We study the equilibrium computation problem in the Fisher market model with constrained piecewise linear concave (PLC) utilities. This general class captures many well-studied special cases, including markets with PLC utilities, markets with satiation, and matching markets. For the special case of PLC utilities, although the problem is PPAD-hard, Devanur and Kannan (FOCS 2008) gave a polynomial-time algorithm when the number of items is constant. Our main result is a fixed parameter approximation scheme for computing an approximate equilibrium, where the parameters are the number of agents and the approximation accuracy. This provides an answer to an open question by Devanur and Kannan for PLC utilities, and gives a simpler and faster algorithm for matching markets as the one by Alaei, Jalaly and Tardos (EC 2017). The main technical idea is to work with the stronger concept of thrifty equilibria, and approximating the input utility functions by `robust' utilities that have favorable marginal properties. With some restrictions, the results also extend to the Arrow--Debreu exchange market model.

翻译：我们研究渔业市场模型中的平衡计算问题,使用有限的片段线性线性电流公用设施(PLC),这一普通类别包含许多研究良好的特殊案例,包括PLC公用设施市场、饱和市场和匹配市场的市场。对于PLC公用设施的特殊案例,尽管问题是PPAD-hard、Devanur和Kannan(FOCS2008)在项目数量不变时给出了多元时间算法。我们的主要结果是为计算近似平衡而制定固定的参数近似计划,其参数是代理商的数量和近似准确性。这为Devanur和Kannan对PLC公用设施的公开问题提供了答案,为将市场与Alaei、Jalary和Tardos(EC 2017)的市场匹配提供了更简单、更快捷的算法。主要的技术理念是,在项目数量不变时采用更强的节能电子平衡概念,并采用“robust”公用设施对输入公用设施功能进行近似边缘特性,结果也有一定的限制,结果还延伸到箭头-Debreueur-Dereu交换市场模式。