Clean Citation Style 002

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Gao, Ziyuan; Jain, Sanjay; Khoussainov, Bakhadyr; Li, Wei; Melnikov, Alexander; Seidel, Karen; Stephan, FrankRandom Subgroups of Rationals. Mathematical Foundations of Computer Science (MFCS) 2019: 25:1-25:14

This paper introduces and studies a notion of algorithmic randomness for subgroups of rationals. Given a randomly generated additive subgroup \((G,+)\) of rationals, two main questions are addressed: first, what are the model-theoretic and recursion-theoretic properties of \((G,+)\); second, what learnability properties can one extract from \(G\) and its subclass of finitely generated subgroups? For the first question, it is shown that the theory of \((G,+)\) coincides with that of the additive group of integers and is therefore decidable; furthermore, while the word problem for \(G\) with respect to any generating sequence for \(G\) is not even semi-decidable, one can build a generating sequence \(\beta\) such that the word problem for \(G\) with respect to \(\beta\) is co-recursively enumerable (assuming that the set of generators of \(G\) is limit-recursive). In regard to the second question, it is proven that there is a generating sequence \(\beta\) for \(G\) such that every non-trivial finitely generated subgroup of \(G\) is recursively enumerable and the class of all such subgroups of \(G\) is behaviourally correctly learnable, that is, every non-trivial finitely generated subgroup can be semantically identified in the limit (again assuming that the set of generators of \(G\) is limit-recursive). On the other hand, the class of non-trivial finitely generated subgroups of \(G\) cannot be syntactically identified in the limit with respect to any generating sequence for \(G\). The present work thus contributes to a recent line of research studying algorithmically random infinite structures and uncovers an interesting connection between the arithmetical complexity of the set of generators of a randomly generated subgroup of rationals and the learnability of its finitely generated subgroups.

Göbel, Andreas; Lagodzinski, J. A. Gregor; Seidel, KarenCounting Homomorphisms to Trees Modulo a Prime. Mathematical Foundations of Computer Science (MFCS) 2018: 49:1-49:13

Many important graph theoretic notions can be encoded as counting graph homomorphism problems, such as partition functions in statistical physics, in particular independent sets and colourings. In this article we study the complexity of~\($\#_p\textsc{HomsTo}H$\), the problem of counting graph homomorphisms from an input graph to a graph \($H$\) modulo a prime number~\($p$\). Dyer and Greenhill proved a dichotomy stating that the tractability of non-modular counting graph homomorphisms depends on the structure of the target graph. Many intractable cases in non-modular counting become tractable in modular counting due to the common phenomenon of cancellation. In subsequent studies on counting modulo~\($2$\), however, the influence of the structure of~\($H$\) on the tractability was shown to persist, which yields similar dichotomies. Our main result states that for every tree~\($H$\) and every prime~\($p$\) the problem \($\#_p\textsc{HomsTo}H$\) is either polynomial time computable or \($\#_p\mathsf{P}$\)-complete. This relates to the conjecture of Faben and Jerrum stating that this dichotomy holds for every graph \($H$\) when counting modulo~2. In contrast to previous results on modular counting, the tractable cases of \($\#_p\textsc{HomsTo}H$\) are essentially the same for all values of the modulo when \($H$\) is a tree. To prove this result, we study the structural properties of a homomorphism. As an important interim result, our study yields a dichotomy for the problem of counting weighted independent sets in a bipartite graph modulo some prime~\($p$\). These results are the first suggesting that such dichotomies hold not only for the one-bit functions of the modulo~2 case but also for the modular counting functions of all primes~\($p$\).

Issac, Davis; van Leeuwen, Erik Jan; Lauri, Juho; Lima, Paloma; Heggernes, PinarRainbow Vertex Coloring Bipartite Graphs and Chordal Graphs. Mathematical Foundations of Computer Science (MFCS) 2018

Given a graph with colors on its vertices, a path is called a rainbow vertex path if all its internal vertices have distinct colors. We say that the graph is rainbow vertex-connected if there is a rainbow vertex path between every pair of its vertices. We study the problem of deciding whether the vertices of a given graph can be colored with at most k colors so that the graph becomes rainbow vertex-connected. Although edge-colorings have been studied extensively under similar constraints, there are significantly fewer results on the vertex variant that we consider. In particular, its complexity on structured graph classes was explicitly posed as an open question. We show that the problem remains NP-complete even on bipartite apex graphs and on split graphs. The former can be seen as a first step in the direction of studying the complexity of rainbow coloring on sparse graphs, an open problem which has attracted attention but limited progress. We also give hardness of approximation results for both bipartite and split graphs. To complement the negative results, we show that bipartite permutation graphs, interval graphs, and block graphs can be rainbow vertex-connected optimally in polynomial time.