Entanglement and the complexity of directed graphs

Dietmar Berwanger, Erich Grädel, Łukasz Kaiser, Roman Rabinovich
2012 Theoretical Computer Science
Entanglement is a parameter for the complexity of finite directed graphs that measures to what extent the cycles of the graph are intertwined. It is defined by way of a game similar in spirit to the cops and robber games used to describe treewidth, directed treewidth, and hypertree width. Nevertheless, on many classes of graphs, there are significant differences between entanglement and the various incarnations of treewidth. Entanglement is intimately related with the computational and
more » ... ve complexity of the modal µ-calculus. The number of fixed-point variables needed to describe a finite graph up to bisimulation is captured by its entanglement. This plays a crucial role in the proof that the variable hierarchy of the µ-calculus is strict. We study complexity issues for entanglement and compare it to other structural parameters of directed graphs. One of our main results is that parity games of bounded entanglement can be solved in polynomial time. Specifically, we establish that the complexity of solving a parity game can be parametrised in terms of the minimal entanglement of subgames induced by a winning strategy. Furthermore, we discuss the case of graphs of entanglement two. While graphs of entanglement zero and one are very simple, graphs of entanglement two allow arbitrary nesting of cycles, and they form a sufficiently rich class for modelling relevant classes of structured systems. We provide characterisations of this class, and propose decomposition notions similar to the ones for treewidth, DAG-width, and Kelly-width. 3 of a complete graph has maximal (undirected) treewidth, in spite of the fact that the directed graph is acyclic, and thus simple. Directed treewidth, the first generalisation of treewidth to directed graphs, is defined by means of an arboreal decomposition similar to the tree decomposition for the undirected case [17] . A variant of the graph searching game for the undirected case, where the robber is restricted to stay in her strongly connected component, characterises directed treewidth only up to a constant additive factor. DAG-width, introduced in [3,23,4], is defined by DAG-decompositions. A DAG-decomposition of width k for a graph G is described by a directed acyclic graph (DAG) D and a map that associates, with every node of the DAG, a set of at most k nodes of G, covering the entire graph G in such a way that, for every d ∈ D, the edges of G leaving a node strictly below d are guarded by nodes in d. DAG-width can also be characterised by a variant of a graph searching game (the directed cops and visible robber game), but with the somewhat unsatisfactory restriction that the cops are only allowed to use robber-monotone strategies, i.e., a move of the cops must never enlarge the portion of the graph in which the robber can move. It has been proved [21] that this restriction is necessary: there exist families of graphs where the difference between the DAG-width and the number of cops that can capture the robber with a non-monotone strategy is unbounded. Kelly-width, introduced in [16], is a similar measure that can be characterised either by a refined notion of decomposition, called Kelly-decomposition, or by a graph searching game in which the robber is invisible to the cops and inert, in the sense that she can move only when a cop is about to land on her current position. Again, the correspondence between decompositions and games only holds with the restriction to monotone strategies [21] . Entanglement, introduced in [5], has been motivated by applications concerning the modal µ-calculus and parity games. It is defined by a game where the moves of both the cops and the robber are more restricted than in other graph searching games: In each move the cops either stay where they are or place one of them on the current position of the robber; here, strategies need not be monotone. Entanglement is, in a sense, more delicate than (directed) treewidth, DAG-width, or Kelly-width [15] . There exist graphs of DAG-width, Kelly-width and directed treewidth three and arbitrarily large entanglement. For a survey of further complexity measures for directed graphs, such as pathwidth [25], cycle rank [11], D-width [27], we refer to [14, 24] . The strengths of entanglement are the close connection with modal logics and bisimulation invariant properties, and the natural game-theoretic characterisation. Thus, entanglement has been instrumental in the proof that the variable hierarchy of the modal µ-calculus is strict [7] . Furthermore, parity games can be solved efficiently on game graphs of bounded