Efficient algorithms for common transversals

Efficient Algorithms for Common Transversals

Suppose we are gIven a set S of n (possibly intersecting) simple objects in the plane, such that for every pair of objects in S, the intersection of the boundaries of these two objects has at most a connected components. The integer a is independent of n, Le. a.=O (1). \Ve consider the problem of detennining whether there exists a straight line that goes through every object in S. We give an 0 ...

Common Transversals

Given t families, each family consisting of s finite sets, we show that if the families “separate points" in a natural way, and if the union of all the sets in all the families contains more than (s+1)t st 1 1 elements, then a common transversal of the t families exists. In case each family is a covering family, the bound is st st 1. Both of these bounds are best possible. This work extends rec...

Algorithms for finding clique-transversals of graphs

A clique-transversal of a graph G is a subset of vertices intersecting all the cliques of G. It is NP-hard to determine the minimum cardinality τc of a clique-transversal of G. In this work, first we propose an algorithm for determining this parameter for a general graph, which runs in polynomial time, for fixed τc. This algorithm is employed for finding the minimum cardinality clique-transvers...

Geometric Permutations and Common Transversals

Common transversals for partitions of a finite set

For s 2, t 1, let A1; : : : ;At be s-cell partitions of a finite set X . Assume that if x;y 2 X , x 6= y, then x;y belong to different cells of at least one of the partitions Ai. For each k 1, let c(s; t;k) be the least integer such that if A1; : : : ;At , X satisfy the preceding conditions, and the smallest of all the cells of all the partitions has exactly k elements, and jX j c(s; t;k), then...