In this paper, we study several graph optimization problems in which the weights of vertices or edges are variables determined by several linear constraints, including maximum matching problem under linear constraints (max-MLC), minimum perfect matching problem under linear constraints (min-PMLC), shortest path problem under linear constraints (SPLC) and vertex cover problem under linear constraints (VCLC). The objective of these problems is to decide the weights that are feasible to the linear constraints, and find the optimal solutions of corresponding graph optimization problems among all feasible choices of weights. We find that these problems are NP-hard and are hard to be approximated in general. These findings suggest us to explore various special cases of them. In particular, we show that when the number of constraints is a fixed constant, all these problems are polynomially solvable. Moreover, if the total number of distinct weights is a fixed constant, then max-MLC, min-PMLC and SPLC are polynomially solvable, and VCLC has a 2-approximation algorithm. In addition, we propose approximation algorithms for various cases of max-MLC.

在本文中，我们研究了几个图优化问题，其中顶点或边的权重是由多个线性约束确定的变量，包括线性约束下的最大匹配问题（max-MLC），线性约束下的最小完美匹配问题（min-PMLC） ），线性约束下的最短路径问题（SPLC）和线性约束下的顶点覆盖问题（VCLC）。这些问题的目的是确定对线性约束可行的权重，并在所有可行的权重选择中找到相应图优化问题的最优解。我们发现这些问题是NP难题，通常很难被近似。这些发现建议我们探索它们的各种特殊情况。特别是，我们表明，当约束的数量为固定常数时，所有这些问题都是多项式可解决的。此外，如果不同权重的总数是一个固定常数，则max-MLC，min-PMLC和SPLC是多项式可解的，并且VCLC具有2近似算法。另外，我们针对max-MLC的各种情况提出了近似算法。