Nemesis, an Escape Game in Graphs

TL;DR

This paper introduces the Nemesis escape game on graphs, analyzes its computational complexity, and presents efficient solutions for specific graph classes while establishing hardness results for general cases.

Contribution

It defines the Nemesis game, provides linear-time algorithms for trees and bounded degree graphs, and proves PSPACE-completeness and NP-hardness for general graphs and related problems.

Findings

Nemesis can be solved in linear time on trees and degree-bounded graphs.

The game is PSPACE-complete for arbitrary graphs.

A variant called Blizzard also admits a linear time solution.

Abstract

We define a new escape game in graphs that we call Nemesis. The game is played on a graph having a subset of vertices labeled as exits and the goal of one of the two players, called the fugitive, is to reach one of these exit vertices. The second player, i.e. the fugitive adversary, is called the Nemesis. Her goal is to trap the fugitive in a connected component which does not contain any exit. At each round of the game, the fugitive moves from one vertex to an adjacent vertex. Then the Nemesis deletes one edge anywhere in the graph. The game ends when either the fugitive reached an exit or when he is in a connected component that does not contain any exit. In trees and graphs of maximum degree bounded by 3, Nemesis can be solved in linear time. We also show that a variant of the game called Blizzard where only edges adjacent to the position of the fugitive can be deleted also admits a linear time solution. For arbitrary graphs, we show that Nemesis is PSPACE-complete, and that it is NP-hard on planar multigraphs. We extend our results to the related Cat Herding problem, proving its PSPACE-completeness. We also prove that finding a strategy based on a full binary escape tree whose leaves are exists is NP-complete.