Branch and Price for Submodular Bin Packing

TL;DR

This paper introduces a branch-and-price algorithm with tailored pricing strategies to efficiently solve the complex Submodular Bin Packing problem, demonstrating promising computational performance on benchmark instances.

Contribution

It develops a novel branch-and-price framework with specialized submodular knapsack pricing for SMBP, advancing solution methods for this challenging problem.

Findings

The proposed algorithm efficiently solves SMBP instances.

Hybrid pricing strategies accelerate column generation.

Computational results validate the effectiveness of the approach.

Abstract

The Submodular Bin Packing (SMBP) problem asks for packing unsplittable items into a minimal number of bins for which the capacity utilization function is submodular. SMBP is equivalent to chance-constrained and robust bin packing problems under various conditions. SMBP is a hard binary nonlinear programming optimization problem. In this paper, we propose a branch-and-price algorithm to solve this problem. The resulting price subproblems are submodular knapsack problems, and we propose a tailored exact branch-and-cut algorithm based on a piece-wise linear relaxation to solve them. To speed up column generation, we develop a hybrid pricing strategy to replace the exact pricing algorithm with a fast pricing heuristic. We test our algorithms on instances generated as suggested in the literature. The computational results show the efficiency of our branch-and-price algorithm and the proposed pricing techniques.