Multiagent MST Cover: Pleasing All Optimally via A Simple Voting Rule

TL;DR

This paper addresses a complex multi-agent graph problem, proving its NP-hardness, and introduces a simple voting algorithm that achieves optimal approximation ratios using only agents' rankings, with extensions to submodular and matroid constraints.

Contribution

The paper proves NP-hardness of the multi-agent MST cover problem and presents a simple, ranking-based voting algorithm with optimal approximation ratio, extending to submodular and matroid cases.

Findings

NP-hardness of the problem established

Voting algorithm achieves optimal approximation ratio

Algorithm requires only agents' rankings

Abstract

Given a connected graph on whose edges we can build roads to connect the nodes, a number of agents hold possibly different perspectives on which edges should be selected by assigning different edge weights. Our task is to build a minimum number of roads so that every agent has a spanning tree in the built subgraph whose weight is the same as a minimum spanning tree in the original graph. We first show that this problem is NP-hard and does not admit better than $((1-o(1))\ln k)$-approximation polynomial-time algorithms unless P=NP, where $k$ is the number of agents. We then give a simple voting algorithm with an optimal approximation ratio. Moreover, our algorithm only needs to access the agents' rankings on the edges. Finally, we extend our results to submodular objective functions and Matroid rank constraints.