General Science 23 JAN, 2025

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

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

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.

💡 Research Summary

The paper addresses a novel two‑dimensional hierarchical rectangle packing (HRP) problem that arises in analog IC layout, facility planning, and logistics. Unlike classic strip or bin packing, the container’s dimensions are not predetermined; each item can be a simple rectangle or a “block occurrence” whose width and height are the solution of a lower‑level packing sub‑problem. This recursive structure creates a multi‑level nesting of packing problems, reflecting real‑world scenarios where components, boxes, or modules must be placed inside higher‑level containers.

The authors first formalize HRP. A hierarchy of L levels is defined, where level ℓ contains a block Bℓ and a set of items Iℓ. Items are either elementary rectangles or references to a block at level ℓ + 1. The objective is to minimize the area of the top‑level block B1 while ensuring non‑overlap and boundary constraints at every level.

Two exact mathematical models are presented. The mixed‑integer linear programming (MILP) formulation introduces continuous coordinates (x_i, y_i) and binary selection variables to linearize non‑overlap via a big‑M approach; block dimensions (W_ℓ, H_ℓ) are linked to the optimal solutions of their sub‑blocks. The constraint‑programming (CP) formulation uses interval variables and global non‑overlap constraints, which naturally handle the piecewise‑linear nature of the problem. Both models become intractable as the number of items (up to 80 per block) and hierarchy depth (up to seven levels) increase; commercial solvers either run out of time or exceed memory limits.

A baseline Bottom‑Up (BU) heuristic is implemented. BU solves the lowest‑level sub‑blocks first, fixes their dimensions, and then proceeds upward. Although simple and fast for small instances, BU suffers from a lack of global coordination: dimensions chosen at lower levels may be sub‑optimal for the overall objective, leading to poor solution quality on deeper hierarchies.

The core contribution is a multi‑level Logic‑based Benders Decomposition (LB‑BD). Traditional Benders decomposition separates a problem into a master and sub‑problems, adding cuts derived from sub‑problem dual information. LB‑BD extends this idea by embedding logical constraints that capture the discrete nature of block dimensions (e.g., only certain width‑height combinations are feasible). The master problem contains variables for all block dimensions and the top‑level area; initially it imposes almost no restrictions on these dimensions. For each block, a sub‑problem is solved with the master‑provided dimensions fixed, using the exact MILP or CP model. If the sub‑problem is feasible, an optimality cut (based on the sub‑problem’s dual values) is added to tighten the master’s objective. If infeasible, a feasibility cut eliminates the current dimension combination. This iterative process continues until no new cuts are generated, at which point the master’s dimensions are optimal and each sub‑problem yields a feasible packing for its block.

Key algorithmic features of LB‑BD include:

  • No need to pre‑select candidate widths or aspect ratios; the master discovers them automatically.
  • Logical cuts enforce discrete dimension choices, preventing the master from proposing impossible sizes.
  • Sub‑problems are independent and can be solved in parallel, offering scalability on multi‑core hardware.

Computational experiments were conducted on 120 synthetic instances with up to seven hierarchy levels and 80 items per block. The methods compared were (a) monolithic MILP, (b) monolithic CP, (c) Bottom‑Up, and (d) the proposed LB‑BD. Results show that LB‑BD consistently outperforms the others: it reduces the top‑level area by an average of 12–18 % relative to BU, and by 30 % or more on the deepest hierarchies. In a one‑hour time limit, the monolithic MILP/CP models failed to solve most instances, often hitting memory limits, whereas LB‑BD converged on every test case. Parallel execution of sub‑problems on eight cores yielded a speed‑up of roughly 2.5×, demonstrating the method’s amenability to modern hardware.

The paper also discusses limitations. Sub‑problems remain NP‑hard, and the strength of Benders cuts depends on the choice of big‑M constants; overly loose constants can generate weak cuts and slow convergence. The current implementation follows a sequential cut‑generation loop, leaving room for advanced cut‑management strategies (e.g., cut pooling, dynamic removal) and for integrating meta‑heuristics within sub‑problems to further accelerate convergence.

Future work suggested includes: (1) adaptive big‑M tuning and cut selection heuristics, (2) exploiting full parallelism across hierarchy levels, (3) applying the framework to real‑world IC layout and facility planning datasets, and (4) extending the model to incorporate additional practical constraints such as alignment, routing, or thermal considerations.

In conclusion, the authors introduce a rigorous definition of hierarchical rectangle packing, provide exact MILP and CP formulations, and develop a novel Logic‑based Benders Decomposition that automatically refines block dimension constraints without manual candidate enumeration. Empirical evidence confirms that LB‑BD delivers superior solution quality and scalability compared to both monolithic approaches and the traditional Bottom‑Up heuristic, offering a powerful tool for complex, multi‑level packing problems encountered in engineering and logistics.