[Paper Review] VPSolver 3: Multiple-choice Vector Packing Solver

Filipe Brandão|arXiv (Cornell University)|Feb 16, 2016

Optimization and Packing Problems7 references3 citations

TL;DR

This paper introduces a new graph construction and compression algorithm for the multiple-choice vector packing problem in VPSolver 3.0.0, enabling efficient generation of compressed arc-flow models that integrate all bin types and item incarnations in a single run. The method significantly reduces model size and computation time compared to prior approaches, while maintaining strong linear relaxation bounds.

ABSTRACT

VPSolver is a vector packing solver based on an arc-flow formulation with graph compression. In this paper, we present the algorithm introduced in VPSolver 3.0.0 for building compressed arc-flow models for the multiple-choice vector packing problem.

Motivation & Objective

• To develop a more efficient graph construction method for the multiple-choice vector packing problem (MVP) that integrates all bin types and item incarnations in a single model.
• To reduce the size and computational cost of arc-flow models for MVP by leveraging graph compression and dynamic programming.
• To improve upon existing arc-flow formulations by enabling a single-pass construction of compressed graphs that include all valid packing patterns across multiple bin types.
• To maintain strong linear relaxation bounds—previously shown to have a maximum gap of 1.0027—while enhancing scalability and performance.
• To provide a scalable, general-purpose solver for MVP that supports diverse applications through reductions to vector packing.

Proposed method

• The method uses a dynamic programming recursion to generate states based on space usage per dimension, item index, and usage count, with lifting to maximize state space utilization.
• It constructs a directed acyclic multi-graph where each path from source to target represents a valid packing pattern, with arcs labeled by item incarnations and weights.
• A final compression step relabels nodes using aggregated state vectors and removes self-loops, reducing graph size while preserving valid paths.
• A transitive reduction is applied to connect internal nodes to only non-dominated bin types, minimizing redundant arcs to target nodes.
• The algorithm unifies the construction of graphs for all bin types in a single run, eliminating the need for separate graphs per bin type as in prior work.
• It integrates loss arcs (0,0) to model unused capacity and enables feedback loops from target to source for flow formulation.

Experimental results

Research questions

• RQ1 Can a single, unified arc-flow graph be constructed for all bin types and item incarnations in the multiple-choice vector packing problem?
• RQ2 How can graph compression be applied effectively to reduce model size while preserving the equivalence to the Gilmore-Gomory formulation?
• RQ3 What is the impact of transitive reduction on the number of arcs required to connect internal nodes to target nodes across multiple bin types?
• RQ4 How does the new algorithm compare in performance and model size to the prior method of constructing separate graphs per bin type?
• RQ5 To what extent does the new method maintain the strong linear relaxation properties observed in previous arc-flow formulations?

Key findings

• The new algorithm produces the same compressed graph as the original method when there is only one bin type and single incarnations per item.
• For multiple bin types, the algorithm constructs a single graph containing all valid packing patterns across all bin types in one execution, reducing complexity.
• The method typically produces smaller graphs than the prior approach of separate graphs per bin type, improving space efficiency.
• The algorithm is significantly faster than the previous method, especially for instances with many bin types.
• The arc-flow formulation remains equivalent to the Gilmore-Gomory model, preserving strong linear relaxation with a maximum absolute gap of 1.0027.
• The use of transitive reduction reduces the number of arcs needed to connect internal nodes to targets, particularly in variable bin size problems, where only one arc per node may be needed.