[Paper Review] A novel and efficient algorithm for scanning all minimal cutsets of a graph

Ahmad R. Sharafat, Omid Reza Marouzi|ArXiv.org|Nov 27, 2002

Complexity and Algorithms in Graphs13 references5 citations

TL;DR

This paper presents a novel, efficient algorithm for enumerating all minimal cutsets in a graph using breadth-first search (BFS) combined with edge contraction, introducing pivot vertices and absorbable clusters to reduce complexity. The algorithm achieves linear time complexity relative to the number of cutsets, significantly outperforming prior methods in simulation results.

ABSTRACT

We propose a novel algorithm for enumerating and listing all minimal cutsets of a given graph. It is known that this problem is NP-hard. We use connectivity properties of a given graph to develop an algorithm with reduced complexity for finding all its cutsets. We use breadth first search (BFS) method in conjunction with edge contraction to develop the algorithm. We introduce the concepts of a pivot vertex and absorbable clusters and use them to develop an enhanced recursive contraction algorithm. The complexity of the proposed algorithm is proportionate to the number of cutsets. We present simulation results to compare the performance of our proposed algorithm with those of existing methods.

Motivation & Objective

• To address the NP-hard problem of enumerating all minimal cutsets in a graph.
• To reduce computational complexity by leveraging connectivity properties and graph structure.
• To develop a recursive contraction algorithm enhanced by pivot vertices and absorbable clusters.
• To achieve a time complexity proportional to the number of minimal cutsets.
• To outperform existing algorithms through simulation-based evaluation.

Proposed method

• The algorithm uses breadth-first search (BFS) to explore graph connectivity and identify potential cutsets.
• Edge contraction is applied recursively to simplify the graph while preserving minimal cutset information.
• Pivot vertices are introduced as strategic reference points to guide the contraction process and avoid redundant computation.
• Absorbable clusters—sets of vertices that can be collapsed without altering cutset structure—are identified and processed efficiently.
• The recursive contraction process is guided by connectivity checks and pivot-based pruning to maintain correctness and efficiency.
• The algorithm ensures all minimal cutsets are generated exactly once through careful control of the contraction and expansion phases.

Experimental results

Research questions

• RQ1 Can a graph traversal-based approach with edge contraction efficiently enumerate all minimal cutsets?
• RQ2 How can pivot vertices and absorbable clusters reduce redundancy and improve algorithmic complexity?
• RQ3 To what extent does the proposed algorithm outperform existing methods in terms of runtime and scalability?
• RQ4 Is the time complexity of the algorithm proportional to the number of minimal cutsets, as claimed?
• RQ5 What structural properties of graphs enable the algorithm’s efficiency?

Key findings

• The proposed algorithm achieves a time complexity that is linearly proportional to the number of minimal cutsets in the graph.
• Simulation results demonstrate that the algorithm significantly outperforms existing methods in terms of execution time and scalability.
• The use of pivot vertices and absorbable clusters effectively reduces redundant exploration and improves pruning efficiency.
• The algorithm correctly enumerates all minimal cutsets without duplication or omission.
• The integration of BFS with recursive edge contraction enables efficient traversal and structure preservation.
• The method is particularly effective on graphs with high connectivity and dense substructures due to its structural optimization.