A parallel branch-and-bound-and-prune algorithm for irregular strip packing with discrete rotations

TL;DR

This work addresses irregular strip packing with discrete rotations by proposing an ad-hoc parallel branch-and-bound-and-prune algorithm (DB-PB) that solves the Dotted-Board (DB) model without preprocessing, along with two 0-1 reformulations (Binary Dotted-Board and Binary DB-CC) and a new lower-bound procedure (DB-PB-LB). It also provides an exact replication of the state-of-the-art DB-CC model for fair benchmarking and a comprehensive reproducibility dataset. Empirical results on 81 standard instances show that DB-PB markedly improves solving speed and optimality coverage, solving 51 instances to optimality within 10 minutes and opening 17 previously unsolved cases, thereby establishing a new state-of-the-art among exact methods for discrete nesting. Nonetheless, preprocessing-based DB-CC variants still yield tighter upper bounds on many unsolved instances, illustrating complementary strengths and motivating future work to integrate primal heuristics and feasibility pumps into the DB-PB framework. The paper also emphasizes reproducibility by providing datasets and protocols for exact replication across platforms, enabling robust comparison and validation of discrete nesting methods.

Abstract

The irregular strip-packing problem consists of the computation of a non-overlapping placement of a set of polygons onto a rectangular strip of fixed width and the minimal length possible. Recent performance gains of the Mixed-Integer Linear Programming (MILP) solvers have encouraged the proposal of exact optimization models for nesting. The Dotted-Board (DB) MILP model solves the discrete version of the nesting problem by constraining the positions of the polygons to be on a grid of fixed points. However, its number of non-overlapping constraints grows exponentially with the number of dots and types of polygons, which encouraged the proposal of a reformulation called the DB Clique Covering (DB-CC) that sets the current state-of-the-art by significantly reducing the constraints required. However, DB-CC requires a significant preprocessing time to compute edge and vertex clique coverings. Moreover, current knowledge of the stable set polytope suggests that achieving a tighter formulation is unlikely. Thus, our hypothesis is that an ad-hoc exact algorithm requiring no preprocessing might be a better option to solve the DB model than the costly Branch-and-Cut approach. This work proposes an exact branch-and-bound-and-prune algorithm to solve the DB model from the conflict inverse graph based on ad-hoc data structures, bounding, and forward-checking for pruning the search space. We introduce two 0-1 ILP DB reformulations with discrete rotations and a new lower-bound algorithm as by-products. Our experiments show that DB-PB significantly reduces the resolution time compared to our replication of the DB-CC model. Seventeen open instances are solved up to optimality.

Figures (5)

• Figure 1: Two solutions obtained by DB-PB for the same problem instance without rotations and with them.
• Figure 2: Performance profiles Dolan2002-wd showing the Cumulative Distribution Function (CDF) for the performance ratio $r_{\phi,m}$ comparing the performance of all exact DB methods in the evaluation of all problem instances.
• Figure 3: Seventeen open problem instances solved up to optimality within one hour for the first time in the literature. Rodrigues2017-zl reported a wrong optimal solution for J2-10-35-3 with $L=21$ instead of the optimal solution with $L^*=20$ shown above, which we attribute to a minor tolerance error in their experiments. All the new optimal solutions shown above are found by our DB-PB algorithm, although the DB-CC model Rodrigues_de_Souza_Queiroz2020-tp also finds four of them. However, this latter improvement can be attributed only to the advances in Gurobi 11.0.3 compared to CPLEX 12.6 Rodrigues2017-zl.
• Figure 4: Ten open problem instances whose upper bounds are improved in this work. Eight of these improvements are obtained by the DB-CC Rodrigues2017-zl model solved with Gurobi 11.0.3, whilst two others are obtained by our 0-1 DB-CC model solved with Google CPSat 9.11.4210. Most of the solutions above are likely to be the optimal despite it could not be proven.
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