Fine-Grained Complexity and Algorithms for the Schulze Voting Method

We study computational aspects of a well-known single-winner voting rule called the Schulze method [Schulze, 2003] which is used broadly in practice. In this method the voters give (weak) ordinal preference ballots which are used to define the weighted majority graph of direct comparisons between pairs of candidates. The choice of the winner comes from indirect comparisons in the graph, and more specifically from considering directed paths instead of direct comparisons between candidates. When the input is the weighted majority graph, to our knowledge, the fastest algorithm for computing all winners in the Schulze method uses a folklore reduction to the All-Pairs Bottleneck Paths (APBP) problem and runs in O(m2.69) time, wherem is the number of candidates. It is an interesting open question whether this can be improved. Our first result is a combinatorial algorithm with a nearly quadratic running time for computing all winners. This running time is essentially optimal as it is nearly linear in the size of the weighted majority graph. If the input to the Schulze winners problem is not the weighted majority graph but the preference profile, then constructing the weighted majority graph is a bottleneck that increases the running time significantly; in the special case when there arem candidates and n = O(m) voters, the running time is O(m2.69), or O(m2.5) if there is a nearly-linear time algorithm for multiplying dense square matrices. To address this bottleneck, we prove a formal equivalence between the well-studied Dominance Product problem and the problem of computing the weighted majority graph. As the Dominance Product problem is believed to require at least time r 2.5-o(1) on r r matrices, our equivalence implies that constructing the weighted majority graph in O(m2.499) time form candidates and n = O(m) voters would imply a breakthrough in the study of intermediate problems [Lincoln et al., 2020] in fine-grained complexity. We prove a similar connection between the so called Dominating Pairs problem and the problem of verifying whether a given candidate is a winner. Our paper is the first to bring fine-grained complexity into the field of computational social choice. Previous approaches say nothing about lower bounds for problems that already have polynomial time algorithms. By bringing fine-grained complexity into the picture we can identify voting protocols that are unlikely to be practical for large numbers of candidates and/or voters, as their complexity is likely, say at least cubic.