The competitive equilibrium (CE) is a fundamental concept for markets, with applications to efficient and fair allocation of resources. We consider the fundamental question of computing a CE in the most basic Fisher market model where agents have linear preferences. While it is known that we can compute a CE in strongly polynomial time with linear goods, there is no known polynomial-time algorithm for CE with linear chores. Computing a CE for linear chores is known to be in CLS with no hardness results. In this paper, we prove that computing an exact CE in Fisher markets with divisible chores and linear disutilities is CLS-hard and thus CLS-complete. To the best of our knowledge, this is the first CLS-completeness result for a market equilibrium problem. On the technical side, we use the nonconvex dual Eisenberg-Gale (EG) program whose Karush-Kuhn-Tucker (KKT) points are CE [Chaudhury, Kroer, Mehta, and Nan, 2024], and the CLS-complete problem of computing a KKT point of a quadratic program [Fearnley, Goldberg, Hollender, and Savani, 2024]. We show that from any exact equilibrium of the constructed market, we can recover in polynomial time an $\varepsilon$-KKT point of the input quadratic program over the unit box.
Web platforms organize collective work, including content review and community maintenance, that imposes heterogeneous effort costs on participants. Fairly distributing these burdens while leaving as few tasks unassigned as possible is a natural objective for the design of such platforms. We study this through the maximin share (MMS) allocation problem with chore disposals under nonnegative additive costs. Our goal is to minimize the number of unassigned chores while guaranteeing every agent her MMS benchmark from the original instance. We prove that, for every instance with $n \ge 4$ agents, at most $n-3$ chore disposals suffice. We improve the bound of $2m/11+n$ disposals established by Barman et al. (2025), where $m$ is the number of chores, and eliminate the dependence on $m$. For four agents, we obtain a tight bound: one disposal always suffices, whereas known instances admit no complete MMS allocation (Ezra and Garbuz, 2026). We also prove the same result for goods: $n-3$ duplications suffice to ensure MMS.