Computing the Yolk in Spatial Voting Games without Computing Median Lines

TL;DR

This paper introduces near-linear time algorithms for computing the yolk in planar spatial voting games without needing median lines, leveraging Megiddo's parametric search to surpass previous computational bounds.

Contribution

The authors develop the first algorithm that computes the yolk without precomputing median lines, breaking the $O(n^{4/3})$ complexity barrier.

Findings

Achieved near-linear time complexity for yolk computation.

First algorithm avoiding median line precomputation.

Potential applicability to other spatial voting problems.

Abstract

The yolk is an important concept in spatial voting games as it generalises the equilibrium and provides bounds on the uncovered set. We present near-linear time algorithms for computing the yolk in the spatial voting model in the plane. To the best of our knowledge our algorithm is the first algorithm that does not require precomputing the median lines and hence able to break the existing $O(n^{4/3})$ bound which equals the known upper bound on the number of median lines. We avoid this requirement by using Megiddo's parametric search, which is a powerful framework that could lead to faster algorithms for many other spatial voting problems.