Generating functions partitioning algorithm for computing power indices in weighted voting games

TL;DR

This paper introduces a novel algorithm combining existing methods to compute power indices in weighted voting games more efficiently, addressing open problems and analyzing complexities.

Contribution

It presents a new partitioning algorithm that improves the computation of power indices, solving an open problem and maintaining pseudopolynomial complexity for integer weights.

Findings

Algorithm has time complexity O(n 2^(n/2))

Pseudopolynomial complexity O(nq) for integer weights

Addresses open problem on computing Banzhaf indices efficiently

Abstract

In this paper new algorithm for calculating power indices is described. The complexity class of the problem is #P-complete and even calculating power index of the biggest player is NP-hard task. Constructed algorithm is a mix of ideas of two algorithms: Klinz & Woeginger partitioning algorithm and Mann & Shapley generating functions algorithm. Time and space complexities of the algorithm are analysed and compared with other known algorithms for the problem. Constructed algorithm has pessimistic time complexity O(n 2^(n/2))and pseudopolynomial complexity O(nq), where q is quota of the voting game. This paper also solves open problem stated by H. Aziz and M. Paterson - existence of the algorithm for calculating Banzhaf power indices of all players with time complexity lower than O(n^2 2^(n/2)). Not only is the answer positive but this can be done keeping the pseudopolynomial complexity of generating functions algorithm in case weights are integers. New open problems are stated.