# Local Search with Dynamic-threshold Configuration Checking and Incremental Neighborhood Updating for Maximum k-plex Problem

national conference on artificial intelligence, 2020.

Keywords:

configuration checkingMaximum Clique Problemlocal search algorithmDynamic-threshold Configuration CheckingDouble-attributes Incremental Neighborhood UpdatingMore(10+)

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Abstract:

The Maximum k-plex Problem is an important combinatorial optimization problem with increasingly wide applications. In this paper, we propose a novel strategy, named Dynamic-threshold Configuration Checking (DCC), to reduce the cycling problem of local search. Due to the complicated neighborhood relations, all the previous local search alg...More

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Introduction

- In social network analysis, detecting a large cohesive subgraph is a fundamental and extensively studied topic with various applications.
- In some real-world applications, the networks of interest may be built based on empirical data with noises and faults.
- In these cases, large cohesive subgraphs hardly appear as ideal cliques.
- Large cohesive subgraphs hardly appear as ideal cliques
- To tackle this problem, many clique relaxation models have been proposed (Luce 1950; Mokken 1979; Pattillo et al 2013).
- The maximum k-plex problem, that is, to find a k-plex of maximum size on a given graph with a given integer k, has received increasing attention from researchers in the fields of social network analysis and data mining (Kondo and Okubo 2012; Xiao et al 2017; Conte et al 2018; Gao et al 2018)

Highlights

- In social network analysis, detecting a large cohesive subgraph is a fundamental and extensively studied topic with various applications
- We evaluate DCCplex and FD-TS on DIMACS and BHOSLIB benchmark with a cutoff time of 1000s
- We evaluate DCCplex on real-world massive graphs from Network Data Repository online (Rossi and Ahmed 2015), which have recently been used in testing the performance of local search algorithms (Rossi et al 2014; Wang, Cai, and Yin 2016; Cai, Lin, and Luo 2017)
- We have proposed a novel variant of Configuration Checking named Dynamic-threshold Configuration Checking (DCC) and the Double-attribute Neighborhood Update (DINU) scheme for the maximum k-plex problem
- Based on the DCC strategy and Double-attributes Incremental Neighborhood Updating (DINU) scheme, we develop a local search algorithm DCCplex
- The experimental result shows that DCCplex achieve good performance across a broad range of problem instances and update the lower bounds on the size of the maximum k-plexes on many hard instances

Results

**Evaluation on DIMACS and BHOSLIB**

The authors evaluate DCCplex and FD-TS on DIMACS and BHOSLIB benchmark with a cutoff time of 1000s.- Within the cutoff time of 1000s, BnB returns a solution and proves its optimality on 76 out of 108 (= 36 × 3) instances.
- For these 76 instances, DCCplex can achieve the same quality solution within the cutoff time of 100s.

Conclusion

- The authors have proposed a novel variant of Configuration Checking named DCC and the Double-attribute Neighborhood Update (DINU) scheme for the maximum k-plex problem.
- Based on the DCC strategy and DINU scheme, the authors develop a local search algorithm DCCplex.
- The experimental result shows that DCCplex achieve good performance across a broad range of problem instances and update the lower bounds on the size of the maximum k-plexes on many hard instances.
- SCCplex DCCplex #Better #Better k=2 1(1) 3(10).

Summary

## Introduction:

In social network analysis, detecting a large cohesive subgraph is a fundamental and extensively studied topic with various applications.- In some real-world applications, the networks of interest may be built based on empirical data with noises and faults.
- In these cases, large cohesive subgraphs hardly appear as ideal cliques.
- Large cohesive subgraphs hardly appear as ideal cliques
- To tackle this problem, many clique relaxation models have been proposed (Luce 1950; Mokken 1979; Pattillo et al 2013).
- The maximum k-plex problem, that is, to find a k-plex of maximum size on a given graph with a given integer k, has received increasing attention from researchers in the fields of social network analysis and data mining (Kondo and Okubo 2012; Xiao et al 2017; Conte et al 2018; Gao et al 2018)
## Results:

**Evaluation on DIMACS and BHOSLIB**

The authors evaluate DCCplex and FD-TS on DIMACS and BHOSLIB benchmark with a cutoff time of 1000s.- Within the cutoff time of 1000s, BnB returns a solution and proves its optimality on 76 out of 108 (= 36 × 3) instances.
- For these 76 instances, DCCplex can achieve the same quality solution within the cutoff time of 100s.
## Conclusion:

The authors have proposed a novel variant of Configuration Checking named DCC and the Double-attribute Neighborhood Update (DINU) scheme for the maximum k-plex problem.- Based on the DCC strategy and DINU scheme, the authors develop a local search algorithm DCCplex.
- The experimental result shows that DCCplex achieve good performance across a broad range of problem instances and update the lower bounds on the size of the maximum k-plexes on many hard instances.
- SCCplex DCCplex #Better #Better k=2 1(1) 3(10).

- Table1: Evaluation on DIMACS and BHOSLIB with k = 2, 3, 4
- Table2: Evaluation on Massive Graphs with k = 2, 3, 4
- Table3: Comparing SCCplex and DCCplex

Funding

- This paper was supported by the National Natural Science Foundation of China (No 61573386, 61976232), National Key R&D Program of China (No 2018YFC0830600), Guangdong Province Natural Science Foundation (No 2016A030313292, 2017A070706010 (soft science), 2018A030313086), Guangdong Province Science and Technology Plan projects (No 2016B030305007 and No 2017B010110011), Guangzhou Science and Technology Project (No 201804010435)
- Shaowei Cai was supported by Youth Innovation Promotion Association, Chinese Academy of Sciences (No.2017150)

Reference

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