Inverse matroid optimization under subset constraints

TL;DR

This paper extends inverse matroid optimization to include subset constraints, developing polynomial-time algorithms for six variants and introducing a refined min-max theorem under the -infinity norm.

Contribution

It introduces new variants of inverse matroid problems with subset constraints and provides efficient combinatorial algorithms for all, along with a novel min-max theorem.

Findings

Polynomial-time algorithms for six inverse matroid variants.

A refined min-max theorem for inverse matroids under -infinity norm.

Broader applicability of inverse optimization on matroids with structural constraints.

Abstract

In the Inverse Matroid problem, we are given a matroid, a fixed basis $B$, and an initial weight function, and the goal is to minimally modify the weights -- measured by some function -- so that $B$ becomes a maximum-weight basis. The problem arises naturally in settings where one wishes to explain or enforce a given solution by minimally perturbing the input. We extend this classical problem by replacing the fixed basis with a subset $S_0$ of the ground set and imposing various structural constraints on the set of maximum-weight bases relative to $S_0$. Specifically, we study six variants: (A) Inverse Matroid Exists, where $S_0$ must contain at least one maximum-weight basis; (B) Inverse Matroid All, where all bases contained in $S_0$ are maximum-weight; and (C) Inverse Matroid Only, where $S_0$ contains exactly the maximum-weight bases, along with their natural negated counterparts. For all variants, we develop combinatorial polynomial-time algorithms under the $\ell_\infty$-norm. A key ingredient is a refined min-max theorem for Inverse Matroid under the $\ell_\infty$-norm, which enables simpler and faster algorithms than previous approaches and may be of independent combinatorial interest. Our work significantly broadens the range of inverse optimization problems on matroids that can be solved efficiently, especially those that constrain the structure of optimal solutions through subset inclusion or exclusion.