We study sparsity in the max-plus algebraic setting. We seek both exact and approximate solutions of the max-plus linear equation with minimum cardinality of support. In the former case, the sparsest solution problem is shown to be equivalent to the minimum set cover problem and, thus, NP-complete. In the latter one, the approximation is quantified by the ℓ1 residual error norm, which is shown to have supermodular properties under some convex constraints, called lateness constraints. Thus, greedy approximation algorithms of polynomial complexity can be employed for both problems with guaranteed bounds of approximation. We also study the sparse recovery problem and present conditions, under which, the sparsest exact solution solves it. Through multi-machine interactive processes, we describe how the present framework could be applied to two practical discrete event systems problems: resource optimization and structure-seeking system identification. We also show how sparsity is related to the pruning problem. Finally, we present a numerical example of the structure-seeking system identification problem and we study the performance of the greedy algorithm via simulations.

中文翻译：

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最大加代数和系统中的稀疏性

我们在最大加代数设置中研究稀疏性。我们寻求具有最小支持基数的最大加线性方程的精确和近似解。在前一种情况下，最稀疏解问题被证明等价于最小集合覆盖问题，因此是 NP 完全问题。在后一个中，逼近由 ℓ1 残差范数量化，在一些凸约束下显示出具有超模特性，称为延迟约束。因此，多项式复杂度的贪婪逼近算法可以用于保证逼近界限的两个问题。我们还研究了稀疏恢复问题和当前条件，在这些条件下，最稀疏精确解可以解决它。通过多机交互流程，我们描述了本框架如何应用于两个实际的离散事件系统问题：资源优化和结构搜索系统识别。我们还展示了稀疏性与修剪问题的关系。最后，我们给出了结构搜索系统识别问题的数值例子，并通过仿真研究了贪心算法的性能。