Abstract

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.

Highlights

  • In the Unsplittable Flow Cover problem (UFP-cover) we are given a path G = (V, E) where each edge e has a demand ue ∈ N, and a set of tasks T where each task i ∈ T has a start vertex si ∈ V and an end vertex ti ∈ V, defining a path P (i), and a size pi ∈ N

  • It is the natural covering version of the well-studied Unsplittable Flow on a Path problem (UFP), see e.g., [22, 21, 9] and references therein. It is a generalization of the knapsack cover problem [11] and it can model general caching in the fault model where we have a cache of fixed size and receive requests for non-uniform size pages, the goal being to minimize the total number of cache misses

  • We present an fixed parameter tractability (FPT)-2-approximation algorithm without resource augmentation, i.e., an algorithm that runs in time f (k)nO(1) and finds a solution of size at most 2k or asserts that there is no solution of size at most k

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Summary

Introduction

The goal is to select a subset of the tasks T ⊆ T of minimum cardinality |T | that covers the demand of each edge, i.e., such that i∈T ∩Te pi ≥ ue for each edge e where Te denotes the set of tasks i ∈ T for which e lies on P (i). It is the natural covering version of the well-studied Unsplittable Flow on a Path problem (UFP), see e.g., [22, 21, 9] and references therein. We show that by allowing such a running time we can compute solutions that are almost optimal

Our contribution
Other related work
Few different task sizes
Resource augmentation
Arbitrary demands
FPT-2-approximation algorithm
Parameterized approximation scheme
Medium intervals
Heavy vertices
Dense intervals
Sparse intervals
Conclusion and open questions
A Reduction from Generalized Caching in the fault model
Full Text
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