Consider the Hitting Set problem where, for a given universe $\mathcal{X} = \left\{ 1, ... , n \right\}$ and a collection of subsets $\mathcal{S}_1, ... , \mathcal{S}_m$, one seeks to identify the smallest subset of $\mathcal{X}$ which has nonempty intersection with every element in the collection. We study a probabilistic formulation of this problem, where the underlying subsets are formed by including each element of the universe with probability $p$, independently of one another. For large enough values of $n$, we rigorously analyse the average case performance of Lov\'asz's celebrated greedy algorithm (Lov\'asz, 1975) with respect to the chosen input distribution. In addition, we study integrality gaps between linear programming and integer programming solutions of the problem.

翻译：考虑击中集问题：对于给定全集 $\mathcal{X} = \left\{ 1, ... , n \right\}$ 和子集族 $\mathcal{S}_1, ... , \mathcal{S}_m$，目标是找到 $\mathcal{X}$ 的最小子集，使其与族中每个元素均有非空交集。我们研究该问题的概率形式，其中底层子集通过以概率 $p$ 独立选取全集中的每个元素构成。对于足够大的 $n$ 值，我们严格分析了 Lovász 著名贪婪算法（Lovász, 1975）在所选输入分布下的平均性能。此外，我们探讨了该问题线性规划与整数规划解之间的积分间隙。