Solving Hard Instances from Knapsack and Bounded Knapsack Problems: A new state-of-the-art solver

TL;DR

This paper introduces RECORD, a novel solver for the Knapsack and Bounded Knapsack Problems, which significantly outperforms existing solvers by integrating advanced dynamic programming and bounding techniques.

Contribution

RECORD enhances prior methods with new strategies like multiplicity reduction, on-the-fly item aggregation, and a divisibility bound, setting a new performance benchmark.

Findings

RECORD outperforms COMBO and BOUKNAP on challenging benchmarks.

It achieves several orders of magnitude speedup on difficult instances.

RECORD maintains near-linear-time behavior on most instances.

Abstract

The Knapsack Problem (KP) and its generalization, the Bounded Knapsack Problem (BKP), are classical NP-hard problems with numerous practical applications, and despite being introduced over 25 years ago, the solvers COMBO and BOUKNAP remain the state of the art due to their highly optimized implementations and sophisticated bounding techniques. In this work, we present RECORD (Refined Core-based Dynamic Programming), a new solver for both problems that builds upon key components of COMBO, including core- and state-based dynamic programming, weak upper bounds, and surrogate relaxation with cardinality constraints, while introducing novel strategies to overcome its limitations. In particular, we propose multiplicity reduction to limit the number of distinct item types, combined with on-the-fly item aggregation, refined fixing-by-dominance techniques, and a new divisibility bound that strengthens item fixing and symmetry breaking. These enhancements allow RECORD to preserve COMBO's near-linear-time behavior on most instances while achieving substantial speedups on more challenging cases, and computational experiments show that it consistently outperforms both COMBO and BOUKNAP on difficult benchmark sets, often by several orders of magnitude, establishing a new state-of-the-art solver for KP and BKP.