A Fast and Scalable Algorithm for Finding the Maximum Clique in Large Sparse Networks

Khái niệm cốt lõi

A novel algorithm based on efficient graph decomposition and the concept of Complete-Upper-Bound-Induced Subgraphs (CUBISs) enables fast and scalable computation of the maximum clique in large sparse networks.

Tóm tắt

The paper presents a fast algorithm for solving the Maximum Clique Problem (MCP) in large sparse networks. The key innovations are:

Pre-pruning: The algorithm first removes peripheral nodes with low degrees that cannot contribute to the maximum clique, reducing the size of the original graph.

Complete-Upper-Bound-Induced Subgraphs (CUBISs): The algorithm constructs one or two small-scale CUBISs, which are node-induced subgraphs that retain the potential structures for forming the maximum clique. This ensures that the search for the maximum clique is performed on a much smaller scale compared to the original graph.

Efficient Search: The Bron-Kerbosch algorithm is used to search for the maximum clique within the CUBISs, with additional pruning techniques to further reduce the search space.

The experiments on 50 empirical networks with up to 20 million nodes demonstrate that the scales of the CUBISs are largely independent of the original network size, enabling approximately linear runtime. This makes the algorithm highly scalable and applicable to large sparse networks. The algorithm also outperforms a benchmark algorithm on the majority of the tested networks.

Thống kê

The maximum clique size in the neighborhood subgraph of the node with the maximum degree is |M|.
The core number of a node i is cn(i).
The largest core number in the graph G is cmax(G).
The second-largest core number in the graph G is c2nd.
The minimum core number c_min that satisfies c_min + 1 >= |M| is determined.

Trích dẫn

"The Maximum Clique Problem (MCP), which involves identifying all complete subgraphs (subnetworks) with the maximum number of nodes, wherein every pair of nodes is adjacent in a general graph (network), stands as a fundamental combinatorial optimization challenge."
"Existing algorithms for MCP, based on their precision, can be broadly categorized into two classes: exact algorithms and heuristic algorithms."
"Compared with the maximum degree of node, the core number serves as a more efficient upper bound for the size of the maximum clique."

Thông tin chi tiết chính được chắt lọc từ

A Fast Maximum Clique Algorithm Based on Network Decomposition for Large Sparse Networks

by Tian... lúc arxiv.org 04-19-2024

https://arxiv.org/pdf/2404.11862.pdf

Yêu cầu sâu hơn

How can the proposed algorithm be extended to handle dynamic networks, where the graph structure changes over time

To extend the proposed algorithm to handle dynamic networks, where the graph structure changes over time, several modifications and additions can be made:

Incremental Updates: Instead of recomputing the entire graph decomposition and maximum clique search from scratch every time the network changes, the algorithm can be adapted to incorporate incremental updates. When a node or edge is added or removed, the algorithm can identify the affected parts of the graph and update the CUBIS structures accordingly.

Dynamic Core Number Calculation: The core number of nodes can be recalculated dynamically as the network evolves. This would involve updating the core numbers of nodes affected by the changes in the network structure and adjusting the CUBISs based on these updated core numbers.

Efficient Pruning Strategies: Develop efficient pruning strategies that take into account the dynamic nature of the network. Nodes that are no longer relevant for forming the maximum clique due to changes in the network structure can be pruned more effectively to reduce the search space.

Adaptive Parameter Tuning: Parameters such as the threshold for constructing CUBISs or the size of the maximum clique obtained from the first CUBIS can be dynamically adjusted based on the evolving network characteristics to optimize the algorithm's performance.

By incorporating these adaptations, the algorithm can effectively handle dynamic networks and efficiently find the maximum clique even as the graph structure changes over time.

What other graph decomposition techniques could be explored to further improve the scalability and efficiency of the maximum clique algorithm

To further improve the scalability and efficiency of the maximum clique algorithm, the following graph decomposition techniques could be explored:

Community Detection: Utilize community detection algorithms to identify dense subgraphs or communities within the network. By focusing on these dense regions, the algorithm can potentially reduce the search space for finding the maximum clique.

Graph Partitioning: Divide the graph into smaller partitions or clusters using graph partitioning techniques. This can enable parallel processing of different parts of the graph, leading to faster computation of the maximum clique.

Hierarchical Decomposition: Implement a hierarchical decomposition approach where the graph is decomposed into multiple levels of subgraphs. This hierarchical structure can help in efficiently identifying potential maximum cliques at different levels of granularity.

Streaming Algorithms: Explore streaming algorithms that can process the network data in a continuous manner, adapting to changes and updates in real-time. This can be particularly useful for handling large-scale dynamic networks.

By incorporating these graph decomposition techniques, the algorithm can be further optimized for scalability and efficiency, especially in the context of large and evolving networks.

What insights from this work on the maximum clique problem could be applied to solve other challenging combinatorial optimization problems in large-scale networks

Insights from this work on the maximum clique problem can be applied to solve other challenging combinatorial optimization problems in large-scale networks in the following ways:

Maximum Independent Set: Techniques used for identifying maximum cliques, such as core number calculations and efficient graph decomposition, can be adapted for finding maximum independent sets in networks. This can help in identifying sets of nodes that are not directly connected, but have no common neighbors.

Graph Coloring: The upper bound constraints derived from core numbers for the maximum clique problem can be utilized for graph coloring problems. By leveraging similar pruning strategies and efficient search algorithms, the algorithm can be extended to find optimal graph colorings in large networks.

Optimization Problems: The branch-and-bound framework and heuristic approaches employed in solving the maximum clique problem can be generalized to tackle other optimization problems in networks, such as minimum vertex cover, maximum cut, or subgraph isomorphism. By adapting the algorithmic principles, these problems can be efficiently addressed in large-scale network settings.

Parallel and Distributed Computing: Techniques used for parallelizing the maximum clique algorithm can be applied to other optimization problems, enabling distributed computing and efficient utilization of computational resources for solving complex network optimization tasks.

By transferring the methodologies and insights gained from solving the maximum clique problem, a wide range of combinatorial optimization challenges in large-scale networks can be effectively addressed.