The Greedy Coin Change Problem

TL;DR

This paper investigates the computational complexity of the greedy coin change problem, proving it is P-complete, which indicates inherent difficulty in parallelization and limited space solutions.

Contribution

It formalizes the decision version of the greedy coin change problem and proves its P-completeness, highlighting computational challenges.

Findings

The greedy coin change problem is P-complete under log-space reductions.

The problem is unlikely to be efficiently parallelizable.

It is computationally hard to determine greedy coin inclusion in general.

Abstract

The Coin Change problem, also known as the Change-Making problem, is a well-studied combinatorial optimization problem, which involves minimizing the number of coins needed to make a specific change amount using a given set of coin denominations. A natural and intuitive approach to this problem is the greedy algorithm. While the greedy algorithm is not universally optimal for all sets of coin denominations, it yields optimal solutions under most real-world coin systems currently in use, making it an efficient heuristic with broad practical applicability. Researchers have been studying ways to determine whether a given coin system guarantees optimal solutions under the greedy approach, but surprisingly little attention has been given to understanding the general computational behavior of the greedy algorithm applied to the coin change problem. To address this gap, we introduce the Greedy Coin Change problem and formalize its decision version: given a target amount $W$ and a set of denominations $C$, determine whether a specific coin is included in the greedy solution. We prove that this problem is $\mathbf P$-complete under log-space reductions, which implies it is unlikely to be efficiently parallelizable or solvable in limited space.