加权k-多数投票的批准差距
Approval Gap of Weighted k-Majority Tournaments
摘要：
有限顶点集$V$上的$k$ -多数竞赛图$T$由$V$上的$2k-1$线性阶的集合定义，如果$u>v$在大多数线性阶中，则边$u \to v$在$T$中。我们认为线性顺序作为选民的喜好和$T$的顶点作为候选人，边$u \to v$在$T$，如果大多数选民更喜欢候选人$u$候选人$v$。在本文中，我们介绍了加权$k$ -多数竞赛，与每个边缘$u \to v$加权的投票者喜欢$u$的数量。我们定义了最大批准差距$γ_w(T)$，一个衡量标准，通过该标准，$T$的任何支配集击败了下一个最受欢迎的候选人。这个参数是类似于以前的工作（未加权）$k$多数竞赛的最小支配集的大小。证明了对任意加权$k$，$k/2 \leq γ_w(T) \leq 2k-1$-多数竞赛图$T$，并对任意有理数$k/2 \leq q \leq 2k-1$，构造了具有$γ_w(T)=q$的竞赛图.我们还考虑了$k$ -多数竞赛图中的最小顶点数$m(q,k)$，其中$γ_w(T)=q$为$γ_w(T)=q$ .
英文摘要：
A $k$-majority tournament $T$ on a finite set of vertices $V$ is defined by a set of $2k-1$ linear orders on $V$, with an edge $u \to v$ in $T$ if $u>v$ in a majority of the linear orders. We think of the linear orders as voter preferences and the vertices of $T$ as candidates, with an edge $u \to v$ in $T$ if a majority of voters prefer candidate $u$ to candidate $v$. In this paper we introduce weighted $k$-majority tournaments, with each edge $u \to v$ weighted by the number of voters preferring $u$. We define the maximum approval gap $γ_w(T)$, a measure by which any dominating set of $T$ beats the next most popular candidate. This parameter is analogous to previous work on the size of minimum dominating sets of (unweighted) $k$-majority tournaments. We prove that $k/2 \leq γ_w(T) \leq 2k-1$ for any weighted $k$-majority tournament $T$, and construct tournaments with $γ_w(T)=q$ for any rational number $k/2 \leq q \leq 2k-1$. We also consider the minimum number of vertices $m(q,k)$ in a $k$-majority tournament with $γ_w(T)=q$.
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