Approximation algorithms for node and element connectivity augmentation problems

• Published in: Theory of computing systems Vol. 68; no. 5; pp. 1468 - 1485
• Main Author: Nutov, Zeev
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.10.2024; Springer Nature B.V
• Subjects: Algorithms; Approximation; Combinatorial analysis; Computer Science; Connectivity; Graph theory; Theory of Computation; Approximation algorithms; Element connectivity; Crossing family; Connectivity augmentation
• ISSN: 1432-4350; 1433-0490
• DOI: 10.1007/s00224-024-10175-x

In connectivity augmentation problems we are given a graph G = ( V , E G ) and an edge set E on V , and seek a min-size edge set J ⊆ E such that G ∪ J has larger connectivity than G . In the 1 -Connectivity Augmentation ( 1 -CA ) problem G is connected and G ∪ J should be 2-connected. In the Leaf to Leaf 1 -CA every edge in E connects two leaves in the block-tree of G . For this version we give a simple combinatorial 5/3-approximation algorithm, improving the 1.892 approximation that applies for the general case. We will also show by a simple proof that if the Steiner Tree problem admits approximation ratio α then 1 -CA admits approximation ratio 1 + ln ( 4 - x ) + ϵ , where x is the solution to the equation 1 + ln ( 4 - x ) = α + ( α - 1 ) x . For the currently best value of α = ln 4 + ϵ this gives approximation ratio 1.942. This is worse than the best known ratio 1.892, but has the advantage of using Steiner Tree approximation as a “black box”. In the Element Connectivity Augmentation problem we are given a graph G = ( V , E ) , S ⊆ V , and connectivity requirements { r ( u , v ) : u , v ∈ S } . The goal is to find a min-size set J of new edges on S such that for all u , v ∈ S the graph G ∪ J contains r ( u ,  v ) uv -paths such that no two of them have an edge or a node in V \ S in common. The problem is NP-hard even when r max = max u , v ∈ S r ( u , v ) = 2 . We obtain approximation ratio 3/2, improving the previous best ratio 7/4. For the case of degree bounds on S we obtain the same ratio with just + 1 degree violation, which is tight, since deciding whether there exists a feasible solution is NP-hard even when r max = 2 .