具有独立权重和长度的Spanner问题的参数化复杂性
Parameterized Complexity of Spanner Problems with Independent Weights and Lengths
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- Institute of Computer Science, Osnabrück University（奥斯纳布吕克大学计算机科学研究所）
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中文总结 AI 辅助
本文首次研究无向图上带独立权重和长度的乘法α-spanner问题的参数化复杂性，证明若干参数化不允许FPT算法，但提出排除和包含两种方法，分别以总移除权重和总权重为参数，并辅以紧致性等次要参数实现FPT算法。
中文摘要 AI 辅助
本文首次考虑了无向图上具有独立权重和长度的乘法α-spanner问题的参数化复杂性。所有先前的FPT结果（除一个关于有向无环图的结果外）都假设基本实例（即单位权重和长度），并以伸缩因子α和（实践中通常非常数的）移除边数为参数。我们表明若干参数化不允许FPT算法。然而，我们的排除方法将现有针对基本实例的算法推广到任意权重和长度。它以总移除权重和一个新的紧致性参数为参数。后者比α更精确，并使我们能够改进基本实例的最佳已知结果。我们的第二种算法称为包含方法，使用spanner总权重的自然参数化。我们证明仅此参数留下一个W[2]-难问题，但也表明当辅以次要参数时存在FPT算法。
英文摘要
In this paper, the parameterized complexity of the multiplicative $α$-spanner problem with independent weights and lengths on undirected graphs is considered for the first time. All prior FPT results (except one on DAGs) assume basic instances (i.e., with unit weights and lengths) and are parameterized in the stretch factor $α$ and the (in practice typically non-constant) number of removed edges. We show that several parameterizations do not allow FPT algorithms. However, our exclusion approach generalizes an existing algorithm for basic instances to arbitrary weights and lengths. It is parameterized by the total removed weight and a new tightness parameter. The latter is more precise than $α$ and allows us to also improve the best known result for basic instances. Our second algorithm, called inclusion approach, uses the natural parameterization in the spanner's total weight. We prove that this sole parameter leaves a W[2]-hard problem, but also show FPT algorithms exist when augmented with secondary parameters.
