Instance-optimality on simple graphs (no parallel edges or self-loops)

Determine whether the instance-optimality guarantees for the shortest s–t path problem in the adjacency-list query model—namely, exact instance-optimality on weighted graphs with strictly positive edge weights and Δ(G)-approximate instance-optimality on unweighted graphs—continue to hold when the input graphs are required to be simple (i.e., without parallel edges and self-loops).

Background

The paper proves that a carefully implemented bidirectional Dijkstra’s algorithm is instance-optimal (with respect to the number of edge queries) on weighted multigraphs with strictly positive edge weights, and that bidirectional BFS is instance-optimal up to a factor of O(Δ(G)) on unweighted multigraphs. These results are established in the adjacency-list model of sublinear algorithms and apply to both directed and undirected multigraphs, allowing parallel edges and self-loops.

In the concluding open problems section, the authors note that their proofs rely on the multigraph setting and explicitly state uncertainty about whether the results extend to simple graphs. They believe self-loops may be removable without affecting the guarantees, but are unsure about the necessity of allowing parallel edges. This leads to the open question of whether the same instance-optimality guarantees can be proved when parallel edges and self-loops are disallowed.