Balanced partitions exist that are O(max{√Δ, k²} ln n)-approximately envy-free and (k + o(k))-approximately core, with efficient computation under relaxed balance and stronger (1.618 + o(1))-core for k=2.
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A new quantile valuation model for indivisible goods is defined and used to derive complexity classifications plus approximation and exact algorithms for utilitarian and egalitarian welfare maximization under balanced vs. unbalanced constraints.
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Some Improved Results on Fair and Balanced Graph Partitions
Balanced partitions exist that are O(max{√Δ, k²} ln n)-approximately envy-free and (k + o(k))-approximately core, with efficient computation under relaxed balance and stronger (1.618 + o(1))-core for k=2.