针对拟阵的Karger收缩算法的去随机化
Derandomizing Karger's Contraction Algorithm for Matroids
AI总结：
该研究针对拟阵的Karger随机收缩算法进行去随机化，得到同指数的确定性算法，解决了Geelen与Kapadia2018年关于秩p扰动图拟阵余圈的问题，还实现了收缩方法的多项确定性扩展应用。
AI中文摘要：
Karger的随机收缩算法在拟阵的余圈密度比有界时，可找到拟阵的最小权重余圈。我们证明，相同假设下可得到具有相同指数的确定性算法。若拟阵M的每个秩至少为r₀的收缩子式的余圈密度比至多为c，且该拟阵的有界秩收缩子式最多有m个平行类，则存在一种无需知晓r₀和c的算法，可在m^{O(r₀)}n^{O(c)}时间内确定性计算M的最小权重余圈。作为推论，我们给出了一种确定性算法，可在2^{O(p²)}n^{O(1)}时间内计算秩p扰动图拟阵的余圈，该算法关于p是固定参数可处理的，解决了Geelen和Kapadia（2018）提出的问题中关于余圈的部分。收缩方法的扩展也可确定性实现：枚举所有近最小1-余圈、计算最小权重k-余圈，以及在多个正准则下计算帕累托前沿。
英文摘要：
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.
