On the Minimum Caterpillar Problem in Digraphs

Taku OKADA  Akira SUZUKI  Takehiro ITO  Xiao ZHOU  

IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E97-A   No.3   pp.848-857
Publication Date: 2014/03/01
Online ISSN: 1745-1337
DOI: 10.1587/transfun.E97.A.848
Print ISSN: 0916-8508
Type of Manuscript: PAPER
Category: Algorithms and Data Structures
bounded treewidth graph,  caterpillar,  dynamic programming,  graph algorithm,  inapproximability,  

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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 KV, 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 show that the problem is NP-hard for any fixed constant number of terminals with |K| ≥ 3, while it is solvable in polynomial time for at most two terminals. We also give an inapproximability result for any fixed constant number of terminals with |K| ≥ 3. Finally, we 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|), and the hidden constant factor of the running time is just a single exponential of the treewidth.