# An efficient local search algorithm for solving maximum edge weight clique problem in large graphs

Journal of Combinatorial Optimization, pp. 933-954, 2020.

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Keywords:

Graph reduction Maximum edge weight clique problem Stochastic local search

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Abstract:

Maximum vertex weight clique problem (MVWCP) and maximum edge weight clique problem (MEWCP) are two significant generalizations of maximum clique problem (MCP), and can be widely used in many real-world applications including molecular biology, broadband network design and pattern recognition. Recently, breakthroughs have been made for so...More

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Introduction

- Due to the theoretical significance and practical application value, considerable efforts have been made to develop algorithms for MCP, MVWCP and MEWCP.
- Algorithms for solving these problems are usually categorized into two classes: complete algorithms and incomplete algorithms

Highlights

- Given an undirected graph G = (V, E), where V is the set of vertices and E is the set of edges, a clique C is a subset of V, such that all vertices in C are connected
- We propose a new algorithm for maximum edge weight clique problem (MEWCP), which works in three phases, i.e., clique construction, local search and graph reduction
- To evaluate the effectiveness of the graph reduction algorithm proposed in Sect. 3.2, as well as the techniques utilized in the clique construction phase and the stochastic local search phase, we disable the graph reduction algorithm in ReConSLS, resulting in another solver named ConSLS
- We present an effective local search algorithm named ReConSLS, which works in three phases, i.e. clique construction, stochastic local search and graph reduction
- We are devoted to improving the performance over the state-of-the-art algorithms for solving MEWCP on large graphs
- We conducted experiments to compare ConSLS against ReConSLS, CERS, LSMR, LSCC and LSCC + Best from Multiple Selections’ (BMS), and the related results show that ConSLS is the second best solver following ReConSLS, indicating the effectiveness of our upper bound function and the techniques utilized in the clique construction phase and the stochastic local search phase

Methods

- The authors include four state-of-the-art SLS solvers as the competitors. CERS (Fan et al 2017b) and LSMR (Li et al 2018) are two efficient solvers for MEWCP in large graphs.
- Breakthroughs have been made in MVWCP solving, resulting in several state-of-the-art solvers, such as LSCC, LSCC + BMS (Wang et al 2016), FastWClq (Cai and Lin 2016) and WLMC (Jiang et al 2017).
- The authors implemented ReConSLS, LSCC and LSCC + BMS in C++.
- CERS was implemented in C++ by its author and could be downloaded online5.
- All the experiments in this paper were carried out on a workstation under the operating system CentOS, with Intel(R) Xeon(R) CPU E5-2620 2.10GHz CPU, 20MB L3 cache and 128GB RAM

Results

- Table 1 presents the comparative results of ReConSLS, CERS, LSMR, LSCC and LSCC + BMS on graphs from Network Repository.
- Table 3 presents the comparative results of ReConSLS, CERS, LSMR, LSCC and LSCC + BMS on 18 graphs from KONECT.
- ReConSLS finds the best clique weight on all 10 runs for all 18 graphs, while this figure for CERS, LSMR, LSCC and LSCC + BMS is only 13, 13, 14 and 13, respectively.
- On 8 of the 18 graphs, ReConSLS’s speed of finding the best averaged clique weight is more than 3 times as fast as the second fastest competitor’s

Conclusion

- The authors present an effective local search algorithm named ReConSLS, which works in three phases, i.e. clique construction, stochastic local search and graph reduction.
- The authors conducted experiments to compare ConSLS against ReConSLS, CERS, LSMR, LSCC and LSCC + BMS, and the related results show that ConSLS is the second best solver following ReConSLS, indicating the effectiveness of the upper bound function and the techniques utilized in the clique construction phase and the stochastic local search phase.

Summary

## Introduction:

Due to the theoretical significance and practical application value, considerable efforts have been made to develop algorithms for MCP, MVWCP and MEWCP.- Algorithms for solving these problems are usually categorized into two classes: complete algorithms and incomplete algorithms
## Methods:

The authors include four state-of-the-art SLS solvers as the competitors. CERS (Fan et al 2017b) and LSMR (Li et al 2018) are two efficient solvers for MEWCP in large graphs.- Breakthroughs have been made in MVWCP solving, resulting in several state-of-the-art solvers, such as LSCC, LSCC + BMS (Wang et al 2016), FastWClq (Cai and Lin 2016) and WLMC (Jiang et al 2017).
- The authors implemented ReConSLS, LSCC and LSCC + BMS in C++.
- CERS was implemented in C++ by its author and could be downloaded online5.
- All the experiments in this paper were carried out on a workstation under the operating system CentOS, with Intel(R) Xeon(R) CPU E5-2620 2.10GHz CPU, 20MB L3 cache and 128GB RAM
## Results:

Table 1 presents the comparative results of ReConSLS, CERS, LSMR, LSCC and LSCC + BMS on graphs from Network Repository.- Table 3 presents the comparative results of ReConSLS, CERS, LSMR, LSCC and LSCC + BMS on 18 graphs from KONECT.
- ReConSLS finds the best clique weight on all 10 runs for all 18 graphs, while this figure for CERS, LSMR, LSCC and LSCC + BMS is only 13, 13, 14 and 13, respectively.
- On 8 of the 18 graphs, ReConSLS’s speed of finding the best averaged clique weight is more than 3 times as fast as the second fastest competitor’s
## Conclusion:

The authors present an effective local search algorithm named ReConSLS, which works in three phases, i.e. clique construction, stochastic local search and graph reduction.- The authors conducted experiments to compare ConSLS against ReConSLS, CERS, LSMR, LSCC and LSCC + BMS, and the related results show that ConSLS is the second best solver following ReConSLS, indicating the effectiveness of the upper bound function and the techniques utilized in the clique construction phase and the stochastic local search phase.

- Table1: Experimental results on graphs from network repository (For the sake of space, we do not report on graphs that all the solvers find the same best clique weight within 25 s, we do not report ‘Wavg’ which is equal to ‘Wmax’)
- Table2: Experimental results on 139 graphs from network repository
- Table3: Comparative results of ReConSLS, CERS, LSMR, LSCC and LSCC + BMS on real world graphs from KONECT
- Table4: Experimental results of ReConSLS, ConSLS, CERS, LSMR, LSCC and LSCC + BMS on all the graphs

Funding

- This work is partially supported by the National Key Research and Development Program of China under Grant 2017YFB0202502
- Shaowei Cai is supported by Youth Innovation Promotion Association, Chinese Academy of Sciences (No 2017150)

Reference

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