[Paper Review] On the Minimum Labelling Spanning bi-Connected Subgraph problem

Perez, J. A. Moreno, Consoli, S.|arXiv (Cornell University)|May 7, 2015

Vehicle Routing Optimization Methods5 references3 citations

TL;DR

This paper introduces the Minimum Labelling Spanning Bi-Connected Subgraph problem (MLSBP), a variant of the Minimum Labelling Spanning Tree problem (MLSTP), where the goal is to find a bi-connected (2-vertex-connected) subgraph using the minimum number of distinct edge labels. The authors adapt the exact A* algorithm and the Maximum Vertex Covering Algorithm (MVCA) heuristic from MLSTP to solve MLSBP, demonstrating that bi-connectivity can be enforced via DFS-based block detection and label selection based on reductions in connected components and blocks.

ABSTRACT

We introduce the minimum labelling spanning bi-connected subgraph problem (MLSBP) replacing connectivity by bi-connectivity in the well known minimum labelling spanning tree problem (MLSTP). A graph is bi-connected if, for every two vertices, there are, at least, two vertex-disjoint paths joining them. The problem consists in finding the spanning bi-connected subgraph or block with minimum set of labels. We adapt the exact method of the MLSTP to solve the MLSTB and the basic greedy constructive heuristic, the maximum vertex covering algorithm (MVCA). This proce- dure is a basic component in the application of metaheuristics to solve the problem.

Motivation & Objective

• To formalize and study the Minimum Labelling Spanning Bi-Connected Subgraph problem (MLSBP), which extends the MLSTP by requiring bi-connectivity instead of simple connectivity.
• To adapt existing exact and heuristic solution methods—specifically the A* search and MVCA—for the new problem class.
• To enable robust network design in transportation, communication, and social networks where node failures must not disconnect the network.
• To lay the foundation for future metaheuristic extensions and real-world applications involving multi-labeled edges.

Proposed method

• Adapt the A* backtracking algorithm used in MLSTP to solve MLSBP by replacing the connectivity check with a bi-connectivity test using DFS.
• Use DFS to detect cut vertices and identify bi-connected components (blocks), leveraging vertex visit order and back edges to determine articulation points.
• Implement a greedy constructive heuristic (MVCA) that selects labels based on minimizing the sum of connected components and blocks in the current subgraph.
• At each step, evaluate label inclusion by computing the change in the number of blocks and connected components, using BFS to find shortest paths between vertices in different blocks.
• Apply pruning rules in the exact method to reduce the search space by eliminating subsets of labels that cannot yield a better solution.
• Use a stack-based DFS traversal to maintain and extract blocks during the search, enabling efficient block detection and component counting.

Experimental results

Research questions

• RQ1 Can the A* exact algorithm used for MLSTP be effectively adapted to solve the more complex MLSBP by replacing connectivity checks with bi-connectivity checks?
• RQ2 How does the performance of the MVCA heuristic, originally designed for MLSTP, transfer to the MLSBP under bi-connectivity constraints?
• RQ3 What are the structural differences in label selection behavior when minimizing blocks and components versus minimizing only components in the context of bi-connectivity?
• RQ4 How do the number of blocks and connected components evolve during label addition, and what rules govern their reduction?
• RQ5 What is the impact of multi-labeled edges—where each edge may carry multiple labels—on the solution space and algorithmic design of MLSBP?

Key findings

• The exact A* algorithm adapted for MLSBP successfully identifies optimal solutions by testing subsets of labels and using DFS to verify bi-connectivity.
• The MVCA heuristic effectively reduces both the number of connected components and blocks by selecting labels that maximize structural consolidation in the subgraph.
• Adding an edge within the same block does not change the number of blocks or components; adding an edge between different components reduces components by one; adding an edge between different blocks in the same component reduces the number of blocks.
• The number of blocks and components can be updated incrementally and efficiently using BFS to compute shortest paths between vertices in different blocks.
• The algorithmic framework allows for future extension to multi-labeled edges, where each edge may have multiple labels representing different providers or communication modes.
• The problem is NP-hard, and the proposed methods provide a foundation for scalable metaheuristic solutions in real-world network design applications.