Strengthening Full Justified Representation in Approval-Based Committee Elections

In approval-based committee elections, _proportional representation_ aims to give voter groups representation in proportion to their size. Full justified representation (FJR) provides a strong guarantee, but checking whether a committee satisfies it is computationally hard, even though such committees can be found efficiently. We introduce full justified representation+ (FJR+), which strengthens both FJR and extended justified representation+ (EJR+) and can always be achieved and verified in polynomial time. The key idea is to allow fractional weights when assessing underrepresentation, making violations detectable through linear programming. A greedy procedure based on equal voter budgets selects a partial committee that can be completed in any way while retaining FJR+. We use this flexibility to obtain two further guarantees. The final committee lies in the sub-core: no group can use its proportional share to give every member all their approved winners and an additional approved candidate. Whenever enough candidates receive approval to fill the committee, it is also priceable, which gives an equal-budget payment interpretation. Finally, we extend the approach to approval-based participatory budgeting with arbitrary project costs, retaining efficient verification and computation together with priceability and the sub-core.