alberto-santini / sgcp-via-cliques

Solving the (Weighted) Selective Graph Colouring Problem via Maximum Cliques

SGCP via Cliques

Solving the (Weighted) Selective Graph Colouring Problem via Maximum Cliques.

Citation

If you use this software, please cite the following paper:

@article{cornaz2019selective,
  title={A note on selective line-graphs and partition colorings},
  author={Cornaz, Denis and Furini, Fabio and Malaguti, Enrico and Santini, Alberto},
  journal={{Operations Research Letters}},
  year=2019,
  volume=47,
  issue=6,
  pages={565--568},
  doi={10.1016/j.orl.2019.08.005}
}

You can also cite this repository via Zenodo.

@misc{sgcp_cliques_github,
    title={Solving the (Weighted) Selective Graph Colouring Problem via Maximum Cliques},
    author={Santini, Alberto},
    date={2019-07-16},
    howpublished={Github repository},
    doi={10.5281/zenodo.3338690},
    url={https://github.com/alberto-santini/sgcp-via-cliques/}
}

Instances

Instances are distributed in folder generator/graphs, which contains both the original VCP instances and the derived SGCP and WSGCP instances. The programe used to generate the instance is also available, in folder generator.

Results

The table below synthetically compares a Mixed-Integer model (see the paper) and the line-graph method. Columns under "MIP" refer to the model, while columns under "LG" refer to the line-graph method. Columns "Opt" give the number of optimal solutions found by each method, while column "Gap%" reports the average gap, computed as UB-LB/UB. UB is the value of the incumbent solutionand LB is the best dual bound. Columns "Time" report the average running time in seconds. We ran both algorithms with a time limit of 1 hour and include the unsolved instances in the computation of the average time. For the line-graph algorithm, we report under columns "nV" and "nE",respectively, the average number of vertices and edges of the final graphs on which we solve the Maximum Weighted Stable Set Problem. The line-graph algorithm is able to solve all instances very quickly, whereas using the Mixed-Integer model leads to having unsolved instances and larger average times. Our approach solves all the 531 instances in less than 12 seconds in average, whereas the direct use of Cplex only solves 509 instances in less than 1 hour.

#V	MIP Opt	MIP Gap%	MIP Time	LG nV	LG nE	LG Opt	LG Gap%	LG Time
10	45	0	0.05	21	132	45	0	0.00
20	54	0	0.20	82	2'115	54	0	0.00
30	54	0	0.79	189	10'795	54	0	0.01
40	54	0	3.02	335	37'078	54	0	0.02
50	53	0.8	90.22	523	87'929	54	0	0.24
60	51	2.1	204.63	750	188'612	54	0	0.28
70	49	3.5	336.07	1021	342'170	54	0	2.45
80	50	3.8	271.67	1324	591'088	54	0	1.48
90	49	5.1	363.18	1688	942'813	54	0	12.37
100	50	3.7	281.82	2073	1'453'197	54	0	5.62