Combinatorial Optimization Problems with Interaction Costs: Complexity and Solvable Cases

TL;DR

This paper introduces COPIC, a general combinatorial optimization problem involving interaction costs, analyzing its complexity and solvable cases across various structures, with implications for real-world applications.

Contribution

It formalizes COPIC, explores its complexity for different structure families, and identifies special cases where it reduces to simpler linear problems.

Findings

COPIC generalizes many classical problems like quadratic assignment.

Complexity varies with interaction cost matrix structure.

Certain matrix structures allow polynomial-time solutions.

Abstract

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.