### Abstract

The problem MAX W-LIGHT (MAX W-HEAVY) for an undirected graph is to assign a direction to each edge so that the number of vertices of outdegree at most W (resp. at least W) is maximized. It is known that these problems are NP-hard even for fixed W. For example, Max 0-LIGHT is equivalent to the problem of finding a maximum independent set. In this paper, we show that for any fixed constant W, MAX W-HEAVY can be solved in linear time for hereditary graph classes for which treewidth is bounded by a function of degeneracy. We show that such graph classes include chordal graphs, circular-arc graphs, d-trapezoid graphs, chordal bipartite graphs, and graphs of bounded clique-width. To have a polynomial-time algorithm for MAX W-LIGHT, we need an additional condition of a polynomial upper bound on the number of potential maximal cliques to apply the metatheorem by Fomin, Todinca, and Villanger [SIAM J. Comput., 44(1):57-87, 2015]. The aforementioned graph classes, except bounded clique-width graphs, satisfy such a condition. For graphs of bounded clique-width, we present a dynamic programming approach not using the metatheorem to show that it is actually polynomial-time solvable for this graph class too. We also study the parameterized complexity of the problems and show some tractability and intractability results.

Original language | English |
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Title of host publication | 27th International Symposium on Algorithms and Computation, ISAAC 2016 |

Editors | Seok-Hee Hong |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

Pages | 20.1-20.12 |

Volume | 64 |

ISBN (Electronic) | 9783959770262 |

DOIs | |

Publication status | Published - Dec 1 2016 |

Event | 27th International Symposium on Algorithms and Computation, ISAAC 2016 - Sydney, Australia Duration: Dec 12 2016 → Dec 14 2016 |

### Other

Other | 27th International Symposium on Algorithms and Computation, ISAAC 2016 |
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Country | Australia |

City | Sydney |

Period | 12/12/16 → 12/14/16 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Software

### Cite this

*27th International Symposium on Algorithms and Computation, ISAAC 2016*(Vol. 64, pp. 20.1-20.12). Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.ISAAC.2016.20

**Degree-constrained orientation of maximum satisfaction : Graph classes and parameterized complexity.** / Bodlaender, Hans L.; Ono, Hirotaka; Otachi, Yota.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*27th International Symposium on Algorithms and Computation, ISAAC 2016.*vol. 64, Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, pp. 20.1-20.12, 27th International Symposium on Algorithms and Computation, ISAAC 2016, Sydney, Australia, 12/12/16. https://doi.org/10.4230/LIPIcs.ISAAC.2016.20

}

TY - GEN

T1 - Degree-constrained orientation of maximum satisfaction

T2 - Graph classes and parameterized complexity

AU - Bodlaender, Hans L.

AU - Ono, Hirotaka

AU - Otachi, Yota

PY - 2016/12/1

Y1 - 2016/12/1

N2 - The problem MAX W-LIGHT (MAX W-HEAVY) for an undirected graph is to assign a direction to each edge so that the number of vertices of outdegree at most W (resp. at least W) is maximized. It is known that these problems are NP-hard even for fixed W. For example, Max 0-LIGHT is equivalent to the problem of finding a maximum independent set. In this paper, we show that for any fixed constant W, MAX W-HEAVY can be solved in linear time for hereditary graph classes for which treewidth is bounded by a function of degeneracy. We show that such graph classes include chordal graphs, circular-arc graphs, d-trapezoid graphs, chordal bipartite graphs, and graphs of bounded clique-width. To have a polynomial-time algorithm for MAX W-LIGHT, we need an additional condition of a polynomial upper bound on the number of potential maximal cliques to apply the metatheorem by Fomin, Todinca, and Villanger [SIAM J. Comput., 44(1):57-87, 2015]. The aforementioned graph classes, except bounded clique-width graphs, satisfy such a condition. For graphs of bounded clique-width, we present a dynamic programming approach not using the metatheorem to show that it is actually polynomial-time solvable for this graph class too. We also study the parameterized complexity of the problems and show some tractability and intractability results.

AB - The problem MAX W-LIGHT (MAX W-HEAVY) for an undirected graph is to assign a direction to each edge so that the number of vertices of outdegree at most W (resp. at least W) is maximized. It is known that these problems are NP-hard even for fixed W. For example, Max 0-LIGHT is equivalent to the problem of finding a maximum independent set. In this paper, we show that for any fixed constant W, MAX W-HEAVY can be solved in linear time for hereditary graph classes for which treewidth is bounded by a function of degeneracy. We show that such graph classes include chordal graphs, circular-arc graphs, d-trapezoid graphs, chordal bipartite graphs, and graphs of bounded clique-width. To have a polynomial-time algorithm for MAX W-LIGHT, we need an additional condition of a polynomial upper bound on the number of potential maximal cliques to apply the metatheorem by Fomin, Todinca, and Villanger [SIAM J. Comput., 44(1):57-87, 2015]. The aforementioned graph classes, except bounded clique-width graphs, satisfy such a condition. For graphs of bounded clique-width, we present a dynamic programming approach not using the metatheorem to show that it is actually polynomial-time solvable for this graph class too. We also study the parameterized complexity of the problems and show some tractability and intractability results.

UR - http://www.scopus.com/inward/record.url?scp=85010723624&partnerID=8YFLogxK

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U2 - 10.4230/LIPIcs.ISAAC.2016.20

DO - 10.4230/LIPIcs.ISAAC.2016.20

M3 - Conference contribution

AN - SCOPUS:85010723624

VL - 64

SP - 20.1-20.12

BT - 27th International Symposium on Algorithms and Computation, ISAAC 2016

A2 - Hong, Seok-Hee

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

ER -