Problem
Research questions and friction points this paper is trying to address.
parameterized complexity
spanner problem
independent weights and lengths
FPT algorithms
Parameterized Complexity of Spanner Problems with Independent Weights and Lengths
🤖 AI Summary
This study presents the first investigation into the parameterized complexity of the multiplicative α-spanner problem for undirected graphs with independent weights and lengths. Methodologically, it proposes two algorithmic frameworks—exclusion and inclusion—and optimizes the exclusion approach by introducing a compactness parameter to improve results on basic instances. The primary contributions are twofold. First, it proves that the problem is W[2]-hard when parameterized solely by total weight, yet becomes fixed-parameter tractable (FPT) when combined with auxiliary parameters. Second, it establishes FPT algorithm bounds for arbitrary weights and lengths, significantly overcoming the limitations of existing methods restricted to unit weights.
📝 Abstract
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.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Parameterized Complexity
Spanner Problem
Independent Weights and Lengths
Fixed-Parameter Tractability
Tightness Parameter
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