More Efforts Towards Fixed-Parameter Approximability of Multiwinner Rules

Multiwinner Elections have emerged as a prominent area of research with numerous practical applications. We contribute to this area by designing parameterized approximation algorithms and also resolving an open question by Yang and Wang [AAMAS'18]. More formally, given a set of candidates, \mathcal{C}, a set of voters,\mathcal{V}, approving a subset of candidates (called approval set of a voter), and an integer $k$, we consider the problem of selecting a ``good'' committee using Thiele rules. This problem is computationally challenging for most Thiele rules with monotone submodular satisfaction functions, as there is no (1-\frac{1}{e}-ε)\footnote{Here, $e$ denotes the base of the natural logarithm.}-approximation algorithm in f(k)(|\mathcal{C}| + |\mathcal{V}|)^{o(k)} time for any fixed $ε> 0$ and any computable function $f$, and no {\sf PTAS} even when the length of approval set is two. Skowron [WINE'16] designed an approximation scheme running in FPT time parameterized by the combined parameter, size of the approval set and $k$. In this paper, we consider a parameter $d+k$ (no $d$ voters approve the same set of $d$ candidates), where $d$ is upper bounded by the size of the approval set (thus, can be much smaller). With respect to this parameter, we design parameterized approximation schemes, a lossy polynomial-time preprocessing method, and show that an extra committee member suffices to achieve the desired score (i.e., $1$-additive approximation). Additionally, we resolve an open question by Yang and Wang~[AAMAS'18] regarding the fixed-parameter tractability of the problem under the PAV rule with the total score as the parameter, demonstrating that it admits an FPT algorithm.