最优公平对应
Optimal Impartial Correspondences
摘要：
我们研究的机制，选择一个有向图的顶点集的子集，以最大限度地提高任何选定的顶点的最小入度，受到公正性约束，一个特定的顶点的选择是独立的，该顶点的传出边缘。对于具有最大出度$d$的图，我们给出了一个至多选择$d+1$顶点且只选择入度至少为图的最大入度减1的顶点的机制.然后，我们表明，这是最好的可能性，在这个意义上，没有公正的机制只能选择顶点的最大程度，即使没有任何限制所选择的顶点的数量。我们最终得到了在所选顶点的最大数目和所选顶点的最小入度之间的以下折衷：当从$n$中选择至多~ $k$顶点时，可以仅选择入度至少为最大入度减去$\lfloor(n-2)/(k-1)\rfloor+1$的顶点。
英文摘要：
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 $\lfloor(n-2)/(k-1)\rfloor+1$.
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