[Paper Review] Digraph Decompositions and Monotonicity in Digraph Searching

Stephan Kreutzer, Sebastian Ordyniak|ArXiv.org|Feb 15, 2008

Advanced Graph Theory Research17 references22 citations

TL;DR

This paper resolves long-standing open problems in digraph searching by proving that two variants of the Cops and Robber game—on directed graphs with an invisible, lazy robber or a visible, fast robber—are non-monotone. The authors construct explicit counterexamples showing that monotone strategies require significantly more cops than non-monotone ones, and further demonstrate that many natural problems remain NP-complete even on graphs of very low DAG-width, limiting the algorithmic scope of directed graph decompositions.

ABSTRACT

We consider monotonicity problems for graph searching games. Variants of these games - defined by the type of moves allowed for the players - have been found to be closely connected to graph decompositions and associated width measures such as path- or tree-width. Of particular interest is the question whether these games are monotone, i.e. whether the cops can catch a robber without ever allowing the robber to reach positions that have been cleared before. The monotonicity problem for graph searching games has intensely been studied in the literature, but for two types of games the problem was left unresolved. These are the games on digraphs where the robber is invisible and lazy or visible and fast. In this paper, we solve the problems by giving examples showing that both types of games are non-monotone. Graph searching games on digraphs are closely related to recent proposals for digraph decompositions generalising tree-width to directed graphs. These proposals have partly been motivated by attempts to develop a structure theory for digraphs similar to the graph minor theory developed by Robertson and Seymour for undirected graphs, and partly by the immense number of algorithmic results using tree-width of undirected graphs and the hope that part of this success might be reproducible on digraphs using a directed tree-width. Unfortunately the number of applications for the digraphs measures introduced so far is still small. We therefore explore the limits of the algorithmic applicability of digraph decompositions. In particular, we show that various natural candidates for problems that might benefit from digraphs having small directed tree-width remain NP-complete even on almost acyclic graphs.

Motivation & Objective

• To resolve the open problem of whether Cops and Robber games on digraphs with an invisible, lazy robber or a visible, fast robber are monotone.
• To investigate the limits of algorithmic tractability for directed graph decompositions, particularly in relation to width measures like DAG-width.
• To determine whether natural problems such as Graph Grundy Numbering and Kernel remain tractable on digraphs of small directed width.
• To establish that monotonicity costs in these games are unbounded, challenging the possibility of constant-factor approximations in monotone strategies.

Proposed method

• Constructing explicit counterexamples using tailored digraphs to demonstrate non-monotonicity in both game variants.
• Reducing known NP-complete problems (3-SAT, FAS, Graph Grundy Numbering, Kernel) to digraphs of bounded DAG-width to prove intractability.
• Designing monotone winning strategies for two cops on specific subgraphs (e.g., variable gadgets and clause cycles) to bound DAG-width.
• Using structural analysis of graph components (e.g., cycles, acyclic subgraphs) to prove that constructed graphs have DAG-width at most two.
• Applying game-theoretic reasoning to show that non-monotone strategies can outperform monotone ones in terms of required cop count.
• Leveraging existing characterisations of width measures via Cops and Robber games to link game monotonicity to decomposition theory.

Experimental results

Research questions

• RQ1 Are Cops and Robber games with an invisible, lazy robber on digraphs monotone?
• RQ2 Are Cops and Robber games with a visible, fast robber on digraphs monotone?
• RQ3 Can problems like Graph Grundy Numbering and Kernel be solved in polynomial time on digraphs of bounded DAG-width?
• RQ4 Is there a constant upper bound on the monotonicity cost (i.e., number of additional cops needed) for these digraph games?
• RQ5 To what extent do directed graph decompositions enable tractable algorithms for natural problems beyond linkage and Hamiltonicity?

Key findings

• The Cops and Robber game with an invisible, lazy robber on digraphs is non-monotone, as demonstrated by explicit counterexamples.
• The Cops and Robber game with a visible, fast robber on digraphs is also non-monotone, with similar counterexamples showing unbounded monotonicity costs.
• The Feedback Arc Set (FAS) problem is NP-complete on digraphs of DAG-width at most four.
• Graph Grundy Numbering is NP-complete on digraphs of DAG-width at most two.
• The Kernel problem is NP-complete on digraphs of DAG-width two.
• Many natural problems remain NP-complete even on almost acyclic digraphs (e.g., DAG-width ≤ 2), indicating fundamental limits to the algorithmic power of directed width measures.