The Ferry Cover Problem

Abstract

In this paper we define and study a family of optimization problems called Ferry problems, which may be viewed as generalizations of the classical wolf-goat-cabbage puzzle. We present the Ferry Cover problem (FC), where the objective is to determine the minimum required boat size to safely transport n items represented by a graph G 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 bipartite graphs with a factor asymptotically equal to  $\frac{4}{3}$ and a 1.56-approximation algorithm for planar graphs. 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.