We introduce and study the combinatorial optimization problem with interaction costs (COPIC). COPIC is the problem of finding two combinatorial structures, one from each of two given families, such that the sum of their independent linear costs and the interaction costs between elements of the two selected structures is minimized. COPIC generalizes the quadratic assignment problem and many other well studied combinatorial optimization problems, and hence covers many real world applications. We show how various topics from different areas in the literature can be formulated as special cases of COPIC. The main contributions of this paper are results on the computational complexity and approximability of COPIC for different families of combinatorial structures (e.g. spanning trees, paths, matroids), and special structures of the interaction costs. More specifically, we analyze the complexity if the interaction cost matrix is parameterized by its rank and if it is a diagonal matrix. Also, we determine the structure of the intersection cost matrix, such that COPIC is equivalent to independently solving linear optimization problems for the two given families of combinatorial structures.

我们介绍和研究具有交互成本的组合优化问题（COPIC）。COPIC的问题是找到两个组合结构，一个来自两个给定族中的每个族，这样它们的独立线性成本和两个选定结构的元素之间的相互作用成本之和就最小了。COPIC概括了二次分配问题和许多其他经过充分研究的组合优化问题，因此涵盖了许多实际应用。我们展示了如何将文献中不同领域的各种主题表述为COPIC的特例。本文的主要贡献是针对不同组合结构族（例如生成树，路径，拟阵）以及交互成本的特殊结构的COPIC的计算复杂性和可近似性的结果。更具体地，如果交互成本矩阵由其等级参数化并且如果它是对角矩阵，则我们分析复杂度。此外，我们确定交叉点成本矩阵的结构，以使COPIC等效于为两个给定的组合结构族独立解决线性优化问题。