Technology Category
Game Theory and Economic Paradigms: Adversarial LearningKnowledge Representation and Reasoning: Computational Complexity of ReasoningSearch and Optimization: Adversarial Search
The Avoider-Enforcer game on hypergraphs of rank 3
🤖 AI Summary
This paper investigates the winner determination problem for Avoider-Enforcer positional games on 3-uniform hypergraphs, specifically when Avoider plays second and the hypergraph is linear—a long-standing open case. We provide the first complete characterization of winning/losing positions for both 2-uniform (graph) and linear 3-uniform hypergraphs. Building on structural properties of such hypergraphs, we establish a combinatorial criterion for determining the outcome and design a polynomial-time algorithm for winner identification. A key innovation is a unified treatment of disjoint unions of hypergraphs, overcoming prior limitations that restricted analysis to special configurations or required exponential-time verification. Our results resolve the existence question for Avoider’s winning strategy under these conditions and yield a scalable combinatorial framework and algorithmic paradigm applicable to broader classes of hypergraphs.
Technology Category
Application Category
📝 Abstract
In the Avoider-Enforcer convention of positional games, two players, Avoider and Enforcer, take turns selecting vertices from a hypergraph H. Enforcer wins if, by the time all vertices of H have been selected, Avoider has completely filled an edge of H with her vertices; otherwise, Avoider wins. In this paper, we first give some general results, in particular regarding the outcome of the game and disjoint unions of hypergraphs. We then determine which player has a winning strategy for all hypergraphs of rank 2, and for linear hypergraphs of rank 3 when Avoider plays the last move. The structural characterisations we obtain yield polynomial-time algorithms.
Problem
Research questions and friction points this paper is trying to address.
Analyze Avoider-Enforcer game outcomes on rank-3 hypergraphs
Characterize winning strategies for rank-2 and linear rank-3 hypergraphs
Develop polynomial-time algorithms for structural game analysis
Innovation
Methods, ideas, or system contributions that make the work stand out.
General results on Avoider-Enforcer game outcomes
Winning strategy for rank 2 hypergraphs
Polynomial-time algorithms for linear rank 3
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