The Ferry Cover Problem

Abstract

In the classical wolf-goat-cabbage puzzle, a ferry boat man must ferry three items across a river using a boat that has room for only one, without leaving two incompatible items on the same bank alone. In this paper we define and study a family of optimization problems called Ferry problems, which may be viewed as generalizations of this familiar puzzle.

In all Ferry problems we are given a set of items and a graph with edges connecting items that must not be left together unattended. We present the Ferry Cover problem (FC), where the objective is to determine the minimum required boat size and demonstrate a close connection with Vertex Cover which leads to hardness and approximation results. We also completely solve the problem on trees. Then we focus on a variation of the same problem with the added constraint that only 1 round-trip is allowed (FC₁). We present a reduction from MAX-NAE-{3}-SAT which shows that this problem is NP-hard and APX-hard. We also provide an approximation algorithm for trees with a factor asymptotically equal to $\frac{4}{3}$. Finally, we generalize the above problem to define FC _m , where at most m round-trips are allowed, and MFT _k , which is the problem of minimizing the number of round-trips when the boat capacity is k. We present some preliminary lemmata for both, which provide bounds on the value of the optimal solution, and relate them to FC.