On the shortest path game: extended version

TL;DR

This paper studies a two-player game version of the shortest path problem, analyzing its computational complexity and providing algorithms for specific graph classes, revealing new complexity results and solution methods.

Contribution

It introduces a game-theoretic variant of shortest path, proves PSPACE-completeness for general graphs, and offers polynomial algorithms for acyclic and cactus graphs.

Findings

Decision problem is PSPACE-complete for bipartite graphs

Polynomial algorithms exist for directed acyclic graphs

Polynomial algorithms exist for cactus graphs

Abstract

In this work we address a game theoretic variant of the shortest path problem, in which two decision makers (players) move together along the edges of a graph from a given starting vertex to a given destination. The two players take turns in deciding in each vertex which edge to traverse next. The decider in each vertex also has to pay the cost of the chosen edge. We want to determine the path where each player minimizes its costs taking into account that also the other player acts in a selfish and rational way. Such a solution is a subgame perfect equilibrium and can be determined by backward induction in the game tree of the associated finite game in extensive form. We show that the decision problem associated with such a path is PSPACE-complete even for bipartite graphs both for the directed and the undirected version. The latter result is a surprising deviation from the complexity status of the closely related game Geography. On the other hand, we can give polynomial time algorithms for directed acyclic graphs and for cactus graphs even in the undirected case. The latter is based on a decomposition of the graph into components and their resolution by a number of fairly involved dynamic programming arrays. Finally, we give some arguments about closing the gap of the complexity status for graphs of bounded treewidth.