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原文摘要

We resolve the central open question in the theory of approval-based committee elections: we prove that the core is always non-empty. This means that there always exists a committee for which no coalition of voters can deviate to an alternative committee of proportional size that every member strictly prefers. This property was introduced by Aziz et al. (2017) to capture proportional representation and coalitional stability, but no known algorithm achieved it except up to a constant approximation factor or in special cases. Our approach begins by strengthening the core to core+, which requires the committee to be robust even to fractional deviations. Using LP duality, this property can equivalently be defined in terms of a payment-based certificate under which no losing candidate gathers too much support. We then define a voting rule that chooses a committee admitting a certificate that maximizes a novel harmonic entropy objective function, which rewards balanced payment distributions. We show that the payment system associated with an optimal committee certifies that the committee satisfies core+, establishing existence. The same is true for local optima, which allows us to design a polynomial-time algorithm computing a core+ committee.