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

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.