Optimality Gap of Decentralized Submodular Maximization under Probabilistic Communication

TL;DR

The paper addresses decentralized submodular maximization under partition matroid constraints with probabilistic inter-agent communication along a chain. It introduces the probabilistic optimality gap $\alpha_p$, tying performance to the clique number of the information graph and providing a polynomial-time method to compute it via generative sequences. It also analyzes how additional communication resources (reinforcement) can improve $\alpha_p$ and the achieved objective, supported by a sensor-deployment empirical study that shows reinforcement decisions depend on chain structure. The work offers a practical framework for designing robust decentralized systems in uncertain networks and points to future directions, including proving submodularity of $\alpha_p$ and extending reinforcement to multiple agents.

Abstract

This paper considers the problem of decentralized submodular maximization subject to partition matroid constraint using a sequential greedy algorithm with probabilistic inter-agent message-passing. We propose a communication-aware framework where the probability of successful communication between connected devices is considered. Our analysis introduces the notion of the probabilistic optimality gap, highlighting its potential influence on determining the message-passing sequence based on the agent's broadcast reliability and strategic decisions regarding agents that can broadcast their messages multiple times in a resource-limited environment. This work not only contributes theoretical insights but also has practical implications for designing and analyzing decentralized systems in uncertain communication environments. A numerical example demonstrates the impact of our results.

Figures (6)

• Figure 1: A directed chain graph where an arrow from one node to another signifies an edge, indicating directional information flow from the tail node (in-neighbor) to the head node (out-neighbor).
• Figure 2: Examples of information sharing topologies. The disconnected edge, indicating failure of message delivery, is shown by red arrows on the message-passing graph.
• Figure 3: Examples of information sharing topologies. The top plots show the communication graph $\mathcal{G}$ along with its consequent maximum clique number. Arrow going from agent $i$ to agent $j$ means that agent $j$ receives agent $i$'s information. In red are represented the communication failures.
• Figure 4: Examples of generative sequences, which generate families of message-passing sequences that have connected components of at least length two (top plot) and length three (bottom plot). A green link indicates an imposed connected edge, and a red link indicates an imposed disconnected edge. For example, generative sequence C in the top plot corresponds to $g(\{3,4\},\{2\})$ and in the bottom plot corresponds to $g(\{3,4,5\},\{2\})$.
• Figure 5: Coverage Problem: (from left to right) initial distribution, coverage considering under no communication failure and coverage under the outcome $[0,1,1,1,0,1,1]$ of probabilistic communication. The disks show the coverage footprint of the sensors; the sensors corresponding to each agent is colored similarly. Note that when communication chain breaks, some agents tend to go to the same placement, reducing the coverage substantially; in the fully successful communication we achieve a solution $f(\mathcal{S})=1879$, while in the disconnection situation $f(\mathcal{S})=1246$.

Theorems & Definitions (6)

• Lemma II.1
• Theorem II.1
• proof
• Lemma II.2
• proof
• Lemma II.3: Probability of a family