In this paper, we introduce the Maximum Matrix Contraction problem, where we aim to contract as much as possible a binary matrix in order to maximize its density. We study the complexity and the polynomial approximability of the problem. Especially, we prove this problem to be NP-Complete and that every algorithm solving this problem is at most a $2\sqrt{n}$-approximation algorithm where n is the number of ones in the matrix. We then focus on efficient algorithms to solve the problem: an integer linear program and three heuristics.

翻译：本文提出最大矩阵收缩问题，旨在通过尽可能压缩二元矩阵以最大化其密度。我们研究了该问题的计算复杂性与多项式近似性。特别地，我们证明该问题是NP完全的，且所有求解该问题的算法至多为$2\sqrt{n}$近似算法，其中n为矩阵中元素1的个数。随后我们聚焦于求解该问题的高效算法：一个整数线性规划与三种启发式方法。