The Degree of (Extended) Justified Representation and Its Optimization

Justified Representation (JR)/Extended Justified Representation (EJR) is a desirable axiom in multiwinner approval voting. In contrast to that (E)JR only requires at least \emph{one} voter to be represented in every cohesive group, we study its optimization version that maximizes the \emph{number} of represented voters in each group. Given an instance, we say a winning committee provides a JR degree (EJR degree, resp.) of $c$ if at least $c$ voters in each $\ell$-cohesive group ($1$-cohesive group, resp.) have approved $\ell$ ($1$, resp.) winning candidates. Hence, every (E)JR committee provides the (E)JR degree of at least $1$. Besides proposing this new property, we propose the optimization problem of finding a winning committee that achieves the maximum possible (E)JR degree, called \MDJR and \MDEJR, corresponding to JR and EJR respectively.We study the computational complexity and approximability of \MDJR of \MDEJR. An (E)JR committee, which can be found in polynomial time, straightforwardly gives a $(k/n)$-approximation. We also show that the original algorithms for finding a JR and an EJR winner committee are also $1/k$ and $1/(k+1)$ approximation algorithms for \MDJR and \MDEJR respectively. On the other hand, we show that it is NP-hard to approximate \MDJR and \MDEJR to within a factor of $\left(k/n\right)^{1-\epsilon}$ and to within a factor of $(1/k)^{1-\varepsilon}$, for any $\epsilon>0$, which complements the positive results. Next, we study the fixed-parameter-tractability of this problem. We show that both problems are W[2]-hard if $k$, the size of the winning committee, is specified as the parameter. However, when $c_{\text{max}}$, the maximum value of $c$ such that a committee that provides an (E)JR degree of $c$ exists, is additionally given as a parameter, we show that both \MDJR and \MDEJR are fixed-parameter-tractable.