# Distributed Submodular Minimization over Networks: a Greedy Column Generation Approach

## TL;DR

This paper introduces a distributed greedy column generation algorithm for submodular minimization over unreliable, directed networks, enabling agents to cooperatively solve complex optimization problems with limited local information.

## Contribution

It presents a novel distributed algorithm that reformulates submodular minimization as a linear program and solves the exponential constraints via a polynomial-time greedy method.

## Key findings

- Algorithm converges in finite time to an optimal solution.
- Proven convergence under asynchronous, unreliable, and time-varying network conditions.
- Numerical experiments validate theoretical convergence and effectiveness.

## Abstract

Submodular optimization is a special class of combinatorial optimization arising in several machine learning problems, but also in cooperative control of complex systems. In this paper, we consider agents in an asynchronous, unreliable and time-varying directed network that aim at cooperatively solving submodular minimization problems in a fully distributed way. The challenge is that the (submodular) objective set-function is only partially known by agents, that is, each one is able to evaluate the function only for subsets including itself. We propose a distributed algorithm based on a proper linear programming reformulation of the combinatorial problem. Our algorithm builds on a column generation approach in which each agent maintains a local candidate basis and locally generates columns with a suitable greedy inner routine. A key interesting feature of the proposed algorithm is that the pricing problem, which involves an exponential number of constraints, is solved by the agents through a polynomial time greedy algorithm. We prove that the proposed distributed algorithm converges in finite time to an optimal solution of the submodular minimization problem and we corroborate the theoretical results by performing numerical computations on instances of the $s$--$t$ minimum graph cut problem.