[Paper Review] A Submodularity-Based Approach for Multi-Agent Optimal Coverage Problems

Xinmiao Sun, Christos G. Cassandras+1 more|arXiv (Cornell University)|Aug 14, 2017

Distributed Control Multi-Agent Systems20 references3 citations

TL;DR

This paper proposes a submodularity-based approach to solve multi-agent optimal coverage problems in environments with obstacles, leveraging the monotone submodular nature of the coverage objective to guarantee a solution within 1−1/e of the global optimum via a greedy algorithm. It further derives tighter lower bounds using total and elemental curvature, which are complementary across sensing capabilities, and combines the greedy solution with gradient-based optimization to achieve superior performance over existing methods.

ABSTRACT

We consider the optimal coverage problem where a multi-agent network is deployed in an environment with obstacles to maximize a joint event detection probability. The objective function of this problem is non-convex and no global optimum is guaranteed by gradient-based algorithms developed to date. We first show that the objective function is monotone submodular, a class of functions for which a simple greedy algorithm is known to be within 0.63 of the optimal solution. We then derive two tighter lower bounds by exploiting the curvature information (total curvature and elemental curvature) of the objective function. We further show that the tightness of these lower bounds is complementary with respect to the sensing capabilities of the agents. The greedy algorithm solution can be subsequently used as an initial point for a gradient-based algorithm to obtain solutions even closer to the global optimum. Simulation results show that this approach leads to significantly better performance relative to previously used algorithms.

Motivation & Objective

• To address the challenge of non-convex, locally optimal solutions in multi-agent optimal coverage problems with obstacles.
• To establish that the joint event detection probability objective is monotone submodular, enabling performance guarantees via greedy algorithms.
• To derive tighter lower bounds on the global optimum using total and elemental curvature of the objective function.
• To develop a hybrid greedy-gradient algorithm that improves solution quality by refining greedy initializations in continuous space.
• To demonstrate through simulation that the proposed method outperforms existing on-line and off-line algorithms in coverage performance.

Proposed method

• Formulate the multi-agent coverage problem with agents restricted to a finite set of feasible positions, enabling submodularity analysis.
• Prove that the joint detection probability objective is monotone submodular, allowing the use of a greedy algorithm with a 1−1/e approximation guarantee.
• Derive two tighter lower bounds using total curvature and elemental curvature of the objective function, which are complementary across different sensing capabilities.
• Select the tighter of the two curvature-based bounds depending on agent sensing range to ensure a near-optimal performance guarantee.
• Use the greedy solution as an initial point for a distributed gradient-based algorithm to refine positions in continuous space.
• Combine the greedy and gradient steps into a greedy-gradient algorithm (GGA) to enhance convergence to high-quality solutions.

Experimental results

Research questions

• RQ1 Can the joint detection probability in multi-agent coverage problems be shown to be monotone submodular, enabling performance guarantees via greedy selection?
• RQ2 How can curvature information—specifically total and elemental curvature—be leveraged to derive tighter lower bounds than the standard 1−1/e guarantee?
• RQ3 Are the performance bounds derived from total curvature and elemental curvature complementary with respect to agent sensing capabilities?
• RQ4 Does combining a greedy initialization with a gradient-based refinement improve coverage performance compared to standalone methods?
• RQ5 Can the proposed greedy-gradient algorithm achieve better convergence and solution quality in complex mission spaces with obstacles?

Key findings

• The joint detection probability objective function is proven to be monotone submodular, enabling a 1−1/e approximation guarantee via the greedy algorithm.
• Two tighter lower bounds are derived using total curvature and elemental curvature, both of which are explicitly computable for the problem’s objective function.
• The tightness of the total curvature and elemental curvature bounds is complementary: one is tight under weak sensing, the other under strong sensing, ensuring a near-optimal bound regardless of agent capabilities.
• Simulation results show that the greedy-gradient algorithm (GGA) achieves significantly higher coverage performance than existing methods, with H(s) values consistently outperforming benchmarks across diverse mission spaces.
• In obstacle-free and maze-like environments, the GGA achieves coverage values up to 2583.5 (H(s)), exceeding the best-performing alternatives by a substantial margin.
• The proposed method reduces the theoretical optimality gap beyond the classical 1−1/e bound, with curvature-based bounds approaching 1 under favorable sensing conditions.