Romeo and Juliet Meeting in Forest Like Regions
Neeldhara Misra, Manas Mulpuri, Prafullkumar Tale, Gaurav Viramgami

TL;DR
This paper investigates the computational complexity of a rendezvous game on graphs, revealing hardness results for various structural parameters and providing efficient algorithms for specific graph classes.
Contribution
It establishes new complexity bounds for the rendezvous game, including hardness for treewidth and feedback vertex set parameters, and offers fixed-parameter algorithms for certain cases.
Findings
Rendezvous is co-NP-hard on graphs with constant treewidth.
The problem is co-W[1]-hard when parameterized by feedback vertex set and number of agents.
Polynomial-time algorithm exists for graphs with treewidth at most two and grids.
Abstract
The game of rendezvous with adversaries is a game on a graph played by two players: Facilitator and Divider. Facilitator has two agents and Divider has a team of agents. While the initial positions of Facilitator's agents are fixed, Divider gets to select the initial positions of his agents. Then, they take turns to move their agents to adjacent vertices (or stay put) with Facilitator's goal to bring both her agents at same vertex and Divider's goal to prevent it. The computational question of interest is to determine if Facilitator has a winning strategy against Divider with agents. Fomin, Golovach, and Thilikos [WG, 2021] introduced this game and proved that it is PSPACE-hard and co-W[2]-hard parameterized by the number of agents. This hardness naturally motivates the structural parameterization of the problem. The authors proved that it admits an FPT algorithm when…
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