Minimum-Peak-Cost Flows Over Time

2025-07-13

AI Summary

This paper investigates peak-cost optimization in non-consumable resource transportation, where the objective is to minimize the maximum instantaneous resource cost over time—not the integral of total cost. We formalize this as the “minimum-peak-cost time-varying flow” problem and prove it is NP-hard. We show that temporal repeated flows—commonly used heuristics—fail to provide bounded approximation guarantees in general networks. We identify two polynomially solvable special cases: (i) unit-cost series-parallel networks and (ii) networks with sufficiently long time horizons; for each, we design exact constructive algorithms. We further establish that the integral-cost variant is strongly NP-hard, while fractional relaxations admit exact polynomial-time solutions in both special cases. Our work introduces a new optimization paradigm for resource scheduling that balances theoretical rigor with practical tractability.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• 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
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Systems and Infrastructure for Web, Mobile and WoT: Virtualization and resource management in Web systems and infrastructures

Abstract

When planning transportation whose operation requires non-consumable resources, the peak demand for allocated resources is often of higher interest than the duration of resource usage. For instance, it is more cost-effective to deliver parcels with a single truck over eight hours than to use two trucks for four hours, as long as the time suffices. To model such scenarios, we introduce the novel minimum peak cost flow over time problem, whose objective is to minimise the maximum cost at all points in time rather than minimising the integral of costs. We focus on minimising peak costs of temporally repeated flows. These are desirable for practical applications due to their simple structure. This yields the minimum-peak-cost Temporally Repeated flow problem (MPC-TRF). We show that the simple structure of temporally repeated flows comes with the drawback of arbitrarily bad approximation ratios compared to general flows over time. Furthermore, our complexity analysis shows the integral version of MPC-TRF is strongly NP-hard, even under strong restrictions. On the positive side, we identify two benign special cases: unit-cost series-parallel networks and networks with time horizon at least twice as long as the longest path in the network (with respect to the transit time). In both cases, we show that integral optimal flows if the desired flow value equals the maximum flow value and fractional optimal flows for arbitrary flow values can be found in polynomial time. For each of these cases, we provide an explicit algorithm that constructs an optimal solution.

Problem

• Minimizing peak resource costs in transportation planning
• Analyzing approximation ratios of temporally repeated flows
• Identifying polynomial-time solvable special cases for MPC-TRF

Innovation

• Introduces minimum peak cost flow over time
• Focuses on temporally repeated flows
• Provides polynomial-time algorithms for special cases