Efficiently Enumerating Hitting Sets of Hypergraphs Arising in Data Profiling. We devise a method to enumerate the inclusionwise minimal hitting sets of a hypergraph. The algorithm has delay O( mk*1 , n2) on nvertex, medge hypergraphs, where k* is the rank of the transversal hypergraph, i.e., the cardinality of the largest minimal solution. In particular, on classes of hypergraphs for which k* is bounded, the delay is polynomial. The algorithm uses space linear in the input size only. The enumeration methods solves the extension problem for minimal hitting sets as a subroutine. We show that this problem, parameterised by the cardinality of the set which is to be extended, is one of the first natural W3complete problems. We give an algorithm for the subroutine that is optimal under the assumption that W2 neq mathrmFPT or the exponential time hypothesis (ETH), respectively. Despite the hardness of the extension problem, we provide empirical evidence indicating that the enumeration outperforms its theoretical worstcase guarantee on hypergraphs arising in the profiling of relational databases, namely, in the detection of unique column combinations. Our analysis suggest that these hypergraphs exhibit structure that allows the subroutine to be fast on average.

The Lexicographic Method for the Threshold Cover Problem The lexicographic method is a technique that was introduced by Hell and Huang Journal of Graph Theory, 20(3): 361xe2x80x93374, 1995 as a way to simplify the problems of recognizing and obtaining representations of comparability graphs, proper circulararc graphs and proper interval graphs. This method gives rise to conceptually simple recognition algorithms and leads to much simpler proofs for some characterization theorems for these classes. Threshold graphs are a class of graphs that have many equivalent definitions and have applications in integer programming and set packing problems. A graph is said to have a threshold cover of size k if its edges can be covered using k threshold graphs. Chvatal and Hammer conjectured in 1977 that given a graph G, a suitably constructed auxiliary graph (Gu0027) has chromatic number equal to the minimum size of a threshold cover of G. Although this conjecture was shown to be false in the general case by Cozzens and Leibowitz, it was shown to be true for graphs having a threshold cover of size 2 by Raschle and Simon Proceedings of the Twentyseventh Annual ACM Symposium on Theory of Computing, STOC xe2x80x9995, pages 650xe2x80x93661, 1995. That is, a graph G has a threshold cover of size 2 if and only if (Gu0027) is bipartitexe2x80x94this is the only known forbidden structure characterization of graphs having a threshold cover of size 2. We show how the lexicographic method can be used to obtain a completely new and much simpler proof for this result. This method also gives rise to a simple new LexBFSbased algorithm for recognizing graphs having a threshold cover of size 2. Although this algorithm is not the fastest known, it is a certifying algorithm that matches the time complexity of the fastest known certifying algorithm for this problem. The algorithm can also be easily adapted to give a certifying recognition algorithm for bipartite graphs that can be covered by two chain subgraphs.