# Distributed Submodular Minimization via Block-Wise Updates and Communications

## TL;DR

This paper introduces a distributed algorithm for submodular minimization that enables networked agents to collaboratively solve the problem while preserving privacy, using block-wise updates and communication based on Lovász extensions.

## Contribution

It presents a novel distributed algorithm employing block-wise updates and Lovász extensions for submodular minimization without central coordination.

## Key findings

- Algorithm converges in expectation to the optimal solution.
- Agents successfully perform distributed image segmentation.
- Reduced computational burden through block-wise greedy updates.

## Abstract

In this paper we deal with a network of computing agents with local processing and neighboring communication capabilities that aim at solving (without any central unit) a submodular optimization problem. The cost function is the sum of many local submodular functions and each agent in the network has access to one function in the sum only. In this \emph{distributed} set-up, in order to preserve their own privacy, agents communicate with neighbors but do not share their local cost functions. We propose a distributed algorithm in which agents resort to the Lov\`{a}sz extension of their local submodular functions and perform local updates and communications in terms of single blocks of the entire optimization variable. Updates are performed by means of a greedy algorithm which is run only until the selected block is computed, thus resulting in a reduced computational burden. The proposed algorithm is shown to converge in expected value to the optimal cost of the problem, and an approximate solution to the submodular problem is retrieved by a thresholding operation. As an application, we consider a distributed image segmentation problem in which each agent has access only to a portion of the entire image. While agents cannot segment the entire image on their own, they correctly complete the task by cooperating through the proposed distributed algorithm.