Efficient Algorithms for Finding Exact Weight Bases in Matroids

Core Concepts

The authors present FPT algorithms for finding a basis of a matroid with weight exactly equal to a given target, where weights can be discrete values or multi-dimensional vectors. They resolve the parameterized complexity of this problem completely by providing new proximity and sensitivity bounds for matroid problems.

Abstract

The paper considers the problem of finding a basis of a matroid with weight exactly equal to a given target. The weights can be discrete values from {-Δ, ..., Δ} or more generally m-dimensional vectors of such discrete values.
The authors present the following key results:

Sensitivity Theorem for Matroids: If A and B are bases of a matroid M, then there exists a basis A' with the same weight as A and the symmetric difference |A' ⊕ B| is bounded by a function of Δ and m.

Proximity Theorem for Matroids: If A is a basis of M and x* is a vertex solution to the polytope {x ∈ PB(M), Wx = W(A)}, then there exists a basis A' that is close to x* in L1 norm, with the distance bounded by a function of Δ and m.

FPT Algorithms: Using the proximity bound, the authors derive FPT algorithms parameterized by Δ and m that can find a basis with weight exactly equal to a given target β, if one exists. The running time is (mΔ)O(Δ)m · nO(1) for general matroids, and can be improved to (mΔ)O(m^2) · nO(1) for linear matroids.

The authors also discuss connections to binary integer linear programming and provide examples of applications where their results can be used to obtain new FPT algorithms.

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Key Insights Distilled From

Sensitivity, Proximity and FPT Algorithms for Exact Matroid Problems

by Frie... at arxiv.org 04-08-2024

https://arxiv.org/pdf/2404.03747.pdf

Deeper Inquiries

What are some concrete applications of the authors' FPT algorithms for exact weight matroid basis problems

The authors' FPT algorithms for exact weight matroid basis problems have several concrete applications in various fields. One application is in Feedback Edge Set with Budget Vectors, where the algorithm can efficiently find a basis with a specific weight target. Another application is in Group-Constrained Matroid Base, where the algorithm can be used to generalize existing algorithms to arbitrary finite groups. Additionally, the algorithms can be applied to combinatorial n-fold integer programs, expanding the range of applications for these types of problems. Overall, the FPT algorithms provide a versatile tool for solving exact weight matroid basis problems efficiently in different scenarios.

How do the proximity and sensitivity bounds compare to those known for other related problems, such as multi-budgeted matroid problems or binary integer linear programming

The proximity and sensitivity bounds presented in the paper for exact weight matroid basis problems are significant compared to those known for other related problems. In comparison to multi-budgeted matroid problems, the bounds in this paper are tailored specifically for exact weight matroid basis problems, providing a more focused and precise analysis. Similarly, when compared to binary integer linear programming, the bounds in this paper offer a unique perspective on the proximity and sensitivity of solutions in the context of matroids with discrete weights. The bounds in this paper contribute to a deeper understanding of the complexity and structure of exact weight matroid problems, setting them apart from other related problems.

Can the optimization variant of the exact weight matroid basis problem also admit strong proximity bounds, similar to the feasibility version shown in the paper

While the feasibility version of the exact weight matroid basis problem exhibits strong proximity bounds, it is not immediately clear if the optimization variant of the problem would also admit similar bounds. The optimization variant involves measuring the distance between an optimal continuous solution and the closest optimal integer solution, which may have different characteristics compared to the feasibility version. Further research and analysis would be needed to determine if the optimization variant of the problem can also achieve strong proximity bounds similar to those demonstrated in the feasibility version presented in the paper.