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.

References

Our proof of \cref{thm:main_informal} works for multigraphs that allow parallel edges and self-loops. While we believe that the possibility of self-loops can be avoided, we are not sure whether the same holds for parallel edges: We believe that whether \cref{thm:main_informal} holds in the setting of simple graphs is an interesting open question.

— Bidirectional Dijkstra's Algorithm is Instance-Optimal  (2410.14638 - Haeupler et al., 2024) in Open problems section

More surprisingly, bidirectional Dijkstra achieves the asymptotically best possible instance-optimality ratio among all algorithms in every setting, except that of order-oblivious directed graphs, where our results are inconclusive.

— Instance-Optimality of Bidirectional Dijkstra on Simple Graphs  (2608.24380 - Bertram et al., 25 Aug 2026) in Section 8, “More Instance-Optimality,” opening discussion

For this reason, the authors of leave the instance optimality status of Algorithm~\ref{BID_DIJ_AUTH} in simple graphs as an open question.

— On the Instance Optimality of Bidirectional Dijkstra's Algorithm  (2608.26952 - Požar, 27 Aug 2026) in Section 6, “Instance Optimality in Simple Graphs”

Is there an explicit uniform deterministic algorithm that, on every rooted directed topology, computes exact labeled with

\operatorname{RAM}_U(G,s,w)=O!\left(m+n+(G,s)\right)

for all nonnegative weightings, and hence uses O((G,s)) charged numerical operations?

— Beyond Distance Ordering: Resource Complexity and Universal Optimality of Exact Labeled Directed Shortest Paths  (2609.04825 - Cai, 4 Sep 2026) in Open Problem 1, Section 7, 'Limitations and open problems'; reiterated in Section 8, 'Conclusion'