A \emph{saddlepoint} of an $n \times n$ matrix is an entry that is the maximum of its row and the minimum of its column. Saddlepoints give the \emph{value} of a two-player zero-sum game, corresponding to its pure-strategy Nash equilibria; efficiently finding a saddlepoint is thus a natural and fundamental algorithmic task. For finding a \emph{strict saddlepoint} (an entry that is the strict maximum of its row and the strict minimum of its column) we recently gave an $O({n\log^*{n}})$-time algorithm, improving the $O({n\log{n}})$ bounds from 1991 of Bienstock, Chung, Fredman, Sch\"affer, Shor, Suri and of Byrne and Vaserstein. In this paper we present an optimal $O({n})$-time algorithm for finding a strict saddlepoint based on random sampling. Our algorithm, like earlier approaches, accesses matrix entries only via unit-cost binary comparisons. For finding a (non-strict) saddlepoint, we extend an existing lower bound to randomized algorithms, showing that the trivial $O(n^2)$ runtime cannot be improved even with the use of randomness.

翻译：一个$n \times n$矩阵的\emph{马鞍点}是指同时为其所在行的最大值和所在列的最小值的元素。马鞍点给出了两人零和博弈的\emph{值}，对应于其纯策略纳什均衡；因此，高效地寻找马鞍点是一项自然而基本的算法任务。对于寻找\emph{严格马鞍点}（即严格为其所在行的最大值和严格为其所在列的最小值的元素），我们最近给出了一种时间复杂度为$O({n\log^*{n}})$的算法，改进了1991年Bienstock、Chung、Fredman、Schäffer、Shor、Suri以及Byrne和Vaserstein提出的$O({n\log{n}})$上界。在本文中，我们提出了一种基于随机抽样的最优$O({n})$时间复杂度算法来寻找严格马鞍点。与早期方法类似，我们的算法仅通过单位成本的二元比较来访问矩阵元素。对于寻找（非严格）马鞍点，我们将现有的下界推广到随机算法，表明即使使用随机化，平凡的$O(n^2)$运行时间也无法被改进。