Utilitarians Without Utilities: Maximizing Social Welfare for Graph Problems using only Ordinal Preferences - Full Version

TL;DR

This paper demonstrates that for a wide range of graph optimization problems, ordinal preference data alone suffices to find near-optimal solutions, eliminating the need for explicit utility values.

Contribution

The paper introduces ordinal approximation algorithms for utility maximization in graph problems, achieving near-optimal solutions using only preference rankings.

Findings

Ordinal greedy algorithm yields a (b+1)-approximation for bounded degree b problems.

Ordinal information suffices for a 2-approximation of Maximum Spanning Tree.

Ordinal algorithms provide a 4-approximation for Max Weight Planar Subgraph.

Abstract

We consider ordinal approximation algorithms for a broad class of utility maximization problems for multi-agent systems. In these problems, agents have utilities for connecting to each other, and the goal is to compute a maximum-utility solution subject to a set of constraints. We represent these as a class of graph optimization problems, including matching, spanning tree problems, TSP, maximum weight planar subgraph, and many others. We study these problems in the ordinal setting: latent numerical utilities exist, but we only have access to ordinal preference information, i.e., every agent specifies an ordering over the other agents by preference. We prove that for the large class of graph problems we identify, ordinal information is enough to compute solutions which are close to optimal, thus demonstrating there is no need to know the underlying numerical utilities. For example, for problems in this class with bounded degree $b$ a simple ordinal greedy algorithm always produces a ($b+1$)-approximation; we also quantify how the quality of ordinal approximation depends on the sparsity of the resulting graphs. In particular, our results imply that ordinal information is enough to obtain a 2-approximation for Maximum Spanning Tree; a 4-approximation for Max Weight Planar Subgraph; a 2-approximation for Max-TSP; and a 2-approximation for various Matching problems.