Search Results

Documents authored by Mann, Kevin

Document
DOI: 10.4230/LIPIcs.IPEC.2024.24
Roman Hitting Functions

Authors: Henning Fernau and Kevin Mann

Published in: LIPIcs, Volume 321, 19th International Symposium on Parameterized and Exact Computation (IPEC 2024)

Abstract
Roman domination formalizes a military strategy going back to Constantine the Great. Here, armies are placed in different regions. A region is secured if there is at least one army in this region or there are two armies in one neighbored region. This simple strategy can be easily translated into a graph-theoretic question. The placement of armies is described by a function which maps each vertex to 0, 1 or 2. Such a function is called Roman dominating if each vertex with value 0 has a neighbor with value 2. Roman domination is one of few examples where the related (so-called) extension problem is polynomial-time solvable even if the original decision problem is NP-complete. This is interesting as it allows to establish polynomial-delay enumeration algorithms for listing minimal Roman dominating functions, while it is open for more than four decades if all minimal dominating sets of a graph or (equivalently) if all hitting sets of a hypergraph can be enumerated with polynomial delay, or even in output-polynomial time. To find the reason why this is the case, we combine the idea of hitting set with the idea of Roman domination. We hence obtain and study a new problem, called Roman Hitting Function, generalizing Roman Domination towards hypergraphs. This allows us to delineate the frontier of polynomial-delay enumerability. Our main focus is on the extension version of this problem, as this was the key problem when coping with Roman domination functions. While doing this, we find some conditions under which the Extension Roman Hitting Function problem is NP-complete. We then use parameterized complexity as a tool to get a better understanding of why Extension Roman Hitting Function behaves in this way. From an alternative perspective, we can say that we use the idea of parameterization to study the question what makes certain enumeration problems that difficult. Also, we discuss another generalization of Extension Roman Domination, where both a lower and an upper bound on the sought minimal Roman domination function is provided. The additional upper bound makes the problem hard (again), and the applied parameterized complexity analysis (only) provides hardness results. Also from the viewpoint of Parameterized Complexity, the studies on extension problems are quite interesting as they provide more and more examples of parameterized problems complete for W[3], a complexity class with only very few natural members known five years ago.
Cite as

Henning Fernau and Kevin Mann. Roman Hitting Functions. In 19th International Symposium on Parameterized and Exact Computation (IPEC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 321, pp. 24:1-24:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{fernau_et_al:LIPIcs.IPEC.2024.24,
  author =	{Fernau, Henning and Mann, Kevin},
  title =	{{Roman Hitting Functions}},
  booktitle =	{19th International Symposium on Parameterized and Exact Computation (IPEC 2024)},
  pages =	{24:1--24:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-353-9},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{321},
  editor =	{Bonnet, \'{E}douard and Rz\k{a}\.{z}ewski, Pawe{\l}},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2024.24},
  URN =		{urn:nbn:de:0030-drops-222504},
  doi =		{10.4230/LIPIcs.IPEC.2024.24},
  annote =	{Keywords: enumeration problems, polynomial delay, domination problems, hitting set, Roman domination}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2023.6
Roman Census: Enumerating and Counting Roman Dominating Functions on Graph Classes

Authors: Faisal N. Abu-Khzam, Henning Fernau, and Kevin Mann

Published in: LIPIcs, Volume 272, 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)

Abstract
The concept of Roman domination has recently been studied concerning enumerating and counting in F. N. Abu-Khzam et al. (WG 2022). More technically speaking, a function that assigns 0,1,2 to the vertices of an undirected graph is called a Roman dominating function if each vertex assigned zero has a neighbor assigned two. Such a function is called minimal if decreasing any assignment to any vertex would yield a function that is no longer a Roman dominating function. It has been shown that minimal Roman dominating functions can be enumerated with polynomial delay, i.e., between any two outputs of a solution, no more than polynomial time will elapse. This contrasts what is known about minimal dominating sets, where the question whether or not these can be enumerated with polynomial delay is open for more than 40 years. This makes the concept of Roman domination rather special and interesting among the many variants of domination problems studied in the literature, as it has been shown for several of these variants that the question of enumerating minimal solutions is tightly linked to that of enumerating minimal dominating sets, see M. Kanté et al. in SIAM J. Disc. Math., 2014. The running time of the mentioned enumeration algorithm for minimal Roman dominating functions (Abu-Khzam et al., WG 2022) could be estimated as 𝒪(1.9332ⁿ) on general graphs of order n. Here, we focus on special graph classes, as has been also done for enumerating minimal dominating sets before. More specifically, for chordal graphs, we present an enumeration algorithm running in time 𝒪(1.8940ⁿ). It is unknown if this gives a tight bound on the maximum number of minimal Roman dominating functions in chordal graphs. For interval graphs, we can lower this time bound further to 𝒪(1.7321ⁿ), which also matches the known lower bound concerning the maximum number of minimal Roman dominating functions. We can also provide a matching lower and upper bound for forests, which is (incidentally) the same, namely 𝒪^*(√3ⁿ). Furthermore, we present an optimal enumeration algorithm running in time 𝒪^*(∛3ⁿ) for split graphs and for cobipartite graphs, i.e., we can also give a matching lower bound example for these graph classes. Hence, our enumeration algorithms for interval graphs, forests, split graphs and cobipartite graphs are all optimal. The importance of our results stems from the fact that, for other types of domination problems, optimal enumeration algorithms are not always found. Interestingly, we use a different form of analysis for the running times of our different algorithms, and the branchings had to be tailored and tweaked to obtain the intended optimality results. Our Roman dominating functions enumeration algorithm for trees and forests is distinctively different from the one for minimal dominating sets by Rote (SODA 2019).Our approach also allows to give concrete formulas for counting minimal Roman dominating functions on more concrete graph families like paths.
Cite as

Faisal N. Abu-Khzam, Henning Fernau, and Kevin Mann. Roman Census: Enumerating and Counting Roman Dominating Functions on Graph Classes. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 6:1-6:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

Copy BibTex To Clipboard

@InProceedings{abukhzam_et_al:LIPIcs.MFCS.2023.6,
  author =	{Abu-Khzam, Faisal N. and Fernau, Henning and Mann, Kevin},
  title =	{{Roman Census: Enumerating and Counting Roman Dominating Functions on Graph Classes}},
  booktitle =	{48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)},
  pages =	{6:1--6:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-292-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{272},
  editor =	{Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2023.6},
  URN =		{urn:nbn:de:0030-drops-185400},
  doi =		{10.4230/LIPIcs.MFCS.2023.6},
  annote =	{Keywords: special graph classes, counting problems, enumeration problems, domination problems, Roman domination}
}
Document
DOI: 10.4230/LIPIcs.ESA.2022.1
Enumerating Minimal Connected Dominating Sets

Authors: Faisal N. Abu-Khzam, Henning Fernau, Benjamin Gras, Mathieu Liedloff, and Kevin Mann

Published in: LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

Abstract
The question to enumerate all (inclusion-wise) minimal connected dominating sets in a graph of order n in time significantly less than 2ⁿ is an open question that was asked in many places. We answer this question affirmatively, by providing an enumeration algorithm that runs in time 𝒪(1.9896ⁿ), using polynomial space only. The key to this result is the consideration of this enumeration problem on 2-degenerate graphs, which is proven to be possible in time 𝒪(1.9767ⁿ). Apart from solving this old open question, we also show new lower bound results. More precisely, we construct a family of graphs of order n with Ω(1.4890ⁿ) many minimal connected dominating sets, while previous examples achieved Ω(1.4422ⁿ). Our example happens to yield 4-degenerate graphs. Additionally, we give lower bounds for the previously not considered classes of 2-degenerate and of 3-degenerate graphs, which are Ω(1.3195ⁿ) and Ω(1.4723ⁿ), respectively. We also address essential questions concerning output-sensitive enumeration. Namely, we give reasons why our algorithm cannot be turned into an enumeration algorithm that guarantees polynomial delay without much efforts. More precisely, we prove that it is NP-complete to decide, given a graph G and a vertex set U, if there exists a minimal connected dominating set D with U ⊆ D, even if G is known to be 2-degenerate. Our reduction also shows that even any subexponential delay is not easy to achieve for enumerating minimal connected dominating sets. Another reduction shows that no FPT-algorithms can be expected for this extension problem concerning minimal connected dominating sets, parameterized by |U|. This also adds one more problem to the still rather few natural parameterized problems that are complete for the class W[3]. We also relate our enumeration problem to the famous open Hitting Set Transversal problem, which can be phrased in our context as the question to enumerate all minimal dominating sets of a graph with polynomial delay by showing that a polynomial-delay enumeration algorithm for minimal connected dominating sets implies an affirmative algorithmic solution to the Hitting Set Transversal problem.
Cite as

Faisal N. Abu-Khzam, Henning Fernau, Benjamin Gras, Mathieu Liedloff, and Kevin Mann. Enumerating Minimal Connected Dominating Sets. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 1:1-1:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

Copy BibTex To Clipboard

@InProceedings{abukhzam_et_al:LIPIcs.ESA.2022.1,
  author =	{Abu-Khzam, Faisal N. and Fernau, Henning and Gras, Benjamin and Liedloff, Mathieu and Mann, Kevin},
  title =	{{Enumerating Minimal Connected Dominating Sets}},
  booktitle =	{30th Annual European Symposium on Algorithms (ESA 2022)},
  pages =	{1:1--1:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-247-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{244},
  editor =	{Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.1},
  URN =		{urn:nbn:de:0030-drops-169390},
  doi =		{10.4230/LIPIcs.ESA.2022.1},
  annote =	{Keywords: enumeration problems, connected domination, degenerate graphs}
}
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact