Matroid Algorithms Under Size-Sensitive Independence Oracles

2026-04-30

AI Summary

This work addresses a fundamental limitation in classical matroid algorithms, which assume independence queries can be answered in constant time, disregarding the actual cost dependence on the size of the queried set. The paper introduces a size-sensitive query model where the cost of an independence query scales linearly with the cardinality of the queried set, and investigates algorithmic complexity for three core tasks under this model: computing a matroid basis, approximating the rank, and partitioning into independent sets. The authors establish the first unconditional query complexity lower bound in this setting, showing that Ω(n²) queries are necessary for general matroids. Moreover, for matroids whose largest circuit has size at most c, they design an explicit algorithm achieving an expected query cost of O(n^{2−1/c} log n), thereby circumventing the general lower bound and yielding nearly tight upper and lower bounds.

Abstract

The standard oracle model for matroid algorithms assumes that each independence query can be answered in constant time, regardless of the size of the queried set. While this abstraction has underpinned much of the theoretical progress in matroid optimization, it masks the true computational effort required by these algorithms. In particular, for natural and widely studied classes such as graphic matroids, even a single independence query can require work linear in the size of the set, making the constant-time assumption implausible. We address this gap by introducing a size-sensitive cost model where the cost of a query $Q$ scales with $|Q|$. Nearly linear-time oracle implementations exist for broad families of matroids, and this refined abstraction therefore captures the true cost of query evaluation while allowing for a more faithful comparison between general matroids and their natural special cases. Within this framework we study three fundamental algorithmic tasks: finding a basis of a matroid, approximating its rank, and approximating its partition size. We establish tight results, proving nearly matching upper and lower bounds that show the optimal query cost is (up to logarithmic factors) quadratic in the size of the matroid. On the algorithmic side, our upper bounds are realized by explicit procedures that construct the desired solution. On the complexity side, our lower bounds are unconditional and already hold even for weaker distinguishing formulations of the problems. Finally, for matroids with maximum circuit size at most $c$, we show that the quadratic barrier can be broken, providing an algorithm that calculates the maximum-weight basis with expected query cost $\mathcal{O}(n^{2-1/c} \log n)$.

Problem

• matroid
• independence oracle
• query complexity
• size-sensitive cost
• algorithmic lower bounds

Innovation

• size-sensitive oracle
• matroid algorithms
• query complexity
• quadratic barrier
• maximum-weight basis