Efficient Algorithms to Augment the Edge-Connectivity of Specified Vertices by One in a Graph

抄録

The k-edge-connectivity augmentation problem for a specified set of vertices (kECA-SV for short) is defined by “Given a graph G=(V, E) and a subset Γ ⊆ V, find a minimum set E' of edges such that G'=(V, E ∪ E') has at least k edge-disjoint paths between any pair of vertices in Γ.” Let σ be the edge-connectivity of Γ (that is, G has at least σ edge-disjoint paths between any pair of vertices in Γ). We propose an algorithm for (σ+1)ECA-SV which is done in O(|Γ|) maximum flow operations. Then the time complexity is O(σ²|Γ||V|+|E|) if a given graph is sparse, or O(|Γ||V||B_G|log(|V|²/|B_G|)+|E|) if dense, where |B_G| is the number of pairs of adjacent vertices in G. Also mentioned is an O(|V||E|+|V|² log |V|) time algorithm for a special case where σ is equal to the edge-connectivity of G and an O(|V|+|E|) time one for σ ≤ 2.