Entanglement and the complexity of directed graphs

Entanglement is a parameter for the complexity of finite directed graphs that measures to which 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 tree width, directed tree width, and hypertree width. Nevertheless, on many classes of graphs, there are significant differences between entanglement and the various incarnations of tree width. Entanglement is intimately related with the computational and descriptive 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 entangement 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 tree width, DAG-width, and Kelly-width.