Optimal universal PMMS approximation guarantees

Determine the optimal universal approximation guarantees for pairwise maximin share (PMMS) allocations under additive goods and additive chores, closing the gaps between the known bounds 0.7808 ≤ α_PMMS ≤ 78/79 for goods and 1.102065 ≤ ρ_PMMS ≤ 4/3 for chores.

Background

The paper establishes that exact PMMS allocations need not exist for both additive goods and additive chores. For additive goods, the paper records a lower bound of 0.7808 from prior work and an upper bound of 78/79 from concurrent work. For additive chores, it proves a lower bound of 1.102065 through an explicit computer-verified instance and an upper bound of 4/3 by constructing an approximate-PMMS allocation for every instance.

The universal approximation factors therefore remain quantitatively unresolved in both settings. Determining their exact values would close the gaps between the stated impossibility and existence guarantees.