Generalizing Alcuin’s River Crossing Problem

More than 1000 years ago Alcuin of York proposed a classical puzzle involving a wolf, a goat and a bunch of cabbages that need to be ferried across a river using a boat that only has enough room for one of them. In this paper we study several generalizations of this problem, called River Crossing problems, involving more items and more complicated incompatibilities. Our study is made from an algorithmic point of view, seeking to minimize either the boat size needed, or the number of trips. We present hardness and approximation results for the case where there is no constraint on the number of trips and the incompatibilities between items are given by a general graph. We show that the same problem can be solved exactly when the graph is a tree. In addition, we present several results when there is a constraint on the number of trips and show how this variation of the problem relates to the unconstrained version.