Inverse Submodular Maximization with Application to Human-in-the-Loop Multi-Robot Multi-Objective Coverage Control

We consider a new type of inverse combinatorial optimization, Inverse Submodular Maximization (ISM), for human-in-the-loop multi-robot coordination. Forward combinatorial optimization, defined as the process of solving a combinatorial problem given the reward (cost)-related parameters, is widely used in multi-robot coordination. In the standard pipeline, the reward (cost)-related parameters are designed offline by domain experts first and then these parameters are utilized for coordinating robots online. What if we need to change these parameters by non-expert human supervisors who watch over the robots during tasks to adapt to some new requirements? We are interested in the case where human supervisors can suggest what actions to take, and the robots need to change the internal parameters based on such suggestions. We study such problems from the perspective of inverse combinatorial optimization, i.e., the process of finding parameters given solutions to the problem. Specifically, we propose a new formulation for ISM, in which we aim to find a new set of parameters that minimally deviate from the current parameters and can make the greedy algorithm output actions the same as those suggested by humans. We show that such problems can be formulated as a Mixed Integer Quadratic Program (MIQP). However, MIQP involves exponentially many binary variables, making it intractable for the existing solver when the problem size is large. We propose a new algorithm under the Branch $\&$ Bound paradigm to solve such problems. In numerical simulations, we demonstrate how to use ISM in multi-robot multi-objective coverage control, and we show that the proposed algorithm achieves significant advantages in running time and peak memory usage compared to directly using an existing solver.