Engineering Weighted Connectivity Augmentation Algorithms

TL;DR

The paper addresses the weighted connectivity augmentation problem (WCAP), where given a $k$-connected graph and a set of weighted links, one seeks a minimum-cost augmentation that raises edge connectivity to $k+1$. It combines theoretical advances with practical algorithms by implementing recent better-than-2 approximations and introducing three heuristics plus an exact ILP, all operating on a cactus graph representation of minimum cuts. Experimental results demonstrate superior solution quality, speed, and memory usage compared to prior state-of-the-art methods, with MSTConnect dominating large-cost instances, GWC excelling on small-cost cases, and LS$(k)$ providing further improvements. An exact solver (eILP) complements the heuristics by solving many real-world instances efficiently, and the work outlines practical guidance and future open-source release plans.

Abstract

Increasing the connectivity of a graph is a pivotal challenge in robust network design. The weighted connectivity augmentation problem is a common version of the problem that takes link costs into consideration. The problem is then to find a minimum cost subset of a given set of weighted links that increases the connectivity of a graph by one when the links are added to the edge set of the input instance. In this work, we give a first implementation of recently discovered better-than-2 approximations. Furthermore, we propose three new heuristic and one exact approach. These include a greedy algorithm considering link costs and the number of unique cuts covered, an approach based on minimum spanning trees and a local search algorithm that may improve a given solution by swapping links of paths. Our exact approach uses an ILP formulation with efficient cut enumeration as well as a fast initialization routine. We then perform an extensive experimental evaluation which shows that our algorithms are faster and yield the best solutions compared to the current state-of-the-art as well as the recently discovered better-than-2 approximation algorithms. Our novel local search algorithm can improve solution quality even further.

Figures (5)

• Figure 1: A graph and its weighted cactus graph with corresponding minimum cuts drawn as dashed lines of same color. Vertex colors encode the function $\Pi$.
• Figure 2: Example of applying MSTConnect (right) and improving its solution with our local search LS(3). All links have weight 1. Green edges represent the current solution, black dashed edges are the edges of the cactus graph representation, light gray edges are the non-solution links. The dashed green ($L_{out}$) and red ($L_{in}$) edges show the swap found in the local search.
• Figure 3: Performance profile comparing solution quality of approximations with $\epsilon=0.15$ on tiny cycle and star instances on the left. The right plot shows the running times.
• Figure 4: Performance profile for the state-of-the-art comparison on solution quality, running time and memory consumption on instances with small cost links.
• Figure 5: Performance profile for the state-of-the-art comparison on solution quality, running time and memory consumption on instances with large cost links.

Theorems & Definitions (4)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4