Florian Chudigiewitsch

Florian Chudigiewitsch, Marlene Gründel, Christian Komusiewicz et al.

A relation modification problem gets a logical structure and a natural number k as input and asks whether k modifications of the structure suffice to make it satisfy a predefined property. We provide a complete classification of the classical and parameterized complexity of relation modification problems - the latter w. r. t. the modification budget k - based on the descriptive complexity of the respective target property. We consider different types of logical structures on which modifications are performed: Whereas monadic structures and undirected graphs without self-loops each yield their own complexity landscapes, we find that modifying undirected graphs with self-loops, directed graphs, or arbitrary logical structures is equally hard w. r. t. quantifier patterns. Moreover, we observe that all classes of problems considered in this paper are subject to a strong dichotomy in the sense that they are either very easy to solve (that is, they lie in paraAC^{0\uparrow} or TC^0) or intractable (that is, they contain W[2]-hard or NP-hard problems).

Florian Chudigiewitsch, Till Tantau, Felix Winkler

A DAG compression of a (typically dense) graph is a simple data structure that stores how vertex clusters are connected, where the clusters are described indirectly as sets of reachable sinks in a directed acyclic graph (DAG). They generalize tree compressions, where the clusters form a tree-like hierarchy, and we give the first proof that DAG compressions can achieve better compressions than tree compressions. Our interest in DAG compression stems from the fact that several simple standard algorithms, like breadth-first search on graphs, can be implemented so that they work directly on the compressed rather than on the original graph and so that, crucially, the runtime is relative to the (typically small) size of the compressed graph. We add another entry to the list of algorithms where this is possible, by showing that Kruskal's algorithm for computing minimum spanning trees can be adapted to work directly on DAG compressions. On the negative side, we answer the central open problem from previous work, namely how hard it is to compute a minimum-size DAG compression for a given graph: This is NP-hard; and this is even the case for the dynamic setting, where we must update the DAG compression optimally when a single edge is added or deleted in the input graph.