Generating Functions For Kernels of Digraphs (Enumeration & Asymptotics for Nim Games)

In this article, we study directed graphs (digraphs) with a coloring constraint due to Von Neumann and related to Nim-type games. This is equivalent to the notion of kernels of digraphs, which appears in numerous fields of research such as game theory, complexity theory, artificial intelligence (default logic, argumentation in multi-agent systems), 0-1 laws in monadic second order logic, combinatorics (perfect graphs)... Kernels of digraphs lead to numerous difficult questions (in the sense of NP-completeness, #P-completeness). However, we show here that it is possible to use a generating function approach to get new informations: we use technique of symbolic and analytic combinatorics (generating functions and their singularities) in order to get exact and asymptotic results, e.g. for the existence of a kernel in a circuit or in a unicircuit digraph. This is a first step toward a generatingfunctionology treatment of kernels, while using, e.g., an approach "a la Wright". Our method could be applied to more general "local coloring constraints" in decomposable combinatorial structures.Comment: Presented (as a poster) to the conference Formal Power Series and Algebraic Combinatorics (Vancouver, 2004), electronic proceeding

Tournaments with kernels by monochromatic paths

In this paper we prove the existence of kernels by monochromatic paths in m-coloured tournaments in which every cyclic tournament of order 3 is atmost 2-coloured in addition to other restrictions on the colouring ofcertain subdigraphs. We point out that in all previous results on kernelsby monochromatic paths in arc coloured tournaments, certain smallsubstructures are required to be monochromatic or monochromatic with atmost one exception, whereas here we allow up to three colours in two smallsubstructures

Dagstuhl Reports : Volume 1, Issue 2, February 2011

Online Privacy: Towards Informational Self-Determination on the Internet (Dagstuhl Perspectives Workshop 11061) : Simone Fischer-Hübner, Chris Hoofnagle, Kai Rannenberg, Michael Waidner, Ioannis Krontiris and Michael Marhöfer Self-Repairing Programs (Dagstuhl Seminar 11062) : Mauro Pezzé, Martin C. Rinard, Westley Weimer and Andreas Zeller Theory and Applications of Graph Searching Problems (Dagstuhl Seminar 11071) : Fedor V. Fomin, Pierre Fraigniaud, Stephan Kreutzer and Dimitrios M. Thilikos Combinatorial and Algorithmic Aspects of Sequence Processing (Dagstuhl Seminar 11081) : Maxime Crochemore, Lila Kari, Mehryar Mohri and Dirk Nowotka Packing and Scheduling Algorithms for Information and Communication Services (Dagstuhl Seminar 11091) Klaus Jansen, Claire Mathieu, Hadas Shachnai and Neal E. Youn

H-absorbence and H-independence in 3-quasi-transitive H-coloured digraphs.

In this paper we prove that if $D$ is a loopless asymmetric 3-quasi-transitive arc-coloured digraph having its arcs coloured with the vertices of a given digraph $H$, and if in $D$ every $C_4$ is an $H$-cycle and every $C_3$ is a quasi-$H$-cycle, then $D$ has an $H$-kernel

Enumeration of Preferred Extensions in Almost Oriented Digraphs

In this paper, we present enumeration algorithms to list all preferred extensions of an argumentation framework. This task is equivalent to enumerating all maximal semikernels of a directed graph. For directed graphs on n vertices, all preferred extensions can be enumerated in O^*(3^{n/3}) time and there are directed graphs with Omega(3^{n/3}) preferred extensions. We give faster enumeration algorithms for directed graphs with at most 0.8004 * n vertices occurring in 2-cycles. In particular, for oriented graphs (digraphs with no 2-cycles) one of our algorithms runs in time O(1.2321^n), and we show that there are oriented graphs with Omega(3^{n/6}) > Omega(1.2009^n) preferred extensions. A combination of three algorithms leads to the fastest enumeration times for various proportions of the number of vertices in 2-cycles. The most innovative one is a new 2-stage sampling algorithm, combined with a new parameterized enumeration algorithm, analyzed with a combination of the recent monotone local search technique (STOC 2016) and an extension thereof (ICALP 2017)

Parameterized Algorithmics for Computational Social Choice: Nine Research Challenges

Computational Social Choice is an interdisciplinary research area involving Economics, Political Science, and Social Science on the one side, and Mathematics and Computer Science (including Artificial Intelligence and Multiagent Systems) on the other side. Typical computational problems studied in this field include the vulnerability of voting procedures against attacks, or preference aggregation in multi-agent systems. Parameterized Algorithmics is a subfield of Theoretical Computer Science seeking to exploit meaningful problem-specific parameters in order to identify tractable special cases of in general computationally hard problems. In this paper, we propose nine of our favorite research challenges concerning the parameterized complexity of problems appearing in this context

Games and Argumentation: Time for a Family Reunion!

The rule "defeated(X) $\leftarrow$ attacks(Y,X), $\neg$ defeated(Y)" states that an argument is defeated if it is attacked by an argument that is not defeated. The rule "win(X) $\leftarrow$ move(X,Y), $\neg$ win(Y)" states that in a game a position is won if there is a move to a position that is not won. Both logic rules can be seen as close relatives (even identical twins) and both rules have been at the center of attention at various times in different communities: The first rule lies at the core of argumentation frameworks and has spawned a large family of models and semantics of abstract argumentation. The second rule has played a key role in the quest to find the "right" semantics for logic programs with recursion through negation, and has given rise to the stable and well-founded semantics. Both semantics have been widely studied by the logic programming and nonmonotonic reasoning community. The second rule has also received much attention by the database and finite model theory community, e.g., when studying the expressive power of query languages and fixpoint logics. Although close connections between argumentation frameworks, logic programming, and dialogue games have been known for a long time, the overlap and cross-fertilization between the communities appears to be smaller than one might expect. To this end, we recall some of the key results from database theory in which the win-move query has played a central role, e.g., on normal forms and expressive power of query languages. We introduce some notions that naturally emerge from games and that may provide new perspectives and research opportunities for argumentation frameworks. We discuss how solved query evaluation games reveal how- and why-not provenance of query answers. These techniques can be used to explain how results were derived via the given query, game, or argumentation framework.Comment: Fourth Workshop on Explainable Logic-Based Knowledge Representation (XLoKR), Sept 2, 2023. Rhodes, Greec