最优度量失真猜想的求解
Resolving the Optimal Metric Distortion Conjecture
摘要：
我们研究了以下度量失真问题：有两个有限的点集，$V$和$C$，位于同一度量空间，我们的目标是选择一个点在$C$的总距离$V$中的点是尽可能小的。然而，不是访问底层距离度量，而是对于$V$中的每个点，我们仅知道其到$C$中的点的距离的排名。我们提出的算法，选择一个点$C$只使用这些排名作为输入，我们提供了他们的\emph{distortion}（最坏情况下的近似比）的界限。这个问题的一个突出动机来自投票理论，其中$V$表示一组投票者，$C$表示一组候选人，并且排名对应于投票者的顺序偏好。在这个框架中的一个主要猜想是，最佳的确定性算法有失真$3$。我们解决这个猜想提供一个多项式时间算法，实现失真$3$，匹配一个已知的下限。我们这样做，证明了一个新的引理匹配选民的候选人，我们称之为\emph{ranking-matching lemma}。这个引理诱导一个家庭的新算法，这可能是独立的利益，我们表明，在这个家庭中的一个特殊的算法实现失真$3$。我们还提供了更精细的，参数化的，边界使用的概念$α$ -决策，量化的程度，选民可能更喜欢她的首选相对于所有其他人。最后，我们引入了一个新的随机算法，与已知的结果相比，改进的失真，也提供了改进的下界的所有确定性和随机算法的失真。
英文摘要：
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 $α$-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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