Bipartite instances of INFLUENCE

TL;DR

This paper studies the scoring game INFLUENCE, providing theoretical insights into its properties, complexity, and specific bipartite cases, including explicit strategies and bounds on scores for certain graph structures.

Contribution

It introduces new theoretical results for scoring games, applies them to INFLUENCE, and analyzes bipartite instances with explicit strategies and complexity results.

Findings

Score computation is PSPACE-complete.

Refined analysis for unions of segments.

Explicit strategies for bipartite graphs like grids and hypercubes.

Abstract

The game INFLUENCE is a scoring combinatorial game that has been introduced in 2020 by Duchene et al. It is a good representative of Milnor's universe of scoring games, i.e. games where it is never interesting for a player to miss his turn. New general results are first given for this universe, by transposing the notions of mean and temperature derived from non-scoring combinatorial games. Such results are then applied to INFLUENCE to refine the case of unions of segments. The computational complexity of the score of the game is also solved and proved to be PSPACE-complete. We finally focus on some specific cases of INFLUENCE when the graph is bipartite, by giving explicit strategies and bounds on the optimal score on structures like grids, hypercubes or torus.