Sensitivity, Proximity and FPT Algorithms for Exact Matroid Problems

TL;DR

This work resolves the parameterized complexity of exactly-weighted matroid bases by introducing novel proximity and sensitivity bounds that are independent of the ground-set size. By combining these bounds with matroid intersection and partition matroids, the authors obtain fixed-parameter tractable algorithms parameterized by the maximum weight magnitude $\Delta$ and the dimension $m$ of the weight vectors, for arbitrary matroids; stronger results hold for linear matroids with randomized running times. The framework extends to multidimensional weight constraints and yields efficient algorithms for a range of applications, including group-constrained bases, budgeted matroid problems, and fairness-aware matchings. The paper also clarifies limitations: proximity and sensitivity do not extend to matroid intersection in general, highlighting fundamental barriers in that setting. Overall, the results significantly advance exact matroid optimization under multiple constraints and provide versatile tools for both theory and applications.

Abstract

We consider the problem of finding a basis of a matroid with weight exactly equal to a given target. Here weights can be discrete values from $\{-Δ,\ldots,Δ\}$ or more generally $m$-dimensional vectors of such discrete values. We resolve the parameterized complexity completely, by presenting an FPT algorithm parameterized by $Δ$ and $m$ for arbitrary matroids. Prior to our work, no such algorithms were known even when weights are in $\{0,1\}$, or arbitrary $Δ$ and $m=1$. Our main technical contributions are new proximity and sensitivity bounds for matroid problems, independent of the number of elements. These bounds imply FPT algorithms via matroid intersection.

Figures (3)

• Figure 1: Schematic overview of proximity, sensitivity and their connection. The vertices of $P_B(M)$ (light gray) are the bases of $M$. The intersection (dark gray) of $P_B(M)$ with the affine subspace $\{x \in \mathbb R^E : Wx = {\hbox{\boldmath$\beta$}}\}$ may contain non-integral vertices. For such a vertex $x^*$ by standard rounding there always exists a close by basis $B$ with $W(B) \approx {\hbox{\boldmath$\beta$}}$. If there is a basis $A$ with $W(A) = {\hbox{\boldmath$\beta$}}$ and the distance to $B$ (equivalently, to $x^*$) is sufficiently large, then by sensitivity there is a closer basis $A'$ also with $W(A') = {\hbox{\boldmath$\beta$}}$. This implies proximity.
• Figure 2: Visualization of the proof of Lemma \ref{['lem:several-exchanges']}
• Figure 3: Example of high proximity and sensitivity in exact matroid intersection

Theorems & Definitions (39)

• Theorem 1: Sensitivity Theorem for Matroids
• Theorem 2: Proximity Theorem for Matroids
• Theorem 3
• Lemma 4: Downsizing
• proof
• Lemma 5
• proof
• proof : Proof of Theorem \ref{['thm:proximity']}
• Theorem 6
• proof : Proof of Theorem \ref{['thr:5']} for $m=1$