On some nonlinear assignment problems

Linear assignment problem (commonly referred to as just assignment problem) is a fundamental problem in combinatorial optimization. The goal is to assign n workers to do n jobs so that the linear sum of corresponding costs is minimized. The linear assignment problem is thoroughly studied and has a O(n^3) solution with Hungarian algorithm. Nevertheless, a wide range of applications involving assignments are naturally modeled with more complex objective functions (for example quadratic sum as in quadratic assignment problem), and are much more computationally challenging. In this thesis we discuss our results on the bilinear assignment problem, which generalizes the quadratic assignment problem, and is also motivated by several unique applications. The focus is on computational complexity, solvable special cases, approximations, linearizations as well as local search algorithms and other heuristic approaches for the problem. We also present our results on few applied projects, where modelling the underlying problem as a nonlinear assignment was instrumental.