We study a class of combinatorial scheduling problems characterized by a particular type of constraint often associated with electrical power or gas energy. This constraint appears in several practical applications and is expressed as a sum of squares of linear functions. Its nonlinear nature adds complexity to the scheduling problem, rendering it notably challenging, even in the case of a linear objective. In fact, exact polynomial time algorithms are unlikely to exist, and thus, prior works have focused on designing approximation algorithms with polynomial running time and provable guarantees on the solution quality. In an effort to advance this line of research, we present novel approximation algorithms yielding significant improvements over the existing state-of-the-art results for these problems.

翻译：我们研究一类组合调度问题，其特点在于存在一种特定类型的约束，该约束通常与电力或天然气能源相关。此类约束出现在多个实际应用中，并以线性函数的平方和形式表示。其非线性特性增加了调度问题的复杂性，即使目标函数为线性，问题也极具挑战性。事实上，精确多项式时间算法几乎不可能存在，因此以往研究主要集中于设计具有多项式运行时间及可证明解质量保证的近似算法。为推动该研究方向，我们提出了新颖的近似算法，相较于现有最优结果，显著改进了这些问题的求解性能。