Partitioned Combinatorial Optimization Games

2025-08-25

AI Summary

This paper investigates the computational complexity of core stability verification and existence in Partitioned Combinatorial Optimization Games (PCOGs), focusing on four classical graph optimization problems: Minimum Vertex Cover, Minimum Dominating Set, Minimum Spanning Tree, and Maximum Matching. We systematically embed these combinatorial optimization problems into a partitioned cooperative game framework, where the value of each coalition is defined as the optimal solution value on its induced subgraph. Leveraging game-theoretic modeling, combinatorial optimization analysis, and rigorous complexity-theoretic reductions, we precisely characterize the complexity boundaries—establishing NP-completeness, polynomial-time solvability, and other fine-grained classifications—for both core existence and core membership verification across all four problem classes. Our results provide a rigorous theoretical foundation and concrete algorithmic criteria for distributed collaborative decision-making in multi-agent systems governed by combinatorial resource constraints.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Game Theory and Economic Paradigms: Cooperative Game Theory
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Systems and Infrastructure for Web, Mobile and WoT: Virtualization and resource management in Web systems and infrastructures

Abstract

We propose a class of cooperative games, called d Partitioned Compbinatorial Optimization Games (PCOGs). The input of PCOG consists of a set of agents and a combinatorial structure (typically a graph) with a fixed optimization goal on this structure (e.g., finding a minimum dominating set on a graph) such that the structure is divided among the agents. The value of each coalition of agents is derived from the optimal solution for the part of the structure possessed by the coalition. We study two fundamental questions related to the core: Core Stability Verification and Core Stability Existence. We analyze the algorithmic complexity of both questions for four classic graph optimization tasks: minimum vertex cover, minimum dominating set, minimum spanning tree, and maximum matching.

Problem

• Analyzing core stability verification in partitioned combinatorial games
• Studying core stability existence for graph optimization problems
• Investigating algorithmic complexity for four classic optimization tasks

Innovation

• Partitioned combinatorial optimization games framework
• Coalition value derived from optimal substructure solutions
• Core stability analysis for four graph problems