On the Hardness of Partially Dynamic Graph Problems and Connections to Diameter

TL;DR

This paper establishes conditional lower bounds for partially dynamic graph problems, demonstrating that efficient algorithms with sublinear amortized update times are unlikely for problems like maximum bipartite matching, maximum flow, and diameter approximation, based on popular conjectures.

Contribution

It provides the first strong lower bounds for partially dynamic maximum flow and diameter approximation, and improves bounds for bipartite matching, highlighting the difficulty of designing efficient algorithms in these settings.

Findings

No sublinear amortized update algorithms for maximum bipartite matching under certain conjectures.

No sublinear amortized update algorithms for maximum flow in directed and weighted graphs.

No sublinear amortized update algorithms for diameter approximation in unweighted graphs.

Abstract

Conditional lower bounds for dynamic graph problems has received a great deal of attention in recent years. While many results are now known for the fully-dynamic case and such bounds often imply worst-case bounds for the partially dynamic setting, it seems much more difficult to prove amortized bounds for incremental and decremental algorithms. In this paper we consider partially dynamic versions of three classic problems in graph theory. Based on popular conjectures we show that: -- No algorithm with amortized update time $O(n^{1-\varepsilon})$ exists for incremental or decremental maximum cardinality bipartite matching. This significantly improves on the $O(m^{1/2-\varepsilon})$ bound for sparse graphs of Henzinger et al. [STOC'15] and $O(n^{1/3-\varepsilon})$ bound of Kopelowitz, Pettie and Porat. Our linear bound also appears more natural. In addition, the result we present separates the node-addition model from the edge insertion model, as an algorithm with total update time $O(m\sqrt{n})$ exists for the former by Bosek et al. [FOCS'14]. -- No algorithm with amortized update time $O(m^{1-\varepsilon})$ exists for incremental or decremental maximum flow in directed and weighted sparse graphs. No such lower bound was known for partially dynamic maximum flow previously. Furthermore no algorithm with amortized update time $O(n^{1-\varepsilon})$ exists for directed and unweighted graphs or undirected and weighted graphs. -- No algorithm with amortized update time $O(n^{1/2 - \varepsilon})$ exists for incremental or decremental $(4/3-\varepsilon')$-approximating the diameter of an unweighted graph. We also show a slightly stronger bound if node additions are allowed. [...]