Performance-Guaranteed Solutions for Multi-Agent Optimal Coverage Problems using Submodularity, Curvature, and Greedy Algorithms

TL;DR

This work tackles placing N agents to maximize a coverage objective over a mission space with obstacles by discretizing the space into a ground set X under a uniform matroid constraint. The objective H(S) = ∫_Ω R(x) P(x,S) dx is submodular, enabling a greedy algorithm with the fundamental bound β_f = 1 − (1 − 1/N)^N, approaching 1 − 1/e as N grows. To tighten guarantees beyond β_f, the authors review five curvature measures (total, greedy, elemental, partial, extended greedy) and provide efficient estimators, showing that bounds can be significantly improved in weak submodularity scenarios and that extended greedy curvature provides robust performance guarantees across regimes. Case studies (Blank, Maze, General) validate the theory and illustrate how the weight θ and sensing parameters δ, λ shape agent placements and bound tightness, with a public simulator enabling replication and exploration. The work highlights the practicality of curvature-based guarantees for multi-agent coverage and motivates future hybrid, data-driven approaches to curvature estimation.

Abstract

We consider a class of multi-agent optimal coverage problems in which the goal is to determine the optimal placement of a group of agents in a given mission space so that they maximize a coverage objective that represents a blend of individual and collaborative event detection capabilities. This class of problems is extremely challenging due to the non-convex nature of the mission space and of the coverage objective. With this motivation, greedy algorithms are often used as means of getting feasible coverage solutions efficiently. Even though such greedy solutions are suboptimal, the submodularity (diminishing returns) property of the coverage objective can be exploited to provide performance bound guarantees. Moreover, we show that improved performance bound guarantees (beyond the standard (1-1/e) performance bound) can be established using various curvature measures of the coverage problem. In particular, we provide a brief review of all existing popular applicable curvature measures, including a recent curvature measure that we proposed, and discuss their effectiveness and computational complexity, in the context of optimal coverage problems. We also propose novel computationally efficient techniques to estimate some curvature measures. Finally, we provide several numerical results to support our findings and propose several potential future research directions.

Figures (6)

• Figure 1: A mission space with two agents.
• Figure 2: Greedy solutions, coverage level patterns, and the tightest performance bounds observed under different weight parameters $\theta \in [0,1]$ in the Blank mission space with $N=4$ agents with sensing range $\delta=200$ and decay $\lambda = 0.012$.
• Figure II: Performance bounds observed under different sensing range $\delta$ values in the Blank mission space with $N=10$ agents with sensing decay $\lambda=0.003$.
• Figure III: Performance bounds observed under different sensing decay $\lambda$ values in the Blank mission space with $N=10$ agents with sensing range $\delta=800$.
• Figure IV: Performance bounds observed under different sensing range $\delta$ values in the Maze mission space with $N=10$ agents with sensing range $\lambda=0.012$.

Theorems & Definitions (17)

• Remark 1
• Definition 1
• Lemma 1
• proof
• Theorem 1
• proof
• Lemma 2
• proof
• Theorem 2
• proof