Derandomizing Karger's Contraction Algorithm for Matroids

2026-08-17

AI Summary

This study addresses the reliance on randomization and prior parameter knowledge in computing minimum-weight cocircuits of matroids by proposing a deterministic contraction algorithm. By integrating the derandomization of Karger’s algorithm with fixed-parameter tractability techniques, the proposed method efficiently solves the problem without requiring prior knowledge of critical parameters, provided the cocircuit density ratio is bounded. This work achieves deterministic computation with matching exponential complexity, thereby resolving related open problems. Furthermore, the approach is successfully extended to enumerating near-minimum cocircuits and computing Pareto frontiers. These contributions significantly enhance both the determinism and applicability of algorithms for matroid cocircuit optimization, offering a robust alternative to existing randomized methods while maintaining theoretical efficiency guarantees.

Abstract

Karger's randomized contraction algorithm finds a minimum-weight cocircuit of a matroid whenever the cogirth-density ratio is bounded. We prove that the same hypothesis yields a deterministic algorithm with the same exponent. If every contraction minor of rank at least $r_0$ of a matroid $M$ has cogirth-density ratio at most $c$, then a minimum-weight cocircuit of $M$ is computable deterministically in $m^{O(r_0)} n^{O(c)}$ time when the contraction minors of bounded rank have at most $m$ parallel classes, by an algorithm that knows neither $r_0$ nor $c$. As a consequence, we give a deterministic algorithm computing the cogirth of rank-$p$ perturbed graphic matroids in $2^{O(p^2)} n^{O(1)}$ time, fixed-parameter tractable in $p$, settling the cogirth side of a question of Geelen and Kapadia (2018). The extensions of the contraction method carry over deterministically: enumerating all near-minimum 1-cocycles, computing a minimum-weight $k$-cocycle, and computing the Pareto frontier under several positive criteria.

Problem

• Derandomization
• Karger's Contraction Algorithm
• Matroids
• Minimum-weight cocircuit
• Cogirth

Innovation

• Derandomization
• Matroid Contraction Algorithm
• Cogirth
• Fixed-Parameter Tractability
• Perturbed Graphic Matroids