# Reduction and Local Search for Weighted Graph Coloring Problem

national conference on artificial intelligence, 2020.

Keywords:

maximum weight clique problemconfiguration checkingconventional benchmarkvertex coloring problemmaximum weight stable set problemMore(10+)

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Abstract:

The weighted graph coloring problem (WGCP) is an important extension of the graph coloring problem (GCP) with wide applications. Compared to GCP, where numerous methods have been developed and even massive graphs with millions of vertices can be solved well, fewer works have been done for WGCP, and no solution is available for solving WGC...More

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Introduction

- In classical GCP, a basic assumption is that vertices in the graph are important, it is hard to hold in many real world scenarios where each vertex is associated with various types of weights.
- WGCP aims to partition all the vertices into several disjoint subsets such that the sum of those subset costs is minimized, where the cost of each subset is given by the maximum weight of a vertex within the current subset.
- WGCP can be directly encoded as the maximum weight stable set problem (MWSS) (Cornaz, Furini, and Malaguti 2017)

Highlights

- Graph coloring problem (GCP) is a well-known combinatorial optimization problem
- Motivated to contribute to solving weighted graph coloring problem (WGCP) problems with massive graphs, in this paper we propose a novel efficient WGCP algorithm, namely RedLS (Reduction plus Local Search)
- RedLS proves the optimal solution for 4 instances from the conventional benchmarks, and the number of reduced vertices is on average 12.45, indicating that the reduction rule is not effective for conventional benchmarks
- This paper introduced a lower bound, a reduction rule, and a local search algorithm for WGCP
- We proposed the reduction rule based on clique sampling to remove some unnecessary vertices
- Experiments on conventional benchmarks and massive graphs indicate that RedLS significantly outperforms the state-of-the-art algorithms

Results

- RedLS finds better values than 2Phase for 96 instances.
- The authors mainly compare RedLS with MWSS and AFISA.
- Most instances are so easy that RedLS, MWSS and AFISA find the same quality values.
- The authors do not report the detailed results of such instances in Table 1, but the authors summarize the run time comparisons in Figure 1.
- The good results of RedLS on conventional benchmarks mainly come from the power of the underlying local search algorithm

Conclusion

- This paper introduced a lower bound, a reduction rule, and a local search algorithm for WGCP.
- The authors proposed the reduction rule based on clique sampling to remove some unnecessary vertices.
- In the local search algorithm, the authors designed the selection rules and the new variant of configuration checking to determine which operation is the candidate selected operation in the local search procedure.
- As for future work, the authors will attempt to further improve RedLS via a few novel reduction rules

Summary

## Introduction:

In classical GCP, a basic assumption is that vertices in the graph are important, it is hard to hold in many real world scenarios where each vertex is associated with various types of weights.- WGCP aims to partition all the vertices into several disjoint subsets such that the sum of those subset costs is minimized, where the cost of each subset is given by the maximum weight of a vertex within the current subset.
- WGCP can be directly encoded as the maximum weight stable set problem (MWSS) (Cornaz, Furini, and Malaguti 2017)
## Results:

RedLS finds better values than 2Phase for 96 instances.- The authors mainly compare RedLS with MWSS and AFISA.
- Most instances are so easy that RedLS, MWSS and AFISA find the same quality values.
- The authors do not report the detailed results of such instances in Table 1, but the authors summarize the run time comparisons in Figure 1.
- The good results of RedLS on conventional benchmarks mainly come from the power of the underlying local search algorithm
## Conclusion:

This paper introduced a lower bound, a reduction rule, and a local search algorithm for WGCP.- The authors proposed the reduction rule based on clique sampling to remove some unnecessary vertices.
- In the local search algorithm, the authors designed the selection rules and the new variant of configuration checking to determine which operation is the candidate selected operation in the local search procedure.
- As for future work, the authors will attempt to further improve RedLS via a few novel reduction rules

- Table1: Results of RedLS, AFISA, MWSS and 2Phase on conventional benchmarks
- Table2: Results of RedLS and AFISA on massive graphs with w1
- Table3: Summary of comparison between RedLS, AFISA, MWSS and 2Phase on massive graph with w2. #Better indicates the number of instances where an algorithm finds better minimal (average) solutions. #N/A denotes the number of instances where an algorithm fails to find a solution under the given time limit

Funding

- This work is supported by the Fundamental Research Funds for the Central Universities 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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