For up to four valuation types, a complete 2/3-EFX allocation always exists; for k types, a (1-ε)-EFX allocation with O~(sqrt(k/ε)) charity exists.
Almost Full EFX Exists for Four Agents
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Almost and Approximate EFX for Few Types of Agents
For up to four valuation types, a complete 2/3-EFX allocation always exists; for k types, a (1-ε)-EFX allocation with O~(sqrt(k/ε)) charity exists.