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 (WMG) 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 WMG, to our knowledge, the fastest algorithm for computing all possible winners in the Schulze method uses a folklore reduction to the All-Pairs Bottleneck Paths (APBP) problem and runs in $O(m^{2.69})$ time, where $m$ 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 possible winners. If the input to the possible winners problem is not the WMG but the preference profile, then constructing the WMG is a bottleneck that increases the running time significantly; in the special case when there are $O(m)$ voters and candidates, the running time becomes $O(m^{2.69})$, or $O(m^{2.5})$ 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 WMG. We prove a similar connection between the so called Dominating Pairs problem and the problem of verifying whether a given candidate is a possible winner. Our paper is the first to bring fine-grained complexity into the field of computational social choice. Using it 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.

HappyDB: A Corpus of 100,000 Crowdsourced Happy Moments

The science of happiness is an area of positive psychology concerned with understanding what behaviors make people happy in a sustainable fashion. Recently, there has been interest in developing technologies that help incorporate the findings of the science of happiness into users' daily lives by steering them towards behaviors that increase happiness. With the goal of building technology that can understand how people express their happy moments in text, we crowd-sourced HappyDB, a corpus of 100,000 happy moments that we make publicly available. This paper describes HappyDB and its properties, and outlines several important NLP problems that can be studied with the help of the corpus. We also apply several state-of-the-art analysis techniques to analyze HappyDB. Our results demonstrate the need for deeper NLP techniques to be developed which makes HappyDB an exciting resource for follow-on research.