A Simple Primal-Dual Approximation Algorithm for 2-Edge-Connected Spanning Subgraphs

TL;DR

This paper introduces a simple, efficient primal-dual approximation algorithm for finding minimum 2-edge-connected spanning subgraphs, improving conceptual simplicity and implementation ease while achieving a 3-approximation ratio.

Contribution

It presents a new primal-dual algorithm with a single growing phase, simplifying previous approaches and addressing Williamson's question.

Findings

Achieves a 3-approximation ratio for the problem.

Runs in near-linear time with simple data structures.

Simplifies the conceptual understanding and implementation of the algorithm.

Abstract

We propose a simple and natural approximation algorithm for the problem of finding a 2-edge-connected spanning subgraph of minimum total edge cost in a graph. The algorithm maintains a spanning forest starting with an empty edge set. In each iteration, a new edge incident to a leaf is selected in a natural greedy manner and added to the forest. If this produces a cycle, this cycle is contracted. This growing phase ends when the graph has been contracted into a single node and a subsequent cleanup step removes redundant edges in reverse order. We analyze the algorithm using the primal-dual method showing that its solution value is at most 3 times the optimum. Although this only matches the ratio of existing primal-dual algorithms, we require only a single growing phase, thereby addressing a question by Williamson. Also, we consider our algorithm to be not only conceptually simpler than the known approximation algorithms but also easier to implement in its entirety. For n and m being the number of nodes and edges, respectively, it runs in O(min{nm, m + n^2 log n}) time and O(m) space without data structures more sophisticated than binary heaps and graphs, and without graph algorithms beyond depth-first search.