An Optimal Randomized Algorithm for Finding the Saddlepoint

TL;DR

The paper tackles the problem of finding a (strict) saddlepoint in an $n\times n$ matrix, which corresponds to the game value in two-player zero-sum games. It introduces a randomization-based pivot-reduction that alternates horizontal and vertical pivots to prune a constant fraction of rows and columns, reducing the problem to a small $s\times s$ submatrix (with $s = n/\log_2 n$) that can be solved with existing deterministic methods. The authors prove a Las Vegas algorithm achieving $O(n)$ expected time with high probability to find a strict saddlepoint (or report nonexistence), and they establish a tight randomized lower bound showing that, for non-strict saddlepoints, no subquadratic randomized algorithm can exist. The work demonstrates near-optimal performance, offers efficient derandomization and parallelization avenues, and clarifies fundamental limits for randomized saddlepoint computation in the presence of duplicates.

Abstract

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, Schäffer, 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.

Figures (3)

• Figure 1: Horizontal pivot $p = A_{ij}$ (framed). Entries denoted $p^{-}$ are smaller than $p$, entries denoted $p^{+}$ are larger than $p$. Columns marked gray cannot contain a strict saddlepoint.
• Figure 2: Reducing the input matrix to size $s \times s$.
• Figure 3: Finding a horizontal pivot of the input matrix.

Theorems & Definitions (15)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Lemma 6: Multiplicative Chernoff dubhashi2009concentration
• Lemma 7
• Lemma 8
• Lemma 9
• Proposition 10