二元代价拟阵上的双目标优化问题
Biobjective Optimization Problems on Matroids with Binary Costs
摘要：
与大多数多目标组合优化问题一样，拟阵上的双目标优化问题一般是难解的，其相应的决策问题一般是NP-难的。在本文中，我们考虑拟阵上的双目标优化问题，其中一个目标函数被限制为二元成本系数。我们表明，在这种情况下，该问题有一个连接的有效集相对于一个自然的定义的邻域结构，因此，可以有效地解决使用邻域搜索方法。据我们所知，这是拟阵的第一个非平凡问题，其中有效集的连通性可以建立。通过数值实验验证了理论结果与双目标最小生成树问题（图形拟阵）和双目标背包问题的基数约束（均匀拟阵）。在最小生成树问题的上下文中，将所有成本为0的边着色为绿色，将所有成本为1的边着色为红色，这将导致一个等价的问题，其中我们希望在帕累托意义上同时最小化一个一般目标和红色边的数量（定义第二个目标）。
英文摘要：
Like most multiobjective combinatorial optimization problems, biobjective optimization problems on matroids are in general intractable and their corresponding decision problems are in general NP-hard. In this paper, we consider biobjective optimization problems on matroids where one of the objective functions is restricted to binary cost coefficients. We show that in this case the problem has a connected efficient set with respect to a natural definition of a neighborhood structure and hence, can be solved efficiently using a neighborhood search approach. This is, to the best of our knowledge, the first non-trivial problem on matroids where connectedness of the efficient set can be established. The theoretical results are validated by numerical experiments with biobjective minimum spanning tree problems (graphic matroids) and with biobjective knapsack problems with a cardinality constraint (uniform matroids). In the context of the minimum spanning tree problem, coloring all edges with cost 0 green and all edges with cost 1 red leads to an equivalent problem where we want to simultaneously minimize one general objective and the number of red edges (which defines the second objective) in a Pareto sense.
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