An Outline of Inverse Combinatorial Optimization Problems

In this chapter, we introduce a unified framework for Inverse Combinatorial Optimization Problems (ICOPs), encompassing a spectrum of problem types including traditional ICOPs, inverse optimal values, partial inverse problems, network improvement, and interdiction problems. This framework elucidates the intricate relationships among different problem classes and examines several polynomial-time-solvable combinatorial optimization problems such as shortest path and minimum spanning tree. We propose a standardized format to delineate the attributes of ICOPs, identifying open questions and exploring the complexity dynamics between forward and inverse problems. We question whether an inverse problem of an N P $$\mathcal {N}\mathcal {P}$$ -hard problem can be polynomially solvable, an unresolved challenge. The chapter highlights the broad applicability of inverse optimization in healthcare, transportation, logistics, and energy systems, emphasizing its role in decision-making and data analytics, with the potential to refine models and predict outcomes in complex systems.