# SCCWalk: An efficient local search algorithm and its improvements for maximum weight clique problem

Artificial Intelligence, pp. 1032302020.

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Keywords:

The maximum weight clique problemLocal searchStrong configuration checkingWalk perturbation procedureMassive graph

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Abstract:

The maximum weight clique problem (MWCP) is an important generalization of the maximum clique problem with wide applications. In this study, we develop two efficient local search algorithms for MWCP, namely SCCWalk and SCCWalk4L, where SCCWalk4L is improved from SCCWalk for large graphs. There are two main ideas in SCCWalk, including stro...More

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Introduction

- The maximum clique problem (MCP) consists of finding a clique with the maximum number of vertices.
- An important generalization of MCP is the maximum weight clique problem (MWCP), in which each vertex is associated with a non-negative integer, and the goal is to find a clique with the largest total weight.
- A novel formulation of selecting object region candidates simultaneously in all frames is proposed, to find the maximum weight cliques in a weighted region graph [6].
- Wu and Hao [8] study the winner determination problem (WDP) in combinatorial auctions by recasting this problem into MWCP

Highlights

- An important generalization of maximum clique problem (MCP) is the maximum weight clique problem (MWCP), in which each vertex is associated with a non-negative integer, and the goal is to find a clique with the largest total weight
- To further demonstrate the performance of SCCWalk on winner determination problem (WDP) benchmarks, we tested SCCWalk on 500 small instances with up to 1500 items and 1500 bids, which can be divided into five groups
- We developed two local search algorithms for the maximum weight clique problem (MWCP)
- SCCWalk4L used two reduction rules with some limitations to reduce the size of the MWCP problem
- Experimental results on massive graphs show that the best from multiple selection (BMS) heuristic and reduction rules significantly improve the performance of the algorithms on massive real-world graphs and large-scaled

Results

**Experimental results with five competitors on the**

DIMACS benchmark.**Experimental results on the**

DIMACS are shown in Table 3.- The results shown in Table 6 illustrate that SCCWalk has a better performance than other competitors in these instances except for Vowel, where WLMC takes a shorter time to find the same solution quality.
- The superiority of SCCWalk on the classical benchmarks is clearly illustrated by Fig. 2, which summarizes the run time distributions of the MWCP algorithms when both SCCWalk and one competitor find the maximum solution values

Conclusion

- The authors developed two local search algorithms for the maximum weight clique problem (MWCP).
- The abovementioned heuristics were used to develop a local search algorithm named SCCWalk.
- SCCWalk4L used two reduction rules with some limitations to reduce the size of the MWCP problem.
- Experimental results on massive graphs show that the BMS heuristic and reduction rules significantly improve the performance of the algorithms on massive real-world graphs and large-scaled frb18-1-2.2-10-0.19748 frb18-2-2.2-10-0.19748 frb18-3-2.2-10-0.19748 frb18-4-2.2-10-0.19748 frb18-5-2.2-10-0.19748 frb19-1-2.2-10-0.19748 frb19-2-2.2-10-0.19748 frb19-3-2.2-10-0.19748 frb19-4-2.2-10-0.19748 frb19-5-2.2-10-0.19748 frb20-1-2.2-10-0.19748 frb20-2-2.2-10-0.19748 frb20-3-2.2-10-0.19748 frb20-4-2.2-10-0.19748 frb20-5-2.2-10-0.19748 frb21-1-2.2-10-0.19748 frb21-2-2.2-10-0.19748 frb21-3-2.2-10-0.19748 frb21-4-2.2-10-0.19748 frb21-5-2.2-10-0.19748

Summary

## Introduction:

The maximum clique problem (MCP) consists of finding a clique with the maximum number of vertices.- An important generalization of MCP is the maximum weight clique problem (MWCP), in which each vertex is associated with a non-negative integer, and the goal is to find a clique with the largest total weight.
- A novel formulation of selecting object region candidates simultaneously in all frames is proposed, to find the maximum weight cliques in a weighted region graph [6].
- Wu and Hao [8] study the winner determination problem (WDP) in combinatorial auctions by recasting this problem into MWCP
## Results:

**Experimental results with five competitors on the**

DIMACS benchmark.**Experimental results on the**

DIMACS are shown in Table 3.- The results shown in Table 6 illustrate that SCCWalk has a better performance than other competitors in these instances except for Vowel, where WLMC takes a shorter time to find the same solution quality.
- The superiority of SCCWalk on the classical benchmarks is clearly illustrated by Fig. 2, which summarizes the run time distributions of the MWCP algorithms when both SCCWalk and one competitor find the maximum solution values
## Conclusion:

The authors developed two local search algorithms for the maximum weight clique problem (MWCP).- The abovementioned heuristics were used to develop a local search algorithm named SCCWalk.
- SCCWalk4L used two reduction rules with some limitations to reduce the size of the MWCP problem.
- Experimental results on massive graphs show that the BMS heuristic and reduction rules significantly improve the performance of the algorithms on massive real-world graphs and large-scaled frb18-1-2.2-10-0.19748 frb18-2-2.2-10-0.19748 frb18-3-2.2-10-0.19748 frb18-4-2.2-10-0.19748 frb18-5-2.2-10-0.19748 frb19-1-2.2-10-0.19748 frb19-2-2.2-10-0.19748 frb19-3-2.2-10-0.19748 frb19-4-2.2-10-0.19748 frb19-5-2.2-10-0.19748 frb20-1-2.2-10-0.19748 frb20-2-2.2-10-0.19748 frb20-3-2.2-10-0.19748 frb20-4-2.2-10-0.19748 frb20-5-2.2-10-0.19748 frb21-1-2.2-10-0.19748 frb21-2-2.2-10-0.19748 frb21-3-2.2-10-0.19748 frb21-4-2.2-10-0.19748 frb21-5-2.2-10-0.19748

- Table1: Information of the DGCA benchmark
- Table2: The parameter settings used in SCCWalk on DIMACS, BHOSLIB, WDP and DGCA benchmarks
- Table3: Experimental results of ReTS1, ReTS2, FastWClq, RRWL, WLMC, and SCCWalk on the DIMACS benchmark
- Table4: Experimental results of ReTS1, ReTS2, FastWClq, RRWL, WLMC and SCCWalk on the BHOSLIB benchmark
- Table5: Experimental results of ReTS1, ReTS2, FastWClq, RRWL, WLMC and SCCWalk on the WDP benchmark
- Table6: Experimental results of ReTS1, ReTS2, FastWClq, RRWL, WLMC and SCCWalk on the DGCA benchmark
- Table7: Experimental results of CCWalk and SCCWalk on the classical benchmarks. A positive δmax or δavg indicates SCCWalk finds better quality clique than CCWalk
- Table8: Comparative results for SCCWalk and five competitors on the classical benchmarks
- Table9: Information of large-scaled FRB instances
- Table10: Experimental results of ReTS1, ReTS2, FastWClq, RRWL, WLMC and SCCWalk4L on the real-world massive graphs I
- Table11: Experimental results of ReTS1, ReTS2, FastWClq, RRWL, WLMC and SCCWalk4L on the real-world massive graphs II
- Table12: Among the five competitors, FastWClq, RRWL, WLMC, and ReTS2 perform badly, so we focus on the comparison between ReTS1 and SCCWalk4L. Overall, SCCWalk4L finds better solutions than ReTS1, on these graphs with massive edges. We observe that ReTS1 can not reach maximum weight clique for 13 graphs, while SCCWalk4L finds maximum weight cliques for all 20 graphs. Experimental results of ReTS1, ReTS2, FastWClq, RRWL, WLMC and SCCWalk4L on the large-scaled FRB instances
- Table13: Experimental results of SCCWalk and SCCWalk4L on representative classical and massive graphs
- Table14: Comparative results for SCCWalk4L and five competitors on the massive graphs

Related work

- As is well known, the decision version of MCP is one of Karp’s prominent 21 NP-complete combinatorial problems [9]. Both MCP and MWCP have been proved to be NP-hard, and the current best-known approximation algorithms can only achieve an approximate ratio of O(n(log log n)2/(log n)3) [10]. Thus, a considerable amount of effort has been devoted to finding a “good” clique within a reasonable time. Up to now, there are mainly two types of algorithms for MCP and MWCP, i.e., exact and heuristic algorithms.
**Related works on exact algorithms for**

MCP and MWCP A number of exact algorithms have been proposed to solve MCP and MWCP. A classic branch and bound algorithm was

MCQ [11], which used a vertex ordering heuristic for independent set partition. The MCQ algorithm was further improved by computing the degrees of vertices dynamically, resulting in the MaxCliqueDyn algorithm [12]. Li and Quan [13] proposed a new encoding from MCP into MaxSAT, and then applied MaxSAT reasoning to improve the upper bound. San et al [14] presented the BBMCSP algorithm, which was based on a leading bit-parallel non-sparse solver, and used a novel sparse encoding for the adjacency matrix. MaxCLQ [15] combined an efficient preprocessing procedure and incremental MaxSAT reasoning in a branch and bound scheme. Recently, MoMC [16] combined a dynamic vertex ordering heuristic and a static vertex ordering heuristic during the search.**Related works on heuristic algorithms for**

MCP and MWCP Although exact algorithms can guarantee the optimality of their solutions, they may fail to solve hard instances of large scale. For solving large-sized instances, a popular approach is local search, which can find an approximate solution within a reasonable time. There are numerous local search algorithms for MCP [24,25,26,27,28,29]. Among these, DLS [25] was a milestone algorithm that used vertex penalties, and dynamically adjusted them during the search. DLS was further improved into a two-phase algorithm called phased local search (PLS) [26]. Wu and Hao proposed a tabu search algorithm with a k-fixed penalty strategy [28]. Benlic and Hao proposed a breakout local search algorithm for MCP [29]. MCP is closely related to minimum vertex cover (MinVC) and maximum independent set problems, and algorithms for these two problems can be directly used to solve MCP.

Funding

- This work is supported by the Fundamental Research Funds for the Central Universities 2412018QD022 and 2412018ZD017, NSFC (under grant Nos. 61806050, 61972063, 61976050, 61972384)
- Shaowei Cai was supported by Youth Innovation Promotion Association, Chinese Academy of Sciences (No 2017150)

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