The Unsplittable Flow Cover problem (UFP-cover) models the well-studied general caching problem and various natural resource allocation settings. We are given a path with a demand on each edge and a set of tasks, each task being defined by a subpath and a size. The goal is to select a subset of the tasks of minimum cardinality such that on each edge e the total size of the selected tasks using e is at least the demand of e. There is a polynomial time 4-approximation for the problem (Bar-Noy et al. STOC 2001) and also a QPTAS (Höhn et al. ICALP 2018). In this paper we study fixed-parameter algorithms for the problem. We show that it is W[1]-hard but it becomes FPT if we can slighly violate the edge demands (resource augmentation) and also if there are at most k different task sizes. Then we present a parameterized approximation scheme (PAS), i.e., an algorithm with a running time of \(f(k)\cdot n^{O_{\epsilon }(1)}\) that outputs a solution with at most (1 + 𝜖)k tasks or asserts that there is no solution with at most k tasks. In this algorithm we use a new trick that intuitively allows us to pretend that we can select tasks from OPT multiple times. We show that the other two algorithms extend also to the weighted case of the problem, at the expense of losing a factor of 1 + 𝜖 in the cost of the selected tasks.

Unsplittable Flow Cover 问题 (UFP-cover) 模拟了经过充分研究的一般缓存问题和各种自然资源分配设置。我们给定了一条路径，每个边上都有一个需求和一组任务，每个任务都由一个子路径和一个大小定义。目标是选择最小基数任务的子集，使得在每条边e 上，使用e的所选任务的总大小至少是e的需求。该问题有多项式时间 4 近似（Bar-Noy 等人 STOC 2001）和 QPTAS（Höhn 等人 ICALP 2018）。在本文中，我们研究了该问题的固定参数算法。我们证明它是 W[1]-hard 但如果我们可以稍微违反边缘需求（资源增强）并且最多有k种不同的任务大小。然后我们提出了一个参数化近似方案（PAS），即一个运行时间为\(f(k)\cdot n^{O_{\epsilon }(1)}\) 的算法，它输出的解最多为 ( 1 + 𝜖 ) k 个任务或断言最多k 个任务没有解决方案。在这个算法中，我们使用了一个新技巧，直观地让我们假装我们可以多次从OPT 中选择任务。我们表明，其他两种算法也扩展到了问题的加权情况，代价是在所选任务的成本中损失了 1 + 𝜖。