Efficient Graph Algorithms for Network Analysis

The GC problem is to identify a pre-determined number of center vertices such that the distances or costs from (or to) the centers to (or from) other vertices is minimized. The bottleneck of a path is the minimum capacity of edges on the path. The Bottleneck Paths (BP) problem is to compute the paths that give us the maximum bottleneck values between pairs of vertices. The Graph Bottleneck (GB) problem is to find the minimum bottleneck value out of bottleneck paths for all possible pairs of vertices. We give two similar algorithms that are based on binary search to solve the 1-center GC problem and the GB problem on directed graphs with unit edge costs. We achieve $\tilde{O}(n^{2.373})$ worst case time complexity for both the 1-center GC problem and the GB problem, where $n$ is the number of vertices in the graph. This is better than the straightforward methods of solving the two problems in $O(n^{2.575})$ and $O(n^{2.688})$ time bounds, respectively. We then combine the Bottleneck Paths (BP) problem with the well known Shortest Paths (SP) problem to compute the shortest paths for all possible flow values. We call this problem the Shortest Paths for All Flows (SP-AF) problem. We show that if the flow demand is uncertain, but between two consecutive capacity values, the unique shortest path can be computed to push that flow. If the uncertainty stretches over two intervals, we need to prepare two shortest paths to accommodate the uncertainty, etc. In introducing this new problem, we define a new semi-ring called the distance/flow semi-ring, and show that the well known algorithm by Floyd can be used over the distance/flow semi-ring to solve the All Pairs Shortest Paths for All Flows (APSP-AF) problem.

💡 Research Summary

The paper addresses two fundamental problems in network analysis—graph‑center (GC) and graph‑bottleneck (GB)—and introduces a unified framework that dramatically improves their worst‑case running times. Both problems are considered on directed graphs with unit edge costs (for GC) and arbitrary non‑negative capacities (for GB).

1‑Center GC Problem
The authors observe that the exact graph center can be found without solving the all‑pairs shortest‑paths (APSP) problem. They encode the adjacency matrix as a Boolean matrix B and compute successive powers B², B⁴, …, B^{n/2} by repeated squaring. In Boolean algebra, B^{ℓ}_{ij}=1 iff there exists a path of length at most ℓ from i to j. Let Δ denote the radius of a candidate center (the maximum distance from the center to any vertex). The condition Δ ≤ ℓ is equivalent to every row of B^{ℓ} containing only 1’s. By binary searching over ℓ in the interval