Quantum algorithms and lower bounds for eccentricity, radius, and diameter in undirected graphs

TL;DR

The paper develops quantum algorithms for fundamental distance-metrics in undirected graphs, delivering subquadratic upper bounds for diameter and radius in the adjacency-list model and a $2/3$-approximation for diameter with $\tilde{O}(\sqrt{m}\,n^{3/4})$ time. It leverages a synthesis of quantum single-source shortest paths, Grover-style search, and quantum partial BFS to reduce the search space and compute witnesses (paths) alongside the numeric results. Complementing the upper bounds, it proves quantum query lower bounds of $\Omega(\sqrt{nm})$ for eccentricity, diameter, and radius via reductions from quantum minima finding, highlighting a divergence from classical APSP-based intuitions. The results demonstrate a notable quantum speedup over classical matrix multiplication-based approaches and open questions about the tightness of lower bounds and the potential for further quantum improvements in graph problems. Overall, the work advances quantum graph algorithms by providing both concrete upper bounds and fundamental limits for core distance measures, with implications for quantum-speedups in network analysis tasks.

Abstract

The problems of computing eccentricity, radius, and diameter are fundamental to graph theory. These parameters are intrinsically defined based on the distance metric of the graph. In this work, we propose quantum algorithms for the diameter and radius of undirected, weighted graphs in the adjacency list model. The algorithms output diameter and radius with the corresponding paths in $\widetilde{O}(n\sqrt{m})$ time. Additionally, for the diameter, we present a quantum algorithm that approximates the diameter within a $2/3$ ratio in $\widetilde{O}(\sqrt{m}n^{3/4})$ time. We also establish quantum query lower bounds of $Ω(\sqrt{nm})$ for all the aforementioned problems through a reduction from the minima finding problem.

Figures (2)

• Figure 1: Reduction for sparse graphs. The graph used for the reduction from Minima Finding ($d$ elements of different types) to Eccentricity. There are $3d+2= n$ nodes, and a total of $m=10d-2$ edges. Vertices in each column are connected by edges of weight 0 to adjacent vertices in the same column, and with edges of some positive weights $w_{ij}$ between vertices in subsequent columns. Edges from $s$ and $t$ are all weights $0$. Finding the eccentricity of vertex $s$ comes down to finding a minimum weight edge between subsequent "columns", required to arrive at vertex $t$. The 0 weight edges between vertices in the same column ensure one can pick truly minimum weight edge between subsequent columns not the minimum weight edge from the vertex one happens to arrive at via the minimum weight edge from the previous column. Since edge weights are positive one has a guarantee that to find the eccentricity of $s$ one has to traverse all columns up to vertices in column $d$. Additionally, we require that in each subsequent column the edge weights connecting to the next column are smaller than all individual edge weights connecting to the current vertex. This requirement prevents a situation of moving back to the previous column of vertices, which could happen if suddenly all weights of edges connecting to the next (right) column are way larger than the weights of edges connecting to the current vertex from the previous (left) column. The value of a constant $3$ (the width of a layer) is an arbitrary choice to make the illustration simple but informative.
• Figure 2: Reduction for dense graphs. The figure depicts a dense weighted graph, with the blue path being the lowest weight path from $s$ (green node) to vertex $t$ (red node) corresponding to the eccentricity of $s$.

Theorems & Definitions (22)

• Theorem 1.4: Upper bounds of computing Diameter and Radius
• Theorem 1.5: Lower bounds of computing Eccentricity, Diameter and Radius
• Theorem 1.7: Upper bound of approximating diameter
• Theorem 2.1: Quantum search, $\mathtt{QSearch}(f)$ boyer1998tight
• Theorem 2.2: Quantum minimum finding, $\mathtt{QMF}(f)$ durr1996quantum
• Theorem 2.3: Quantum threshold finding, $\mathtt{QTF}(f,t)$ buhrman1999bounds
• Theorem 2.4: Quantum breadth-first search, $\mathtt{QBFS}(G,v)$ durr2006quantum
• Theorem 2.5: Quantum single source shortest paths, $\mathtt{QSSSP}(G,v)$ durr2006quantum
• proof
• proof