An Algorithm with a Delay of O(kΔ) for Enumerating All Connected Induced Subgraphs of Size k

Core Concepts

The proposed algorithm enumerates all connected induced subgraphs of size k in a graph with a delay of O(kΔ), where k is the subgraph size and Δ is the maximum degree of the graph.

Abstract

The paper presents a new algorithm for enumerating all connected induced subgraphs of size k in an undirected graph. The key highlights are:

1. The proposed algorithm has a delay of O(kΔ), which improves upon the current best delay bound of O(k²Δ) in the literature.
2. The algorithm works by visiting the vertices in reverse order of a depth-first search and expanding the subgraphs by adding neighboring vertices in a depth-first manner. Visited vertices are marked as closed to avoid redundant enumeration.
3. The algorithm uses global data structures like arrays and boolean flags to efficiently track the visited vertices and enable fast lookups during the enumeration process.
4. The correctness of the algorithm is proven, showing that each connected induced subgraph of size k is enumerated exactly once.
5. The space complexity of the algorithm is shown to be O(|V| + |E|), where |V| is the number of vertices and |E| is the number of edges in the input graph.

Stats

The number of connected induced subgraphs of size k in a graph G with n vertices and maximum degree Δ is upper bounded by n * (eΔ)^k / (Δ-1)^k.
The delay of the proposed algorithm is O(kΔ).

Quotes

"The delay of these algorithms is O(k²Δ)."
"The delay of the proposed algorithm is O(kΔ)."

Key Insights Distilled From

An algorithm with a delay of $\mathcal{O}(kΔ)$ for enumerating connected induced subgraphs of size $k$

by Chenglong Xi... at arxiv.org 04-22-2024

https://arxiv.org/pdf/2404.12559.pdf

Deeper Inquiries

How can the proposed algorithm be extended to handle directed graphs or weighted graphs

To extend the proposed algorithm to handle directed graphs, we can modify the neighbor set calculation to consider the direction of edges. In a directed graph, the neighbor set of a vertex would consist of both its incoming and outgoing neighbors. Additionally, we would need to adjust the traversal and expansion of subgraphs to account for the directed nature of the edges. For weighted graphs, we can incorporate edge weights into the algorithm by considering them during the subgraph expansion process. The algorithm can be adapted to prioritize or exclude certain edges based on their weights when adding vertices to the subgraph.

What are the practical implications of the improved delay bound in real-world applications like bioinformatics or social network analysis

The improved delay bound of the algorithm has significant practical implications in real-world applications such as bioinformatics and social network analysis. In bioinformatics, where analyzing biological networks is crucial, the ability to efficiently enumerate connected induced subgraphs can aid in identifying functional modules or motifs within complex biological systems. This can lead to insights into protein interactions, gene regulatory networks, and disease mechanisms. Similarly, in social network analysis, the algorithm's improved efficiency can help in identifying cohesive groups or communities within social networks, facilitating targeted marketing, influence analysis, or community detection.

Can the techniques used in this algorithm be applied to other graph enumeration problems beyond connected induced subgraphs

The techniques used in this algorithm for enumerating connected induced subgraphs can be applied to other graph enumeration problems beyond this specific case. For instance, the approach of incrementally expanding a subgraph by adding neighboring vertices in a depth-first search manner can be adapted for enumerating other types of subgraphs, such as induced paths, cycles, or trees. By adjusting the criteria for adding vertices and defining the termination conditions, the algorithm can be tailored to enumerate various types of subgraphs based on different structural constraints or properties. This flexibility makes the algorithm versatile and applicable to a wide range of graph enumeration problems beyond connected induced subgraphs.