Execution Order Matters in Greedy Algorithms with Limited Information

TL;DR

This paper investigates how the order of agents affects the efficiency of greedy algorithms in limited communication settings, proposing a distributed method to optimize ordering and improve distributed submodular maximization.

Contribution

It characterizes the impact of agent ordering on communication complexity and introduces a distributed algorithm to find efficient orderings for greedy execution.

Findings

Best ordering yields O(n) communication complexity.

Worst ordering increases complexity to O(n^2).

Proposed algorithm improves distributed submodular maximization.

Abstract

In this work, we study the multi-agent decision problem where agents try to coordinate to optimize a given system-level objective. While solving for the global optimal is intractable in many cases, the greedy algorithm is a well-studied and efficient way to provide good approximate solutions - notably for submodular optimization problems. Executing the greedy algorithm requires the agents to be ordered and execute a local optimization based on the solutions of the previous agents. However, in limited information settings, passing the solution from the previous agents may be nontrivial, as some agents may not be able to directly communicate with each other. Thus the communication time required to execute the greedy algorithm is closely tied to the order that the agents are given. In this work, we characterize interplay between the communication complexity and agent orderings by showing that the complexity using the best ordering is O(n) and increases considerably to O(n^2) when using the worst ordering. Motivated by this, we also propose an algorithm that can find an ordering and execute the greedy algorithm quickly, in a distributed fashion. We also show that such an execution of the greedy algorithm is advantageous over current methods for distributed submodular maximization.