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On the Approximability of Unsplittable Flow on a Path with Time Windows

Alexander Armbruster, Fabrizio Grandoni, Edin Husić, Antoine Tinguely, Andreas Wiese

TL;DR

This work investigates TwUFP, the Time-Windows Unsplittable Flow on a Path, proving APX-hardness even for spanUFP and providing strong approximation results under resource augmentation. The main algorithm delivers a quasi-polynomial-time $(2+\varepsilon)$-approximation for twUFP with $(1+O(\varepsilon))$-augmenting capacity, and a $(1+\varepsilon)$-approximation for the special spanUFP case under augmentation; the framework relies on a hierarchical interval decomposition, box packing, and harmonic-grouping to manage time windows and edge capacities. The paper also establishes APX-hardness for spanUFP via a reduction from 3DM, ruling out PTAS or QPTAS without augmentation. These results delineate the approximability landscape of twUFP, showing a clear separation from UFP when time windows are present and clarifying the role of resource augmentation in achieving near-optimal schedules. The techniques—boxes, artificial tasks, and left-right recursion with careful coupling—may inform related scheduling and packing problems with time windows and resource constraints.

Abstract

In the Time-Windows Unsplittable Flow on a Path problem (twUFP) we are given a resource whose available amount changes over a given time interval (modeled as the edge-capacities of a given path $G$) and a collection of tasks. Each task is characterized by a demand (of the considered resource), a profit, an integral processing time, and a time window. Our goal is to compute a maximum profit subset of tasks and schedule them non-preemptively within their respective time windows, such that the total demand of the tasks using each edge $e$ is at most the capacity of $e$. We prove that twUFP is $\mathsf{APX}$-hard which contrasts the setting of the problem without time windows, i.e., Unsplittable Flow on a Path (UFP), for which a PTAS was recently discovered [Grandoni, Mömke, Wiese, STOC 2022]. Then, we present a quasi-polynomial-time $2+\varepsilon$ approximation for twUFP under resource augmentation. Our approximation ratio improves to $1+\varepsilon$ if all tasks' time windows are identical. Our $\mathsf{APX}$-hardness holds also for this special case and, hence, rules out such a PTAS (and even a QPTAS, unless $\mathsf{NP}\subseteq\mathrm{DTIME}(n^{\mathrm{poly}(\log n)})$) without resource augmentation.

On the Approximability of Unsplittable Flow on a Path with Time Windows

TL;DR

This work investigates TwUFP, the Time-Windows Unsplittable Flow on a Path, proving APX-hardness even for spanUFP and providing strong approximation results under resource augmentation. The main algorithm delivers a quasi-polynomial-time -approximation for twUFP with -augmenting capacity, and a -approximation for the special spanUFP case under augmentation; the framework relies on a hierarchical interval decomposition, box packing, and harmonic-grouping to manage time windows and edge capacities. The paper also establishes APX-hardness for spanUFP via a reduction from 3DM, ruling out PTAS or QPTAS without augmentation. These results delineate the approximability landscape of twUFP, showing a clear separation from UFP when time windows are present and clarifying the role of resource augmentation in achieving near-optimal schedules. The techniques—boxes, artificial tasks, and left-right recursion with careful coupling—may inform related scheduling and packing problems with time windows and resource constraints.

Abstract

In the Time-Windows Unsplittable Flow on a Path problem (twUFP) we are given a resource whose available amount changes over a given time interval (modeled as the edge-capacities of a given path ) and a collection of tasks. Each task is characterized by a demand (of the considered resource), a profit, an integral processing time, and a time window. Our goal is to compute a maximum profit subset of tasks and schedule them non-preemptively within their respective time windows, such that the total demand of the tasks using each edge is at most the capacity of . We prove that twUFP is -hard which contrasts the setting of the problem without time windows, i.e., Unsplittable Flow on a Path (UFP), for which a PTAS was recently discovered [Grandoni, Mömke, Wiese, STOC 2022]. Then, we present a quasi-polynomial-time approximation for twUFP under resource augmentation. Our approximation ratio improves to if all tasks' time windows are identical. Our -hardness holds also for this special case and, hence, rules out such a PTAS (and even a QPTAS, unless ) without resource augmentation.

Paper Structure

This paper contains 13 sections, 21 theorems, 7 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Our Results and Techniques
  3. Related work
  4. A $(2+\varepsilon)$-approximation for twUFP
  5. Root subproblem
  6. Arbitrary subproblems
  7. Running time
  8. Approximation guarantee
  9. Proof of Theorem \ref{['thr:tw-apx']} and \ref{['thr:qptas']}
  10. Hardness of approximation of spanUFP
  11. Preliminaries
  12. The reduction from 3DM
  13. Proof of Lemma \ref{['lem:preprocessing']}

Key Result

Theorem 1

spanUFP (thus also twUFP) is $\mathsf{APX}$-hard and does not admit a (polynomial-time) $\frac{2755}{2754}$-approximation algorithm (unless $\mathsf{P=NP}$), even in the cardinality case and if demands and the number of edges is polynomially bounded in $n$.

Figures (4)

  • Figure 1: The capacity profile (red) of an instance used in our hardness result for spanUFP.
  • Figure 2: The interval $I$ and its subdivision together with a task $i$.
  • Figure 3: A box of height $h(b)$, demand $d(b)$ and a path $pb(b)$ with a set of five tasks fitting into the box.
  • Figure 4: The capacity profile (red) of one interval and the 8 tasks corresponding to one hyperedge $h_\ell=(x_i, y_j, z_k)$ scheduled in this interval.

Theorems & Definitions (39)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 3
  • Lemma 3
  • proof
  • Lemma 3
  • proof
  • Lemma 3
  • proof
  • ...and 29 more