Gehrlein stability in committee selection: parameterized hardness and algorithms

In a multiwinner election based on the Condorcet criterion, we are given a set of candidates, and a set of voters with strict preference rankings over the candidates. A committee is weakly Gehrlein stable (WGS) if each committee member is preferred to each non-member by at least half of the voters. Recently, Aziz et al. [IJCAI 2017] studied the computational complexity of finding a WGS committee of size k. They show that this problem is NP-hard in general and polynomial-time solvable when the number of voters is odd. In this article, we initiate a systematic study of the problem in the realm of parameterized complexity. We first show that the problem is W[1]-hard when parameterized by the size of the committee. To overcome this intractability result, we use a known reformulation of WGS as a problem on directed graphs and then use parameters that measure the “structure” of these directed graphs. We show that the problem is fixed parameter tractable and admits linear kernels with respect to these parameters; and also present an exact-exponential time algorithm with running in time O(1. 2207 ⁿn^{O ( 1 )}) , where n denotes the number of candidates.