Improving the Bounds on Approximating Weighted Maximin Share for Chores

We investigate the problem of allocating indivisible chores among agents with arbitrary positive weights and additive nonnegative cost functions. An allocation is said to be $\alpha$-weighted-maximin-share (WMMS) fair if the cost incurred by each agent is at most $\alpha$ times her weighted maximin share. We establish that every instance admits a $4.124234528$-WMMS allocation. Moreover, for instances in which all weights and costs are rational numbers represented in binary, we give an algorithm that computes a $4.12424$-WMMS allocation in time polynomial in the number of agents, the number of chores, and the total binary encoding length of the input. Complementing these upper bounds, we construct a family of instances with strictly positive, identically ordered costs for which the optimal WMMS approximation factors converge to $e$. Together, our results substantially narrow the previously best-known gap of $[2,12]$ established by Li et al.~\cite{LWX26} to $[e,4.12424]$.