Optimal Impartial Correspondences

TL;DR

This paper introduces mechanisms for selecting vertices in directed graphs to maximize minimum indegree while maintaining impartiality, establishing optimal bounds and trade-offs for the selection process.

Contribution

It provides the first mechanisms with provable guarantees for impartial vertex selection based on indegree, including optimal bounds and trade-offs.

Findings

Selected at most d+1 vertices for graphs with max outdegree d.

Selected vertices have indegree at least max indegree minus one.

Optimal trade-off between number of vertices and minimum indegree achieved.

Abstract

We study mechanisms that select a subset of the vertex set of a directed graph in order to maximize the minimum indegree of any selected vertex, subject to an impartiality constraint that the selection of a particular vertex is independent of the outgoing edges of that vertex. For graphs with maximum outdegree $d$, we give a mechanism that selects at most $d+1$ vertices and only selects vertices whose indegree is at least the maximum indegree in the graph minus one. We then show that this is best possible in the sense that no impartial mechanism can only select vertices with maximum degree, even without any restriction on the number of selected vertices. We finally obtain the following trade-off between the maximum number of vertices selected and the minimum indegree of any selected vertex: when selecting at most~$k$ vertices out of $n$, it is possible to only select vertices whose indegree is at least the maximum indegree minus $\lfloor(n-2)/(k-1)\rfloor+1$.