一句话：为 valued promise CSP 建立代数框架，用 valued minion 同态统一导出近似归约与不可近似性。
本文研究有值 promise CSP（valued promise CSP）的近似复杂度，旨在为这类约束满足问题建立类似代数方法的统一框架。作者为每个有值 promise CSP 关联一个代数对象——valued minion，并证明若两个关联的 valued minion 之间存在同态，则原 CSP 之间存在多项式时间归约。该一般归约定理可推出若干重要不可近似性结果，例如几乎可解的线性方程组在超过随机赋值阈值后不可近似。工作以代数同态刻画归约，为 promise CSP 的近似下界提供了通用工具。
Following the success of the so-called algebraic approach to the study of decision constraint satisfaction problems (CSPs), exact optimization of valued CSPs, and most recently promise CSPs, we propose an algebraic framework for valued promise CSPs. To every valued promise CSP we associate an algebraic object, its so-called valued minion. Our main result shows that the existence of a homomorphism between the associated valued minions implies a polynomial-time reduction between the original CSPs. We also show that this general reduction theorem includes important inapproximability results, for instance, the inapproximability of almost solvable systems of linear equations beyond the random assignment threshold.