Jannes Malanowski | Searcharxiv

arXiv2026

Cost-Free Neutrality for the River Method

Recently, the River Method was introduced as novel refinement of the Split Cycle voting rule. The decision-making process of River is closely related to the well established Ranked Pairs Method. Both methods consider a margin graph computed from the voters' preferences and eliminate majority cycles in that graph to choose a winner. As ties can occur in the margin graph, a tiebreaker is required along with the preferences. While such a tiebreaker makes the computation efficient, it compromises the fundamental property of neutrality: the voting rule should not favor alternatives in advance. One way to reintroduce neutrality is to use Parallel-Universe Tiebreaking (PUT), where each alternative is a winner if it wins according to any possible tiebreaker. Unfortunately, computing the winners selected by Ranked Pairs with PUT is NP-complete. Given the similarity of River to Ranked Pairs, one might expect River to suffer from the same complexity. Surprisingly, we show the opposite: We present a polynomial-time algorithm for computing River winners with PUT, highlighting significant structural advantages of River over Ranked Pairs. Our Fused-Universe (FUN) algorithm simulates River for every possible tiebreaking in one pass. From the resulting FUN diagram one can then directly read off both the set of winners and, for each winner, a certificate that explains how this alternative dominates the others.

cs.DS↗

arXiv2025

Conquering the Multiverse: The River Voting Method with Efficient Parallel Universe Tiebreaking

Democracy relies on making collective decisions through voting. In addition, voting procedures have further applications, for example in the training of artificial intelligence. An essential criterion for determining the winner of a fair election is that all alternatives are treated equally: this is called neutrality. The established Ranked Pairs voting method cannot simultaneously guarantee neutrality and be computationally tractable for election with ties. River, the recently introduced voting method, shares desirable properties with Ranked Pairs and has further advantages, such as a new property related to resistance against manipulation. Both Ranked Pairs and River use a weighted margin graph to model the election. Ties in the election can lead to edges of equal margin. To order the edges in such a case, a tiebreaking scheme must be employed. Many tiebreaks violate neutrality or other important properties. A tiebreaking scheme that preserves neutrality is Parallel Universe Tiebreaking (PUT). Ranked Pairs with PUT is NP-hard to compute. The main result of this thesis shows that River with PUT can be computed in polynomial worst-case runtime: We can check whether an alternative is a River PUT winner, by running River with a specially constructed ordering of the edges. To construct this ordering, we introduce the semi-River diagram which contains the edges that can appear in any River diagram for some arbitrary tiebreak. On this diagram we can compute the River winners, by applying a variant of Prims algorithm per alternative. Additionally, we give an algorithm improve the previous naive runtime of River from $\mathcal{O}(n^4)$ to $\mathcal{O}(n^2 \log n)$, where n is the number of alternatives.

cs.DS↗