3 research outputs found
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 winners in the Schulze method uses a folklore reduction to the
All-Pairs Bottleneck Paths problem and runs in time, where 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. If the input to the Schulze 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
candidates and voters, the running time is , or
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 finding a winner in the Schulze method.
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.Comment: 19 pages, 2 algorithms, 2 tables. A previous version of this work
appears in EC 2021. In this version we strengthen Theorem 6.2 which now holds
also for the problem of finding a Schulze winne
Multi-Winner Voting with Approval Preferences
Approval-based committee (ABC) rules are voting rules that output a
fixed-size subset of candidates, a so-called committee. ABC rules select
committees based on dichotomous preferences, i.e., a voter either approves or
disapproves a candidate. This simple type of preferences makes ABC rules widely
suitable for practical use. In this book, we summarize the current
understanding of ABC rules from the viewpoint of computational social choice.
The main focus is on axiomatic analysis, algorithmic results, and relevant
applications.Comment: This is a draft of the upcoming book "Multi-Winner Voting with
Approval Preferences
The Schulze Method of Voting
We propose a new single-winner election method ("Schulze method") and prove
that it satisfies many academic criteria (e.g. monotonicity, reversal symmetry,
resolvability, independence of clones, Condorcet criterion, k-consistency,
polynomial runtime). We then generalize this method to proportional
representation by the single transferable vote ("Schulze STV") and to methods
to calculate a proportional ranking ("Schulze proportional ranking").
Furthermore, we propose a generalization of the Condorcet criterion to
multi-winner elections. This paper contains a large number of examples to
illustrate the proposed methods