Change-making problems revisited: a parameterized point of view

Abstract

The change-making problem is the problem of representing a given amount of money with the fewest number of coins possible from a given set of coin denominations. In the general version of the problem, an upper bound for the availability of every coin value is given. Even the special case, where for each value an unlimited number of coins is available, is NP-hard. Since in the original problem some amounts can not be represented, especially if no coin of value one exists, we introduce generalized problems that look for approximations of the given amount such that a cost function is minimized. We recall algorithms for the change-making problem and present new algorithms for the generalized version of the problem. Motivated by the NP-hardness we study fixed-parameter tractability of all these problems. We show that some of these problems are fixed-parameter tractable and that some are $$\hbox {W}[1]$$ -hard. In order to show the existence of polynomial and constant-size kernels we prove some general results and apply them to several parameterizations of the change-making problems.