[Paper Review] A General Framework for Computing the Nucleolus Via Dynamic Programming

Jochen Koenemann, Justin Toth|arXiv (Cornell University)|May 21, 2020

Game Theory and Voting Systems34 references4 citations

TL;DR

This paper presents a general dynamic programming framework for efficiently computing the nucleolus in cooperative games, showing that if the minimum excess coalition problem can be solved via an integral dynamic program, the nucleolus can be computed in polynomial time. The key contribution is a polynomial-time algorithm for computing the nucleolus of b-matching games on graphs with bounded treewidth, extending prior work beyond weighted voting games.

ABSTRACT

This paper defines a general class of cooperative games for which the nucleolus is efficiently computable. This class includes new members for which the complexity of computing their nucleolus was not previously known. We show that when the minimum excess coalition problem of a cooperative game can be formulated as a hypergraph dynamic program its nucleolus is efficiently computable. This gives a general technique for designing efficient algorithms for computing the nucleolus of a cooperative game. This technique is inspired by a recent result of Pashkovich (2018) on weighted voting games. However our technique significantly extends beyond the capabilities of previous work. We demonstrate this by applying it to give an algorithm for computing the nucleolus of b-matching games in polynomial time on graphs of bounded treewidth.

Motivation & Objective

• To develop a general framework for computing the nucleolus in cooperative games using dynamic programming.
• To extend previous results on weighted voting games to broader classes of combinatorial optimization games.
• To establish conditions under which the nucleolus is efficiently computable via dynamic programming with congruency constraints.
• To demonstrate the applicability of the framework to b-matching games on graphs of bounded treewidth.
• To formalize dynamic programming on directed acyclic hypergraphs to support congruency-constrained optimization.

Proposed method

• Formalize dynamic programming using directed acyclic hypergraphs to model feasible solutions and constraints.
• Introduce a technique to incorporate congruency constraints into dynamic programs with only a polynomial increase in complexity.
• Prove that adding congruency constraints to hypergraph-based dynamic programs increases complexity by at most a polynomial factor.
• Reduce nucleolus computation to solving the minimum excess coalition problem via dynamic programming.
• Construct a dynamic programming formulation for the minimum excess coalition in b-matching games on bounded-treewidth graphs.
• Apply the general framework to show that the nucleolus of such games is computable in polynomial time.

Experimental results

Research questions

• RQ1 Under what conditions can the nucleolus of a cooperative game be computed in polynomial time?
• RQ2 Can the dynamic programming approach for weighted voting games be generalized to other classes of cooperative games?
• RQ3 What is the computational impact of adding congruency constraints to dynamic programs in the context of cooperative game solution concepts?
• RQ4 Is the nucleolus of b-matching games on bounded-treewidth graphs computable in polynomial time?
• RQ5 Can the hypergraph-based dynamic programming model support complex constraints like those in b-matching games?

Key findings

• The nucleolus of any cooperative game can be computed in polynomial time if the minimum excess coalition problem is solvable via an integral dynamic program.
• Adding congruency constraints to dynamic programs on directed acyclic hypergraphs increases computational complexity by only a polynomial factor.
• The nucleolus of b-matching games on graphs with bounded treewidth is computable in polynomial time.
• The framework generalizes and significantly extends prior work on weighted voting games by Pashkovich.
• A dynamic programming formulation for the minimum excess coalition in b-matching games was successfully constructed using hypergraph-based dynamic programming with congruency constraints.
• The framework provides a new, systematic method for identifying tractable classes of cooperative games based on structural properties like treewidth.