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**Published in:** LIPIcs, Volume 117, 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)

We study a family of generalizations of Edge Dominating Set on directed graphs called Directed (p,q)-Edge Dominating Set. In this problem an arc (u,v) is said to dominate itself, as well as all arcs which are at distance at most q from v, or at distance at most p to u.
First, we give significantly improved FPT algorithms for the two most important cases of the problem, (0,1)-dEDS and (1,1)-dEDS (that correspond to versions of Dominating Set on line graphs), as well as polynomial kernels. We also improve the best-known approximation for these cases from logarithmic to constant. In addition, we show that (p,q)-dEDS is FPT parameterized by p+q+tw, but W-hard parameterized just by tw, where tw is the treewidth of the underlying graph of the input.
We then go on to focus on the complexity of the problem on tournaments. Here, we provide a complete classification for every possible fixed value of p,q, which shows that the problem exhibits a surprising behavior, including cases which are in P; cases which are solvable in quasi-polynomial time but not in P; and a single case (p=q=1) which is NP-hard (under randomized reductions) and cannot be solved in sub-exponential time, under standard assumptions.

Rémy Belmonte, Tesshu Hanaka, Ioannis Katsikarelis, Eun Jung Kim, and Michael Lampis. New Results on Directed Edge Dominating Set. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 67:1-67:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{belmonte_et_al:LIPIcs.MFCS.2018.67, author = {Belmonte, R\'{e}my and Hanaka, Tesshu and Katsikarelis, Ioannis and Kim, Eun Jung and Lampis, Michael}, title = {{New Results on Directed Edge Dominating Set}}, booktitle = {43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)}, pages = {67:1--67:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-086-6}, ISSN = {1868-8969}, year = {2018}, volume = {117}, editor = {Potapov, Igor and Spirakis, Paul and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2018.67}, URN = {urn:nbn:de:0030-drops-96490}, doi = {10.4230/LIPIcs.MFCS.2018.67}, annote = {Keywords: Edge Dominating Set, Tournaments, Treewidth} }

Document

**Published in:** LIPIcs, Volume 101, 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018)

A graph is d-orientable if its edges can be oriented so that the maximum in-degree of the resulting digraph is at most d. d-orientability is a well-studied concept with close connections to fundamental graph-theoretic notions and applications as a load balancing problem. In this paper we consider the d-Orientable Deletion problem: given a graph G=(V,E), delete the minimum number of vertices to make G d-orientable. We contribute a number of results that improve the state of the art on this problem. Specifically:
- We show that the problem is W[2]-hard and log n-inapproximable with respect to k, the number of deleted vertices. This closes the gap in the problem's approximability.
- We completely characterize the parameterized complexity of the problem on chordal graphs: it is FPT parameterized by d+k, but W-hard for each of the parameters d,k separately.
- We show that, under the SETH, for all d,epsilon, the problem does not admit a (d+2-epsilon)^{tw}, algorithm where tw is the graph's treewidth, resolving as a special case an open problem on the complexity of PseudoForest Deletion.
- We show that the problem is W-hard parameterized by the input graph's clique-width. Complementing this, we provide an algorithm running in time d^{O(d * cw)}, showing that the problem is FPT by d+cw, and improving the previously best know algorithm for this case.

Tesshu Hanaka, Ioannis Katsikarelis, Michael Lampis, Yota Otachi, and Florian Sikora. Parameterized Orientable Deletion. In 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 101, pp. 24:1-24:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{hanaka_et_al:LIPIcs.SWAT.2018.24, author = {Hanaka, Tesshu and Katsikarelis, Ioannis and Lampis, Michael and Otachi, Yota and Sikora, Florian}, title = {{Parameterized Orientable Deletion}}, booktitle = {16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018)}, pages = {24:1--24:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-068-2}, ISSN = {1868-8969}, year = {2018}, volume = {101}, editor = {Eppstein, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2018.24}, URN = {urn:nbn:de:0030-drops-88506}, doi = {10.4230/LIPIcs.SWAT.2018.24}, annote = {Keywords: Graph orientations, FPT algorithms, Treewidth, SETH} }

Document

**Published in:** LIPIcs, Volume 92, 28th International Symposium on Algorithms and Computation (ISAAC 2017)

In (k,r)-Center we are given a (possibly edge-weighted) graph and are asked to select at most k vertices (centers), so that all other vertices are at distance at most r from a center. In this paper we provide a number of tight fine-grained bounds on the complexity of this problem with respect to various standard graph parameters. Specifically:
- For any r>=1, we show an algorithm that solves the problem in O*((3r+1)^cw) time, where cw is the clique-width of the input graph, as well as a tight SETH lower bound matching this algorithm's performance. As a corollary, for r=1, this closes the gap that previously existed on the complexity of Dominating Set parameterized by cw.
- We strengthen previously known FPT lower bounds, by showing that (k,r)-Center is W[1]-hard parameterized by the input graph's vertex cover (if edge weights are allowed), or feedback vertex set, even if k is an additional parameter. Our reductions imply tight ETH-based lower bounds. Finally, we devise an algorithm parameterized by vertex cover for unweighted graphs.
- We show that the complexity of the problem parameterized by tree-depth is 2^Theta(td^2) by showing an algorithm of this complexity and a tight ETH-based lower bound.
We complement these mostly negative results by providing FPT approximation schemes parameterized by clique-width or treewidth which work efficiently independently of the values of k,r. In particular, we give algorithms which, for any epsilon>0, run in time O*((tw/epsilon)^O(tw)), O*((cw/epsilon)^O(cw)) and return a (k,(1+epsilon)r)-center, if a (k,r)-center exists, thus circumventing the problem's W-hardness.

Ioannis Katsikarelis, Michael Lampis, and Vangelis Th. Paschos. Structural Parameters, Tight Bounds, and Approximation for (k,r)-Center. In 28th International Symposium on Algorithms and Computation (ISAAC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 92, pp. 50:1-50:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{katsikarelis_et_al:LIPIcs.ISAAC.2017.50, author = {Katsikarelis, Ioannis and Lampis, Michael and Paschos, Vangelis Th.}, title = {{Structural Parameters, Tight Bounds, and Approximation for (k,r)-Center}}, booktitle = {28th International Symposium on Algorithms and Computation (ISAAC 2017)}, pages = {50:1--50:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-054-5}, ISSN = {1868-8969}, year = {2017}, volume = {92}, editor = {Okamoto, Yoshio and Tokuyama, Takeshi}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2017.50}, URN = {urn:nbn:de:0030-drops-82441}, doi = {10.4230/LIPIcs.ISAAC.2017.50}, annote = {Keywords: FPT algorithms, Approximation, Treewidth, Clique-width, Domination} }

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