The paper studies a family of matroid optimization problems with a linear constraint (MOL). These problems seek to optimize (maximize or minimize) a linear objective function subject to (i) a matroid independent set or basis constraint, and (ii) an additional linear constraint.
Key highlights:
The authors introduce the Π-matroid family, which carefully exploits the interaction between a weight function and the matroid constraint to hide a specific property Π within the independent sets. This allows them to show the unconditional hardness of the Exact Matroid Basis (EMB) problem.
Using the Π-matroid construction, the authors prove that none of the non-trivial MOL problems, including well-studied problems like Budgeted Matroid Independent Set (BM), Budgeted Matroid Intersection (BMI), and Constrained Minimum Basis of a Matroid (CMB), admit a randomized Fully PTAS. This resolves the complexity status of these problems.
The authors complement their oracle model lower bounds by showing that similar hardness results hold in the standard computational model, assuming P ≠ NP, even when the matroid is encoded as part of the input.
The authors' results distinguish MOL problems with arbitrary matroids from special cases with simpler matroid constraints, for which Fully PTAS are known. This promotes future research to design (or rule out) Fully PTAS for MOL problems on restricted matroid classes.
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by Ilan Doron-A... um arxiv.org 04-23-2024
https://arxiv.org/pdf/2307.07773.pdfTiefere Fragen