Consider a committee election consisting of (i) a set of candidates who are divided into arbitrary groups each of size \emph{at most} two and a diversity constraint that stipulates the selection of \emph{at least} one candidate from each group and (ii) a set of voters who are divided into arbitrary populations each approving \emph{at most} two candidates and a representation constraint that stipulates the selection of \emph{at least} one candidate from each population who has a non-null set of approved candidates. The DiRe (Diverse + Representative) committee feasibility problem (a.k.a. the minimum vertex cover problem on unweighted undirected graphs) concerns the determination of the smallest size committee that satisfies the given constraints. Here, for this problem, we discover an unconditional deterministic polynomial-time algorithm that is an amalgamation of maximum matching, breadth-first search, maximal matching, and local minimization.

翻译：考虑一个委员会选举问题，其包含以下两部分：(i) 一组候选人，他们被任意划分为若干小组，每组最多两人，并设有多样性约束，要求从每个小组中至少选出一名候选人；(ii) 一组选民，他们被任意划分为若干群体，每个群体最多批准两名候选人，并设有代表性约束，要求从每个至少批准一名候选人的群体中至少选出一名候选人。DiRe（多样性与代表性）委员会可行性问题（即无权重无向图上的最小顶点覆盖问题）旨在确定满足给定约束的最小规模委员会。针对该问题，我们提出了一种无条件的确定性多项式时间算法，该算法融合了最大匹配、广度优先搜索、极大匹配及局部最小化策略。