  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.
