Unsplittable Flow on a Short Path

TL;DR

This work studies UFP and Bag-UFP on short paths, parameterized by the path length $m$, and delivers parameterized approximation schemes with strong guarantees. The authors introduce a p-EPTAS for Bag-UFP and a substantially faster p-EPTAS for UFP, underpinned by a partition-matroid LP relaxation, heavy/light decompositions, and a novel use of representative sets to bound heavy-task choices. They also prove no p-FPTAS for either problem (via reductions from Partition and SSM), establishing qualitative tightness and highlighting a clear separation between Bag-UFP and UFP in parameterized complexity. The results leverage LP-based rounding, matroid theory, and color-coding-inspired hardness to achieve near-optimal parameterized approximations for short-path instances and point to directions for matroid-constrained UFP variants. Overall, the paper advances the understanding of parameterized approximation in flow problems and demonstrates a powerful combination of LP techniques and representative-set tools for combinatorial optimization on paths.

Abstract

In the Unsplittable Flow on a Path problem UFP, we are given a path graph with edge capacities and a collection of tasks. Each task is characterized by a demand, a profit, and a subpath. Our goal is to select a maximum profit subset of tasks such that the total demand of the selected tasks that use each edge $e$ is at most the capacity of $e$. Bag-UFP is the generalization of UFP where tasks are partitioned into bags, and we are allowed to select at most one task per bag. UFP admits a PTAS [Grandoni,M{ö}mke,Wiese'22] but not an EPTAS [Wiese'17]. Bag-UFP is APX-hard [Spieksma'99] and the current best approximation is $O(\log n/\log\log n)$ [Grandoni,Ingala,Uniyal'15], where $n$ is the number of tasks. In this paper, we study the mentioned two problems when parameterized by the number $m$ of edges in the graph, with the goal of designing faster parameterized approximation algorithms. We present a parameterized EPTAS for Bag-UFP, and a substantially faster parameterized EPTAS for UFP (which is an FPTAS for $m=O(1)$). We also show that a parameterized FPTAS for UFP (hence for BagUFP) does not exist, therefore our results are qualitatively tight.

Figures (2)

• Figure 1: An illustration of the graph $\mathcal{G}$ and the maximum matching $M$ (in red). Every edge $(i,B)$ in the graph indicates that bag $B$ belongs to $\textsf{fit}(i)$; that is, the representative from $B$ in the class of $i$ belongs to $R$ and the demand of this representative is at most the demand of $i$. Note that even though $i_1$ and $i_2$ are both connected to bag $B_2$, $i_1$ and $i_2$ may belong to different classes.
• Figure 2: An illustration of the construction. The path $v'_0,v_1,v'_1,\ldots, v_k,v'_k, v_{k+1}$ of the interleaving sequences of edges $e_0,f_1,e_1,f_2, \ldots, f_k,e_k$ is shown along with the subpaths of the tasks $\delta_1,\ldots, \delta_k$ (in dashed brown), the subpaths of the tasks $z^1_j,\ldots, z^k_j$ (in dashed blue, for some $j$), and the subpaths of the tasks $q^1_j,\ldots, q^k_j$ (in dashed red). Tasks with larger weights have thicker lines for their subpaths.

Theorems & Definitions (62)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Definition 2.1
• Lemma 2.2
• Lemma 2.3
• proof
• Lemma 2.4