Multi-objective unconstrained combinatorial optimization problems (MUCO) are in general hard to solve, i.e., the corresponding decision problem is NP-hard and the outcome set is intractable. In this paper we explore special cases of MUCO problems that are actually easy, i.e., solvable in polynomial time. More precisely, we show that MUCO problems with up to two ordinal objective functions plus one real-valued objective function are tractable, and that their complete nondominated set can be computed in polynomial time. For MUCO problems with one ordinal and a second ordinal or real-valued objective function we present an even more efficient algorithm that applies a greedy strategy multiple times.

翻译：多目标无约束组合优化问题（MUCO）通常难以求解，即对应的决策问题是NP难的且结果集难以处理。本文探讨了实际上易于求解的MUCO问题特例，即可在多项式时间内求解的情形。具体而言，我们证明了具有最多两个序数目标函数加上一个实值目标函数的MUCO问题是可处理的，并且其完整非支配集可在多项式时间内计算。对于包含一个序数目标函数及第二个序数或实值目标函数的MUCO问题，我们提出了一种更为高效的算法，该算法通过多次应用贪心策略实现求解。