via Chip Games

Erdös has shown that for all k-hypergraphs with fewer than 2 k−1 edges, there exists a 2-coloring of the nodes so that no edge is monochromatic. Erdös has also shown that when the number of edges is greater than k 2 2 k+1, there exist k-hypergraphs with no such 2-coloring. These bounds are not constructive, however. In this paper, we take an “on-line ” look at this problem, showing constructive upper and lower bounds on the number of edges of a hypergraph which allow it to be 2-colored on-line. These bounds become particularly interesting for degree-k k-hypergraphs which always have a good 2-coloring for all k ≥ 10 by the Lovász Local Lemma. In this case, our upper bound demonstrates an inherent weakness of on-line strategies by constructing an adversary which defeats any on-line 2-coloring algorithm using degree-k k-hypergraph with (3 + 2 √ 2) k edges

Noise tolerant algorithms for learning and searching

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 1995.Includes bibliographical references (p. 109-112).by Javed Alexander Aslam.Ph.D