Solving the Canonical Representation and Star System Problems for Proper Circular-Arc Graphs in Log-Space

We present a logspace algorithm that constructs a canonical intersection model for a given proper circular-arc graph, where `canonical' means that models of isomorphic graphs are equal. This implies that the recognition and the isomorphism problems for this class of graphs are solvable in logspace. For a broader class of concave-round graphs, that still possess (not necessarily proper) circular-arc models, we show that those can also be constructed canonically in logspace. As a building block for these results, we show how to compute canonical models of circular-arc hypergraphs in logspace, which are also known as matrices with the circular-ones property. Finally, we consider the search version of the Star System Problem that consists in reconstructing a graph from its closed neighborhood hypergraph. We solve it in logspace for the classes of proper circular-arc, concave-round, and co-convex graphs.Comment: 19 pages, 3 figures, major revisio

Parameterized Complexity of Small Weight Automorphisms

We show that checking if a given hypergraph has an automorphism that moves exactly k vertices is fixed parameter tractable, using k and additionally either the maximum hyperedge size or the maximum color class size as parameters. In particular, it suffices to use k as parameter if the hyperedge size is at most polylogarithmic in the size of the given hypergraph. As a building block for our algorithms, we generalize Schweitzer\u27s FPT algorithm [ESA 2011] that, given two graphs on the same vertex set and a parameter k, decides whether there is an isomorphism between the two graphs that moves at most k vertices. We extend this result to hypergraphs, using the maximum hyperedge size as a second parameter. Another key component of our algorithm is an orbit-shrinking technique that preserves permutations that move few points and that may be of independent interest. Applying it to a suitable subgroup of the automorphism group allows us to switch from bounded hyperedge size to bounded color classes in the exactly-k case

Welche Hemmnisse sehen derzeit sächsische Landwirte bei einer Umstellung auf ökologischen Landbau? - Erste Ergebnisse einer Befragung -

Zwar verzeichnete der ökologische Landbau in Europa in den letzten Jahren hohe jährliche Zuwachsraten von durchschnittlich 25 %, in Deutschland einschließlich Sachsen war die Entwicklung bis Ende 2000 jedoch vergleichsweise sehr verhalten. Um das derzeit abschätzbare Potential des ökologischen Landbaus nutzen zu können, ist eine enorme Steigerung des bisherigen Wachstums der ökologisch bewirtschafteten Fläche auch in Sachsen erforderlich. In der vorliegenden Studie sollte die Sicht der sächsischen Landwirte als zentrale Träger der weiteren Entwicklung des ökologischen Landbaus auf diese Wirtschaftsweise und die wahrgenommenen Hemmnisse bezüglich einer Umstellung des eigenen Betriebes mit Hilfe einer Befragung untersucht werden. Es zeigte sich ein bereits hoher Bekanntheitsgrad des ökologischen Landbaus innerhalb der sächsischen Landwirtschaftsbetriebe. Das Stimmungsbild stellt sich sehr differenziert dar. Eher gering ist der Wissensstand zu den Besonderheiten dieser Wirtschaftsweise einzuschätzen. Jedoch ist das Interesse an weiteren Fachinformationen sehr groß

The Parameterized Complexity of Fixing Number and Vertex Individualization in Graphs

In this paper we study the complexity of the following problems: 1. Given a colored graph X=(V,E,c), compute a minimum cardinality set of vertices S (subset of V) such that no nontrivial automorphism of X fixes all vertices in S. A closely related problem is computing a minimum base S for a permutation group G <= S_n given by generators, i.e., a minimum cardinality subset S of [n] such that no nontrivial permutation in G fixes all elements of S. Our focus is mainly on the parameterized complexity of these problems. We show that when k=|S| is treated as parameter, then both problems are MINI[1]-hard. For the dual problems, where k=n-|S| is the parameter, we give FPT~algorithms. 2. A notion closely related to fixing is called individualization. Individualization combined with the Weisfeiler-Leman procedure is a fundamental technique in algorithms for Graph Isomorphism. Motivated by the power of individualization, in the present paper we explore the complexity of individualization: what is the minimum number of vertices we need to individualize in a given graph such that color refinement "succeeds" on it. Here "succeeds" could have different interpretations, and we consider the following: It could mean the individualized graph becomes: (a) discrete, (b) amenable, (c)compact, or (d) refinable. In particular, we study the parameterized versions of these problems where the parameter is the number of vertices individualized. We show a dichotomy: For graphs with color classes of size at most 3 these problems can be solved in polynomial time, while starting from color class size 4 they become W[P]-hard