We study the problem of computing approximate market equilibria in Fisher markets with separable piecewise-linear concave (SPLC) utility functions. In this setting, the problem was only known to be PPAD-complete for inverse-polynomial approximations. We strengthen this result by showing PPAD-hardness for constant approximations. This means that the problem does not admit a polynomial time approximation scheme (PTAS) unless PPAD$=$P. In fact, we prove that computing any approximation better than $1/11$ is PPAD-complete. As a direct byproduct of our main result, we get the same inapproximability bound for Arrow-Debreu exchange markets with SPLC utility functions.

翻译：我们研究了在具有可分离分段线性凹（SPLC）效用函数的Fisher市场中计算近似市场均衡的问题。在该设定下，此前仅已知该问题对于反多项式近似是PPAD完全的。我们通过证明该问题对于常数近似也是PPAD难的，强化了这一结果。这意味着该问题不存在多项式时间近似方案（PTAS），除非PPAD$=$P。事实上，我们证明了计算任何优于$1/11$的近似都是PPAD完全的。作为我们主要结果的直接副产品，我们得到了具有SPLC效用函数的Arrow-Debreu交换市场中相同的不可近似性界限。