Optimal Scalability-Aware Allocation of Swarm Robots: From Linear to Retrograde Performance via Marginal Gains

TL;DR

The paper addresses how to optimally allocate a finite swarm of agents across multiple tasks when task performance follows concave scalability curves. It develops a polynomial-time, greedy marginal-gain algorithm that proves optimality for any concave task function by comparing per-task marginal gains \delta(d_i,n_i) and assigning agents to maximize collective performance \ C(\\mathbf{D},\\mathbf{N}). The authors explore linear, saturating, and retrograde scalability using models from parallel computing and crowd dynamics, and validate the approach through extensive swarm-robotics simulations including cases with and without physical interference. The work demonstrates size-dependent allocations, reveals intuitive allocation patterns (even splits for linear, targeting harder tasks for large swarms, and finite peaks for retrograde), and offers a practical framework for deploying large-scale multi-robot systems with efficient offline task planning.

Abstract

In collective systems, the available agents are a limited resource that must be allocated among tasks to maximize collective performance. Computing the optimal allocation of several agents to numerous tasks through a brute-force approach can be infeasible, especially when each task's performance scales differently with the increase of agents. For example, difficult tasks may require more agents to achieve similar performances compared to simpler tasks, but performance may saturate nonlinearly as the number of allocated agents increases. We propose a computationally efficient algorithm, based on marginal performance gains, for optimally allocating agents to tasks with concave scalability functions, including linear, saturating, and retrograde scaling, to achieve maximum collective performance. We test the algorithm by allocating a simulated robot swarm among collective decision-making tasks, where embodied agents sample their environment and exchange information to reach a consensus on spatially distributed environmental features. We vary task difficulties by different geometrical arrangements of environmental features in space (patchiness). In this scenario, decision performance in each task scales either as a saturating curve (following the Condorcet's Jury Theorem in an interference-free setup) or as a retrograde curve (when physical interference among robots restricts their movement). Using simple robot simulations, we show that our algorithm can be useful in allocating robots among tasks. Our approach aims to advance the deployment of future real-world multi-robot systems.

Figures (11)

• Figure 1: Overview of the proposed algorithm, showing the steps from input to final agent–task assignment. In this representative example, a total of $N = 24$ agents are allocated across $T = 4$ tasks, each characterized by a distinct scalability curve $C(d_i,n_i)$. Based on marginal performance gains, the algorithm iteratively computes the optimal allocation as $N^* = [11, 9, 3, 1]$. In the illustrated collective decision-making scenario (collective sensing of environmental features shown as black and white floor tiles), the robot allocation is biased in favor of the easiest tasks, that is, tasks where robots can measure the correct state of the world with increased individual accuracy. In contrast, when swarms are large (resources to be allocated are abundant), the optimal solution is biased in favor of the more difficult tasks (Sec.\ref{['sec:res-many-tasks']}).
• Figure 2: Schematic plot of three types of scalability functions---with linear, saturating, and retrograde trends---describing the group performance as a function of group size. Each task, depending on the particular application and scenario, can scale differently.
• Figure 3: Simulated binary classification tasks with fill ratio $f=0.52$. The environment floor (sized [$36\times36$]su) is composed of black and white tiles (sized [$1\times1$]su). The robots are tasked with reaching a consensus on the most frequent color. We consider four spatial distributions: (a) checkerboard, (b) striped, (c) four rectangles and (d) halved environments.
• Figure 4: (a) Centralized controller: All robots start in the exploration state, during which they gather environmental data and form their initial opinion. After exploring, they communicate through all-to-all interactions to reach a majority-based consensus. (b) Decentralized controller: Robots form their initial opinion similarly to the centralized controller. However, communication is restricted to local interactions and robots disseminate their opinions to neighbors until the entire swarm reaches a unanimous decision. (c) Iterative controller: This controller follows an iterative process of exploration and dissemination where they repeatedly combine personal and social information. Robots simultaneously disseminate their opinions, listen to neighbors, and continue exploring. The iterative process continues until the swarm reaches a unanimous decision.
• Figure 5: Scaling of the group accuracy (y-axis) in making majority decisions for different group sizes (x-axis) in the standard checkerboard environment (see Fig. \ref{['fig:env']}(a)). The different colors show results for different fill ratios. Solid line shows the CJT predictions and, in (a), the dotted, dash-dotted, and dashed lines show the multi-agent results for the centralized, decentralized, and iterative controllers, respectively. There is a good agreement between the CJT predictions and the multi-agent results. In (b), the CJT lines are extended to larger swarm sizes, using the same values of $p$ as in (a).

Theorems & Definitions (2)

• Theorem 1
• proof