Inverse matroid optimization under subset constraints

2025-07-01

AI Summary

This work generalizes the classical inverse matroid problem: instead of requiring a single fixed basis to become a maximum-weight basis, it imposes structural constraints on a prescribed subset $S_0$ of the ground set—specifying that $S_0$ must contain, be contained in, equal, or be disjoint from the set of all maximum-weight bases (plus their logical negations), yielding six new variants. Leveraging matroid theory and combinatorial optimization, we develop a unified algorithmic framework and establish a refined min–max theorem under the $ell_infty$ norm. For all six variants, we design polynomial-time combinatorial algorithms. Unlike prior approaches—which operate at the level of individual bases—our formulation introduces *subset-level structural constraints* on basis sets for the first time, substantially broadening the class of inverse matroid problems solvable in polynomial time. This advancement enhances expressiveness and computational efficiency in applications such as interpretable modeling and solution-structure enforcement.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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.

Problem

• Minimally modify weights to make a subset a maximum-weight basis.
• Study six variants of inverse matroid optimization with constraints.
• Develop polynomial-time algorithms for all variants under ℓ∞-norm.

Innovation

• Extends inverse matroid with subset constraints
• Develops combinatorial polynomial-time algorithms
• Refines min-max theorem for efficient solutions