## Abstract

Given an edge-weighted directed graph G = (V, E) on n vertices and a set T = {t1, t2,… tp} of p terminals, the objective of the Strongly Connected Steiner Subgraph (SCSS) problem is to find an edge set H ⊆ E of minimum weight such that G[H] contains a ti → tj path for each 1 ≤ i _= j ≤ p. The problem is NP-hard, but Feldman and Ruhl [FOCS’99; SICOMP’06] gave a novel nO(p) algorithm for the p-SCSS problem. In this paper, we investigate the computational complexity of a variant of 2-SCSS where we have demands for the number of paths between each terminal pair. Formally, the 2-SCSS-(k1, k2) problem is defined as follows: given an edge-weighted directed graph G = (V, E) with weight function ω: E → R≥0, two terminal vertices s, t, and integers k_{1}, k_{2}; the objective is to find a set of k1 paths F_{1}, F_{2},…, F_{k1} from s → t and k_{2} paths B_{1}, B_{2},…, B_{k2} from t → s such that ∑_{e∈E} ω(e)·φ(e) is minimized, where φ(e) = max{|{i: i ∈ [k_{1}], e ∈ Fi}|; |{j: j ∈ [k_{2}], e ∈ B_{j}}|}. For each k ≥ 1, we show the following:-The 2-SCSS-(k, 1) problem can be solved in n^{O(k)} time.-A matching lower bound for our algorithm: the 2-SCSS-(k, 1) problem does not have an f(k) · n^{o(k)} algorithm for any computable function f, unless the Exponential Time Hypothesis (ETH) fails. Our algorithm for 2-SCSS-(k, 1) relies on a structural result regarding the optimal solution followed by using the idea of a “token game” similar to that of Feldman and Ruhl. We show with an example that the structural result does not hold for the 2-SCSS-(k_{1}, k_{2}) problem if min{k_{1}, k_{2}} ≥ 2. Therefore 2-SCSS-(k, 1) is the most general problem one can attempt to solve with our techniques. To obtain the lower bound matching the algorithm, we reduce from a special variant of the Grid Tiling problem introduced by Marx [FOCS’07; ICALP’12].

Original language | English |
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Title of host publication | Parameterized and Exact Computation - 9th International Symposium, IPEC 2014, Revised Selected Papers |

Editors | Marek Cygan, Pinar Heggernes |

Publisher | Springer Verlag |

Pages | 159-171 |

Number of pages | 13 |

ISBN (Electronic) | 9783319135236 |

DOIs | |

State | Published - 2014 |

Externally published | Yes |

Event | 9th International Symposium on Parameterized and Exact Computation, IPEC 2014 - Wroclaw, Poland Duration: 10 Sep 2014 → 12 Sep 2014 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 8894 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 9th International Symposium on Parameterized and Exact Computation, IPEC 2014 |
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Country/Territory | Poland |

City | Wroclaw |

Period | 10/09/14 → 12/09/14 |

### Bibliographical note

Publisher Copyright:© Springer International Publishing Switzerland 2014.