Risk-Aware Submodular Optimization for Multi-Robot Coordination

Robotics optimization has evolved from simple task allocation to complex decision-making under uncertainty. Early works like influence maximization [Kempe et al., 2003] and robot-target tracking [Yamashita et al., 2014] focused on maximizing information or utility. Recent trends incorporate risk measures, notably CVaR, to handle rare but impactful events. However, most existing methods address linear or continuous models, leaving a gap in discrete, combinatorial settings. Submodular functions, capturing information gain and coverage, are central to many applications. Despite progress, integrating risk metrics like CVaR into discrete submodular maximization under complex constraints remains underexplored, limiting robustness in real-world deployments.

The core challenge is to maximize a stochastic, monotone submodular utility function under matroid constraints, considering the tail risk quantified by CVaR. Traditional greedy algorithms optimize expected utility but fail to address rare adverse events. Extending CVaR to discrete submodular functions introduces computational complexity and theoretical hurdles, such as lack of polynomial-time guarantees. The problem becomes more intricate when constraints are complex, and the uncertainty model involves independent random variables. Addressing these issues requires novel algorithms with provable approximation bounds, capable of balancing risk and utility efficiently.

This work introduces several innovations: 1) Formulating CVaR maximization for discrete submodular functions under matroid constraints, filling a theoretical gap. 2) Developing the Sequential Greedy Algorithm (SGA), which combines parameter search with greedy subroutines, leveraging submodular curvature for performance guarantees. 3) Theoretically bounding the approximation ratio, explicitly relating it to the function's curvature. 4) Applying the framework to practical problems like vehicle scheduling and sensor placement, demonstrating effectiveness. 5) Incorporating adaptive re-planning via event-triggered mechanisms, reducing unnecessary computation. These innovations collectively advance risk-sensitive combinatorial optimization in robotics.

• �� Define the utility function \(f(S, y)\) with stochastic variables, and formulate CVaR as the optimization target.

• �� Design the Sequential Greedy Algorithm (SGA):
• Search over τ in [0, Γ] with step ∆, where Γ bounds the maximum utility.
• For each τ, run a greedy subroutine to select set \(S_{G}\) that maximizes auxiliary function \(H(S, τ)\).
• Store pairs \((S_{G}, τ)\) for all τ.
• Select the pair with the highest \(H(S, τ)\) as the solution.

• �� Use sampling to estimate \(H(S, τ)\), ensuring approximation accuracy with high probability.

• �� Analyze the algorithm's performance via submodular curvature, deriving bounds on the approximation ratio.

• �� Extend to adaptive online scenarios, triggering re-planning only when risk thresholds are crossed, using event-based mechanisms.

• �� Validate through simulations on vehicle routing and sensor deployment, comparing CVaR-based solutions with expectation-based baselines.

Simulations employ traffic data for vehicle assignment and environmental maps for sensor placement. Metrics include CVaR values, average wait times, and coverage errors. Baselines include traditional greedy algorithms optimizing expected utility. Hyperparameters such as sampling size and risk level α are tuned to balance accuracy and computational cost. Multiple scenarios test robustness under varying uncertainty levels. Results show the proposed algorithm consistently outperforms baselines in risk mitigation, with CVaR improvements around 20% in vehicle scheduling and 15% in sensor coverage. Adaptive re-planning reduces unnecessary computation, confirming practicality for real-time applications.

The algorithm achieved a 20% increase in CVaR in vehicle scheduling, significantly reducing extreme wait times. In sensor deployment, considering failure risks, the coverage error decreased by 15%. The approximation ratio closely matched theoretical bounds, validating the curvature-based analysis. Adaptive mechanisms cut re-planning frequency by 30%, improving efficiency without sacrificing robustness. These results demonstrate the method's capacity to handle complex uncertainties effectively, providing safer and more reliable robotic operations.

Applicable in urban vehicle dispatch, environmental monitoring, disaster response, and autonomous exploration. Requires task-specific utility functions with submodular properties and uncertainty models. Suitable for scenarios demanding high robustness against rare events. The framework supports integration with existing planning systems, enhancing safety and performance. Long-term, it can enable autonomous systems to adaptively manage risks in dynamic, uncertain environments, fostering broader deployment in critical sectors.

Performance depends on the submodular function's curvature; high curvature reduces approximation quality. Effectiveness diminishes when risk level α approaches zero, increasing error margins. Assumes independence among random variables, which may not hold in correlated environments. Computational complexity scales with problem size and sampling requirements, challenging real-time deployment in large-scale systems. Future work should address correlated uncertainties, improve scalability, and incorporate online learning for dynamic risk adjustment.