On the minimum caterpillar problem in digraphs

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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 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 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.

Original languageEnglish
Pages (from-to)848-857
Number of pages10
JournalIEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences
Issue number3
Publication statusPublished - 2014


  • Bounded treewidth graph
  • Caterpillar
  • Dynamic programming
  • Graph algorithm
  • Inapproximability

ASJC Scopus subject areas

  • Signal Processing
  • Computer Graphics and Computer-Aided Design
  • Electrical and Electronic Engineering
  • Applied Mathematics


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