### 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 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 language | English |
---|---|

Pages (from-to) | 848-857 |

Number of pages | 10 |

Journal | IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences |

Volume | E96-A |

Issue number | 3 |

DOIs | |

Publication status | Published - 2014 Jan 1 |

### Keywords

- 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