In the Arc Disjoint Cycle Packing problem, we are given a simple directed graph (digraph) G, a positive integer k, and the task is to decide whether there exist k arc disjoint cycles. The problem is known to be W[1]-hard on general digraphs parameterized by the standard parameter k. In this paper we show that the problem admits a polynomial kernel on α-bounded digraphs. That is, we give a polynomial-time algorithm, that given an instance (D,k) of Arc Disjoint Cycle Packing, outputs an equivalent instance \((D^{\prime },k^{\prime })\) of Arc Disjoint Cycle Packing, such that \(k^{\prime }\leq k\) and the size of \(D^{\prime }\) is upper-bounded by a polynomial function of k. For any integer α ≥ 1, the class of α-bounded digraphs, denoted by \({\mathcal D}_{\alpha }\), contains a digraph D such that the maximum size of an independent set in D is at most α. That is, in D, any set of α + 1 vertices has an arc with both end-points in the set. For α = 1, this corresponds to the well-studied class of tournaments. Our results generalize the recent result by Bessy et al. [MFCS, 2019] about Arc Disjoint Cycle Packing on tournaments.

在Arc Disjoint Cycle Packing问题中，给定一个简单的有向图（digraph）G，一个正整数k，任务是判断是否存在k个不相交的弧循环。已知该问题在由标准参数k参数化的一般有向图上是 W[1]-hard 问题。在本文中，我们表明该问题在α有界有向图上允许多项式核。也就是说，我们给出一个多项式时间算法，给定Arc Disjoint Cycle Packing的一个实例 ( D , k ) ，输出一个等效实例\((D^{\prime },k^{\prime })\ )圆弧不相交循环填料，使得\(k^{\prime }\leq k\)和\(D^{\prime }\)的大小上界为k的多项式函数。对于任何整数α ≥ 1，由\({\mathcal D}_{\alpha }\)表示的α有界有向图的类包含一个有向图D ，使得D中独立集的最大大小最多为α。也就是说，在D中，任何α + 1 个顶点的集合都有一条弧，其两个端点都在集合中。对于α = 1，这对应于经过充分研究的锦标赛类别。我们的结果概括了 Bessy 等人最近的结果。[MFCS, 2019] 关于锦标赛中的Arc Disjoint Cycle Packing。