The Shortest Connection Game

TL;DR

The paper introduces the Shortest Connection Game, analyzing its computational complexity and providing algorithms for specific graph classes, revealing it is generally hard but tractable in certain cases.

Contribution

It formally defines the Shortest Connection Game and analyzes its complexity, offering a polynomial-time algorithm for cactus graphs with simple paths.

Findings

Game is computationally hard on bipartite, acyclic, and cactus graphs.

Polynomial-time algorithm exists for cactus graphs with simple paths.

Complexity results guide efficient algorithm design for specific graph classes.

Abstract

We introduce Shortest Connection Game, a two-player game played on a directed graph with edge costs. Given two designated vertices in which they start, the players take turns in choosing edges emanating from the vertex they are currently located at. In this way, each of the players forms a path that origins from its respective starting vertex. The game ends as soon as the two paths meet, i.e., a connection between the players is established. Each player has to carry the cost of its chosen edges and thus aims at minimizing its own total cost. In this work we analyze the computational complexity of Shortest Connection Game. On the negative side, the game turns out to be computationally hard even on restricted graph classes such as bipartite, acyclic and cactus graphs. On the positive side, we can give a polynomial time algorithm for cactus graphs when the game is restricted to simple paths.