Economical Graph Discovery

Consider a weighted n-vertex, m-edge graph G with designated source s and destination t. The topology of G is known, while the edge weights are hidden. Our goal is to discover either the edge weights in the graph or a shortest (s, t)-path. This is done by means of agents that traverse different (s, t)-paths in multiple rounds and report back the total cost they incurred. Various cost models are considered, differing from each other in their approach to congestion effects. We seek bounds on the number of rounds and the number of agents required to complete the discovery of the edge weights or a shortest path. A host of results concerning such bounds for both directed and undirected graphs are established. Among these results, we show that: (1) for undirected graphs, all edge weights can be discovered within a single round consisting of m agents; (2) discovering a shortest path in either undirected or directed acyclic graphs requires at least m−n+1 agents; and (3) the edge weights in a directed acyclic graph can be discovered in m rounds with m+ n− 2 agents under congestion-aware cost models. Our study introduces a new setting of graph discovery under uncertainty and provides fundamental understanding of the problem.