Robust Extensible Bin Packing and Revisiting the Convex Knapsack Problem

2025-10-27

AI Summary

This paper studies the robust scalable bin packing problem under budgeted uncertainty, where item sizes reside in an uncertainty set defined by the intersection of box constraints and an ℓ₁-norm ball. Addressing the NP-hard separation subproblem, we first formulate it as a continuous convex knapsack problem and prove its strong NP-hardness; we then design a pseudo-polynomial dynamic programming (DP) algorithm and an FPTAS with guaranteed approximation ratio. To solve the overall robust optimization problem, we propose a master–slave framework for scenario generation, integrating robust optimization, SOS constraints, and mixed-integer programming in an iterative scheme. Experiments demonstrate that our DP algorithm significantly outperforms commercial MIP solvers in computational efficiency. In a real-world surgical scheduling case study, the proposed robust model substantially reduces both schedule delay risk and resource wastage compared to both current practice and the nominal optimal solution.

Technology Category

• Planning, Routing, and Scheduling: Scheduling under Uncertainty
• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

We study a robust extensible bin packing problem with budgeted uncertainty, under a budgeted uncertainty model where item sizes are defined to lie in the intersection of a box with a one-norm ball. We propose a scenario generation algorithm for this problem, which alternates between solving a master robust bin-packing problem with a finite uncertainty set and solving a separation problem. We first show that the separation is strongly NP-hard given solutions to the continuous relaxation of the master problem. Then, focusing on the separation problem for the integer master problem, we show that this problem becomes a special case of the continuous convex knapsack problem, which is known to be weakly NP-hard. Next, we prove that our special case when each of the functions is piecewise linear, having only two pieces, remains NP-hard. We develop a pseudo-polynomial dynamic program (DP) and a fully polynomial-time approximation scheme (FPTAS) for our special case whose running times match those of a binary knapsack FPTAS. Finally, our computational study shows that the DP can be significantly more efficient in practice compared with solving the problem with specially ordered set (SOS) constraints using advanced mixed-integer (MIP) solvers. Our experiments also demonstrate the application of our separation problem method to solving the robust extensible bin packing problem, including the evaluation of deferring the exact solution of the master problem, separating based on approximate master solutions in intermediate iterations. Finally, a case-study, based on real elective surgery data, demonstrates the potential advantage of our model compared with the actual schedule and optimal nominal schedules.

Problem

• Studying robust extensible bin packing with budgeted uncertainty constraints
• Developing algorithms for scenario generation and separation problems
• Solving NP-hard convex knapsack problems with dynamic programming approaches

Innovation

• Scenario generation algorithm with master and separation problems
• Dynamic program and FPTAS for piecewise linear knapsack problem
• Separation method applied to robust extensible bin packing