线性参数最小平均环问题的一种算法

中文摘要 AI 辅助

加权有向图上的最小平均环问题（MCM）是寻找所有环中环平均（即成本与长度之比）最小值的问題。尽管该问题在离散事件系统中有着广泛的应用，但与网络优化中的其他参数问题不同，MCM的参数化对应形式在文献中受到的关注相对较少。在本文中，我们考虑线性参数化MCM，其中每条边$e$的成本为$a(e)-b(e)t$，参数为$t$。我们提出了一种算法，在$O((m+n\log n)n^2W)$时间内求解线性参数化MCM，其中$n$是顶点数，$m$是边数，$W$是系数$b(e) \in \mathbb{Z}$的最大绝对值。所提方法的核心技术是参数化最短路径问题的算法。MCM与热带半环中的谱理论密切相关，其中“$\min$”运算被视为加法，“$+$”被视为乘法。通过利用它们之间的联系，我们提供了一种计算热带参数矩阵的特征值和特征向量的方法。

英文摘要

The minimum cycle mean problem (MCM) on weighted digraphs is the problem of finding the minimum value of the cycle mean, that is, the ratio of the cost to the length, over all cycles. Despite its wide range of applications to discrete event systems, the parametric counterpart of the MCM has received relatively little attention in the literature, unlike other parametric problems in network optimization. In this paper, we consider the linear parametric MCM, where all edges $e$ have cost $a(e)-b(e)t$ with parameter $t$. We propose an algorithm to solve the linear parametric MCM in $O((m+n\log n)n^2W)$ time, where $n$ is the number of vertices, $m$ is the number of edges, and $W$ is the maximum absolute value of the coefficients $b(e) \in \mathbb{Z}$. The central technique of the proposed method is the algorithm for the parametric shortest path problem. The MCM is closely related to spectral theory in the tropical semiring, where the ``$\min$'' operation is regarded as addition and ``$+$'' as multiplication. By exploiting the connection between them, we provide a method to compute the eigenvalues and eigenvectors of tropical parametric matrices.

发表机构

• Kyoto Prefectural University（京都府立大学）

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