Risk-Aware Submodular Optimization for Multi-Robot Coordination

TL;DR

This paper introduces a risk-aware approach to discrete submodular optimization using CVaR, with a novel approximation algorithm and applications in multi-robot coordination and environmental monitoring.

Contribution

It extends CVaR-based optimization to discrete submodular problems and proposes a polynomial-time approximation algorithm with guarantees.

Findings

The Sequential Greedy Algorithm achieves near-optimal solutions within a constant factor.

The algorithms are effective in vehicle assignment and sensor selection problems.

Simulation results validate the approach's efficiency and robustness.

Abstract

We study the problem of incorporating risk while making combinatorial decisions under uncertainty. We formulate a discrete submodular maximization problem for selecting a set using Conditional-Value-at-Risk (CVaR), a risk metric commonly used in financial analysis. While CVaR has recently been used in optimization of linear cost functions in robotics, we take the first step towards extending this to discrete submodular optimization and provide several positive results. Specifically, we propose the Sequential Greedy Algorithm that provides an approximation guarantee on finding the maxima of the CVaR cost function under a matroidal constraint. The approximation guarantee shows that the solution produced by our algorithm is within a constant factor of the optimal and an additive term that depends on the optimal. Our analysis uses the curvature of the submodular set function, and proves that the algorithm runs in polynomial time. This formulates a number of combinatorial optimization problems that appear in robotics. We use two such problems, vehicle assignment under uncertainty for mobility-on-demand and sensor selection with failures for environmental monitoring, as case studies to demonstrate the efficacy of our formulation. In particular, for the mobility-on-demand study, we propose an online triggering assignment algorithm that triggers a new assignment only can potentially lead to reducing the waiting time at demand locations. We verify the performance of the Sequential Greedy Algorithm and the online triggering assignment algorithm through simulations.