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

2025-03-22

AI Summary

This paper studies the time-windowed Unsplittable Flow Problem on a path (twUFP): given a path graph with edge capacities varying dynamically over time, schedule a subset of non-preemptive tasks—each with a time window, demand, and profit—to maximize total profit, subject to the constraint that aggregate demand on each edge never exceeds its instantaneous capacity. We first establish that twUFP is APX-hard. Then, we present the first quasi-polynomial-time (2+ε)-approximation algorithm with (1+ε)-resource augmentation; when all tasks share identical time windows, the approximation ratio improves to (1+ε). Moreover, we rigorously rule out the existence of a PTAS or QPTAS without resource augmentation. These results settle the computational complexity landscape of twUFP, achieve the best-known approximation guarantees, and provide both fundamental limits and an efficient algorithmic framework for time-constrained resource allocation.

Technology Category

• Planning, Routing, and Scheduling: Scheduling under Uncertainty
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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, M""omke, 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 $mathsf{NP}subseteqmathrm{DTIME}(n^{mathrm{poly}(log n)})$) without resource augmentation.

Problem

• Maximize profit by scheduling tasks within time windows under capacity constraints
• Prove APX-hardness for Time-Windows Unsplittable Flow on a Path problem
• Develop quasi-polynomial-time approximation algorithms with resource augmentation

Innovation

• APX-hard proof for twUFP
• Quasi-polynomial-time approximation algorithm
• Resource augmentation improves ratio