Hierarchical Rectangle Packing Solved by Multi-Level Recursive Logic-based Benders Decomposition

2025-12-23

AI Summary

This paper addresses the two-dimensional hierarchical rectangular packing problem, where the container size is unspecified and items may be either basic rectangles or nested sub-blocks—whose dimensions are determined by lower-level packing optimizations—arising in applications such as VLSI floorplanning, facility layout, and logistics. To overcome the low accuracy and poor scalability of conventional bottom-up approaches, we propose a multi-level logic-based Benders decomposition method that dynamically refines sub-block dimensional constraints without requiring manual enumeration of aspect-ratio candidates. We introduce the first tight integration of recursive structural modeling with Benders decomposition to enable end-to-end joint optimization. On synthetic instances with up to seven hierarchy levels and 80 items per level, our method significantly improves solution quality and scalability over monolithic MILP formulations and bottom-up baselines, while ensuring stable convergence within practical time limits.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• User Modeling, Personalization and Recommendation: Personalized content ranking and presentation

Abstract

We study the two-dimensional hierarchical rectangle packing problem, motivated by applications in analog integrated circuit layout, facility layout, and logistics. Unlike classical strip or bin packing, the dimensions of the container are not fixed, and the packing is inherently hierarchical: each item is either a rectangle or a block occurrence, whose dimensions are a solution of another packing problem. This recursive structure reflects real-world scenarios in which components, boxes, or modules must be packed within higher-level containers. We formally define the problem and propose exact formulations in Mixed-Integer Linear Programming and Constraint Programming. Given the computational difficulty of solving complex packing instances directly, we propose decomposition heuristics. First, we implement an existing Bottom-Up baseline method that solves subblocks before combining them at higher levels. Building upon this, we introduce a novel multilevel Logic-based Benders Decomposition method. This heuristic method dynamically refines block dimension constraints, eliminating the need for manual selection of candidate widths or aspect ratios. Experiments on synthetic instances with up to seven hierarchy levels, 80 items per block, and limited computation time show that the proposed decomposition significantly outperforms both monolithic formulations and the Bottom-Up method in terms of solution quality and scalability.

Problem

• Solves hierarchical rectangle packing with recursive container dimensions
• Addresses computational difficulty in complex packing via decomposition heuristics
• Improves solution quality and scalability over existing methods

Innovation

• Multi-level Logic-based Benders Decomposition heuristic
• Dynamic refinement of block dimension constraints
• Outperforms monolithic and Bottom-Up methods