Improved Approximate EFX Guarantees for Multigraphs

In recent years, a new line of work in fair allocation has focused on EFX allocations for $(p,q)$-bounded valuations, where each good is relevant to at most $p$ agents, and any pair of agents share at most $q$ relevant goods. For the case $p=2$ and $q=\mathrm{\infty }$, such instances can be equivalently represented as multigraphs whose vertices are the agents and whose edges represent goods, each edge incident to exactly the one or two agents for whom the good is relevant. A recent result of \citet{amanatidis2024pushing} shows that for additive $(2,\mathrm{\infty })$ bounded valuations, a $(\text{nicefrac}23)$-EFX allocation always exists. In this paper, we improve this bound by proving the existence of a $(\text{nicefrac}1\sqrt{2})$-$\text{efx}$ allocation for additive $(2,\mathrm{\infty })$-bounded valuations.