Optimal Metric Distortion for Matching on the Line

TL;DR

The paper investigates metric distortion for matching on the line under ordinal preferences. It shows a deterministic ordinal algorithm achieves the optimal $k$-centrum distortion of $3$ for the one-sided problem across all $k$, and proves this bound is best possible; for the two-sided problem, ordinal information suffices to compute an exact optimal matching, with a structured query model further reducing information needs. Techniques hinge on representing matchings as permutation graphs and exploiting line-structure to derive tight bounds, including a reduction to graphs without forward edges. These results advance understanding of distortion in metric matching, highlight the special role of the line metric, and offer practical guidance for designing fair and efficient line-based matching mechanisms.

Abstract

We study the distortion of one-sided and two-sided matching problems on the line. In the one-sided case, $n$ agents need to be matched to $n$ items, and each agent's cost in a matching is their distance from the item they were matched to. We propose an algorithm that is provided only with ordinal information regarding the agents' preferences (each agent's ranking of the items from most- to least-preferred) and returns a matching aiming to minimize the social cost with respect to the agents' true (cardinal) costs. We prove that our algorithm simultaneously achieves the best-possible approximation of $3$ (known as distortion) with respect to a variety of social cost measures which include the utilitarian and egalitarian social cost. In the two-sided case, where the agents need be matched to $n$ other agents and both sides report their ordinal preferences over each other, we show that it is always possible to compute an optimal matching. In fact, we show that this optimal matching can be achieved using even less information, and we provide bounds regarding the sufficient number of queries.

Figures (4)

• Figure 1: The metrics used in the proof of \ref{['thm:lower']} to give a lower bound of $3$ on the distortion of (randomized) algorithms for (a) $k=1$ and (b) $k\geq 2$. Circles correspond to agents, rectangles correspond to items, and $i \in [n-1]$. The weight of each edge is the distance between its endpoints.
• Figure 2: A snapshot of a graph $P_M$ induced by some matching $M$, with its nodes appearing in the order of the corresponding agents' true locations on the line. The dashed lines below the nodes exhibit how these agents are partitioned into $A_{\text{out}}^\ell$, $A_1$, $A_2$, and $A_{\text{out}}^r$. A subset of the graph's edges appear above the nodes, labeled by their type (forward, backward, internal, and inward).
• Figure 3: The two metrics that establish the lower bound shown in \ref{['thm:fullprefLB']}.
• Figure 4: The metrics used in the proof of \ref{['obs:tiebreak']} to give a lower bound of 5 and 7 (depending on the value of $k$) on the distortion of \ref{['algname:order_match']} where in Step 1 of the algorithm $a_\ell$ and $a_r$ are chosen arbitrarily among agents whose favorite item is $g_\ell$ and $g_r$, respectively, rather than being chosen to maximize the size of $G_{\text{in}}$ for (a) $k=1$ and (b) $k \geq 2$. Circles correspond to agents, rectangles correspond to items, and for (a) $i \in \{2,\dots,n-2\}$ and (b) $i \in [n-2]$. The weight of each edge is the distance between its endpoints.

Theorems & Definitions (43)

• Theorem 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Theorem 3.1
• proof
• Remark 1
• Lemma 3.2