On Efficient Computation of DiRe Committees

Consider a committee election consisting of (i) a set of candidates who are divided into arbitrary groups each of size ${at~most}$ two and a diversity constraint that stipulates the selection of ${at~least}$ one candidate from each group and (ii) a set of voters who are divided into arbitrary populations each approving ${at~most}$ two candidates and a representation constraint that stipulates the selection of ${at~least}$ 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 propose an algorithm that is an amalgamation of maximum matching, breadth-first search, maximal matching, and local minimization. We prove the algorithm terminates in polynomial-time. We conjecture the algorithm is an unconditional deterministic polynomial-time algorithm.