Resolving the Optimal Metric Distortion Conjecture

• URL: http://arxiv.org/abs/2004.07447v2
• Date: Mon, 7 Sep 2020 16:52:59 GMT
• Title: Resolving the Optimal Metric Distortion Conjecture
• Authors: Vasilis Gkatzelis, Daniel Halpern, and Nisarg Shah
• Abstract summary: We study the following metric distortion problem. There are two finite sets of points, $V$ and $C$, that lie in the same metric space. We propose algorithms that choose a point in $C$ using only these rankings as input.
• Score: 27.344649091365067
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We study the following metric distortion problem: there are two finite sets of points, $V$ and $C$, that lie in the same metric space, and our goal is to choose a point in $C$ whose total distance from the points in $V$ is as small as possible. However, rather than having access to the underlying distance metric, we only know, for each point in $V$, a ranking of its distances to the points in $C$. We propose algorithms that choose a point in $C$ using only these rankings as input and we provide bounds on their \emph{distortion} (worst-case approximation ratio). A prominent motivation for this problem comes from voting theory, where $V$ represents a set of voters, $C$ represents a set of candidates, and the rankings correspond to ordinal preferences of the voters. A major conjecture in this framework is that the optimal deterministic algorithm has distortion $3$. We resolve this conjecture by providing a polynomial-time algorithm that achieves distortion $3$, matching a known lower bound. We do so by proving a novel lemma about matching voters to candidates, which we refer to as the \emph{ranking-matching lemma}. This lemma induces a family of novel algorithms, which may be of independent interest, and we show that a special algorithm in this family achieves distortion $3$. We also provide more refined, parameterized, bounds using the notion of $\alpha$-decisiveness, which quantifies the extent to which a voter may prefer her top choice relative to all others. Finally, we introduce a new randomized algorithm with improved distortion compared to known results, and also provide improved lower bounds on the distortion of all deterministic and randomized algorithms.

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