Suppose that we have $n$ agents and $n$ items which lie in a shared metric space. We would like to match the agents to items such that the total distance from agents to their matched items is as small as possible. However, instead of having direct access to distances in the metric, we only have each agent's ranking of the items in order of distance. Given this limited information, what is the minimum possible worst-case approximation ratio (known as the distortion) that a matching mechanism can guarantee? Previous work by Caragiannis et al. proved that the (deterministic) Serial Dictatorship mechanism has distortion at most $2^n - 1$. We improve this by providing a simple deterministic mechanism that has distortion $O(n^2)$. We also provide the first nontrivial lower bound on this problem, showing that any matching mechanism (deterministic or randomized) must have worst-case distortion $\Omega(\log n)$. In addition to these new bounds, we show that a large class of truthful mechanisms derived from Deferred Acceptance all have worst-case distortion at least $2^n - 1$, and we find an intriguing connection between thin matchings (analogous to the well-known thin trees conjecture) and the distortion gap between deterministic and randomized mechanisms.

翻译：假设我们有 $n$ 个代理和 $n$ 个物品，它们位于同一个度量空间中。我们希望将代理匹配到物品，使得代理与其匹配物品之间的总距离尽可能小。然而，我们无法直接获取度量空间中的距离，只能得到每个代理关于物品距离的排序。在这种有限信息下，匹配机制所能保证的最小最坏情况近似比（即失真）是多少？先前Caragiannis等人的工作证明，（确定性）串行独裁机制的失真最多为 $2^n - 1$。我们通过提出一个具有 $O(n^2)$ 失真的简单确定性机制，改进了这一结果。同时，我们给出了该问题的首个非平凡下界，证明任何匹配机制（无论是确定性还是随机性）的最坏情况失真至少为 $\Omega(\log n)$。除了这些新界限，我们还证明了源自延迟接受机制的一大类诚实机制的最坏情况失真至少为 $2^n - 1$，并发现薄匹配（类似于著名的薄树猜想）与确定性和随机机制之间的失真差距之间存在有趣的联系。