[Paper Review] Graph Searching, Parity Games and Imperfect Information

Bernd Puchala, Roman Rabinovich|RWTH Publications (RWTH Aachen)|Oct 25, 2011

Advanced Graph Theory Research12 references3 citations

TL;DR

This paper introduces a new graph searching game where r cops simultaneously pursue r visible robbers, generalizing directed path-width and DAG-width. It proves that r robbers can be captured monotonously using at most r × DAG-width(G) cops, establishing a hierarchy of complexity measures that does not collapse and enabling PTIME solvability of parity games with bounded imperfect information on graphs of bounded DAG-width.

ABSTRACT

We investigate the interrelation between graph searching games and games with imperfect information. As key consequence we obtain that parity games with bounded imperfect information can be solved in PTIME on graphs of bounded DAG-width which generalizes several results for parity games on graphs of bounded complexity. We use a new concept of graph searching where several cops try to catch multiple robbers instead of just a single robber. The main technical result is that the number of cops needed to catch r robbers monotonously is at most r times the DAG-width of the graph. We also explore aspects of this new concept as a refinement of directed path-width which accentuates its connection to the concept of imperfect information.

Motivation & Objective

• To formalize a new graph searching game where multiple visible robbers are pursued by cops, refining directed path-width and DAG-width.
• To establish a hierarchy of complexity measures dw₁(G) ≤ dw₂(G) ≤ ... ≤ dpw(G) that generalizes existing width measures.
• To prove that dwᵣ(G) ≤ r × dw(G), showing bounded imperfect information enables efficient solution of parity games.
• To demonstrate that the hierarchy does not collapse, proving the refinement is non-trivial and provides a proper approximation of directed path-width.

Proposed method

• Define dwᵣ(G) as the minimum number of cops needed to capture r visible robbers monotonously on graph G.
• Use a monotone strategy where cops are placed to prevent recontamination, ensuring progress toward capture.
• Prove that applying a single-robber winning strategy r times in parallel captures r robbers, leveraging the boundedness of imperfect information.
• Construct a family of graphs Gₖʳ = Tᵣ ⊕ Kₖ to show the hierarchy does not collapse, using lexicographic product to amplify complexity.
• Use induction and path-width arguments to prove dpw(Gₖʳ) = k(r + 1) and dwᵢ(Gₖʳ) ≥ i·(k−1)/2 for i ≤ r.
• Leverage the connection between graph searching and imperfect information games to transfer strategies from original to powerset graphs.

Experimental results

Research questions

• RQ1 Can the concept of graph searching be generalized to multiple visible robbers to refine directed path-width and DAG-width?
• RQ2 Is there a non-trivial hierarchy of complexity measures dw₁(G) ≤ dw₂(G) ≤ ... ≤ dpw(G) that does not collapse?
• RQ3 Can the number of cops needed to capture r visible robbers monotonously be bounded by r × DAG-width(G)?
• RQ4 Does this new framework enable PTIME solvability of parity games with bounded imperfect information on graphs of bounded DAG-width?
• RQ5 Can the hierarchy of width measures be shown to be proper through explicit graph constructions?

Key findings

• The paper proves that dwᵣ(G) ≤ r × dw(G), establishing a tight upper bound on the number of cops needed to capture r visible robbers monotonously.
• The hierarchy of width measures dw₁(G) ≤ dw₂(G) ≤ ... ≤ dpw(G) does not collapse, as shown by constructing graphs where dwᵢ(Gₖʳ) ≥ i·(k−1)/2 for all i ≤ r.
• For every k > 0, there exists a family of graphs Gₖʳ such that dpw(Gₖʳ) = k(r + 1), demonstrating the exponential growth of path-width relative to the number of robbers.
• The construction Gₖʳ = Tᵣ ⊕ Kₖ, where Tᵣ is a full tree of height r+1 and branching degree at least 3, achieves the required path-width and robber-capture complexity.
• The result generalizes previous PTIME solvability results for parity games with bounded imperfect information on graphs of bounded directed path-width.
• The framework provides a natural bridge between graph searching games and games of imperfect information, showing that bounded uncertainty in games corresponds to bounded visibility in searching games.