Abstract
Suppose that each arc in a digraph D = (V, A) has two costs of non-negative integers, called a spine cost and a leaf cost. A caterpillar is a directed tree consisting of a single directed path (of spine arcs) and leaf vertices each of which is incident to the directed path by exactly one incoming arc (leaf arc). For a given terminal set K ⊆ V, we study the problem of finding a caterpillar in D such that it contains all terminals in K and its total cost is minimized, where the cost of each arc in the caterpillar depends on whether it is used as a spine arc or a leaf arc. In this paper, we first study the complexity status of the problem with respect to the number of terminals: the problem is solvable in polynomial time for any digraph with two terminals, while it is NP-hard for three terminals. We then give a linear-time algorithm to solve the problem for digraphs with bounded treewidth, where the treewidth for a digraph D is defined as the one for the underlying graph of D. Our algorithm runs in linear time even if |K| = O(|V|).
This work is partially supported by JSPS Grant-in-Aid for Scientific Research, Grant Numbers 24.3660(A. Suzuki), 22700001(T. Ito) and 23500001(X. Zhou).
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Okada, T., Suzuki, A., Ito, T., Zhou, X. (2013). On the Minimum Caterpillar Problem in Digraphs. In: Du, DZ., Zhang, G. (eds) Computing and Combinatorics. COCOON 2013. Lecture Notes in Computer Science, vol 7936. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38768-5_66
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DOI: https://doi.org/10.1007/978-3-642-38768-5_66
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