CP and MIP approaches for soccer analysis

Robinson Duque; Alejandro Arbelaez; Juan Francisco Díaz

doi:10.3934/jimo.2018109

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October 2019, 15(4): 1535-1564. doi: 10.3934/jimo.2018109

CP and MIP approaches for soccer analysis

Robinson Duque ^1,, , Alejandro Arbelaez ^2, and Juan Francisco Díaz ^1,

1.	AVISPA Research Group, Escuela de Ingeniería de Sistemas y Computación, Universidad del Valle, Colombia
2.	Riomh Research Group, Department of Computer Science, Cork Institute of Technology, Ireland

* Corresponding author: Robinson Duque

Received December 2016 Revised August 2017 Published July 2018

Soccer is one of the most popular sports in the world with millions of fans that usually raise interesting questions when the competition is partially completed. One interesting question relates to the elimination problem which consists in checking at some stage of the competition if a team i still has a theoretical chance to finish first in a league or be within the first k teams in a tournament to qualify to the playoffs (e.g., become the champion if k = 1). Some other interesting problems from the literature are the guaranteed qualification problem, the possible qualification problem, the score vector problem, promotion and relegation problem. These problems are NP-complete for the actual FIFA pointing rule system (0 points-loss, 1 point-tie, 3 points-win). SABIO is an online platform that helps users discover information related to soccer by letting them formulate questions in form of constraints and go beyond the classical soccer computational problems. In this paper we considerably improve the performance of an existing constraint programming (CP) model and combine the use of mixed integer programming (MIP) and CP to answer general soccer queries in a real-time application.

Keywords: Constraint programming, mixed integer programming, machine learning, soccer analysis, elimination problem.

Mathematics Subject Classification: Primary: 90C11, 62F30; Secondary: 68T20, 90C90, 68T05, 97R99.

Citation: Robinson Duque, Alejandro Arbelaez, Juan Francisco Díaz. CP and MIP approaches for soccer analysis. Journal of Industrial & Management Optimization, 2019, 15 (4) : 1535-1564. doi: 10.3934/jimo.2018109

References:

[1]	I. Adler, A. L. Erera, D. S. Hochbaum and E. V. Olinick, Baseball, optimization, and the world wide web, Interfaces, 32 (2002), 12-22. doi: 10.1287/inte.32.2.12.67. Google Scholar
[2]	F. Alarcón, G. Durán, M. Guajardo, J. Miranda, H. Muñoz, L. Ramírez, M. Ramírez, D. Sauré, M. Siebert, S. Souyris, A. Weintraub, R. Wolf-Yadlin and G. Zamorano, Operations research transforms the scheduling of Chilean soccer leagues and South American World Cup qualifiers, Interfaces, 47 (2017), 52-69. Google Scholar
[3]	T. Bartsch, A. Drexl and S. Kröger, Scheduling the professional soccer leagues of Austria and Germany, Computers & Operations Research, 33 (2006), 1907-1937. doi: 10.1016/j.cor.2004.09.037. Google Scholar
[4]	T. Bernholt, A. Gälich, T. Hofmeister and N. Schmitt, Football elimination is hard to decide under the 3-point-rule, International Symposium on Mathematical Foundations of Computer Science, (1999), 410-418. doi: 10.1007/3-540-48340-3_37. Google Scholar
[5]	J. Borrett, E. P. Tsang and N. R. Walsh, Adaptive constraint satisfaction: The quickest first principle, in European Conference on Artificial Intelligence, (1996). Google Scholar
[6]	F. Boussemart, F. Hemery, C. Lecoutre and L. Sais, Boosting systematic search by weighting constraints, in ECAI, (2004), 146–150. Google Scholar
[7]	S. K. Card, G. G. Robertson and J. D. Mackinlay, The information visualizer, an information workspace, in SIGCHI Conference on Human Factors in Computing Systems (ACM), (1991), 181–186. Google Scholar
[8]	J. Christensen, A. N. Knudsen and K. S. Larsen, Soccer is harder than football, International Journal of Foundations of Computer Science, 26 (2015), 477-486. doi: 10.1142/S0129054115500264. Google Scholar
[9]	M. Consortium, The mozart programming system, 2013. Available from http://mozart.github.io/. Google Scholar
[10]	D. Costa, An evolutionary tabu search algorithm and the NHL scheduling problem, INFOR: Information Systems and Operational Research, 33 (1995), 161-178. doi: 10.1080/03155986.1995.11732279. Google Scholar
[11]	R. Duque, J. F. Díaz and A. Arbelaez, Constraint programming and machine learning for interactive soccer analysis, in Learning and Intelligent Optimization, (2016), 240–246. doi: 10.1007/978-3-319-50349-3_18. Google Scholar
[12]	R. Duque, J. F. Díaz and A. Arbelaez, SABIO: An implementation of MIP and CP for interactive soccer queries, in International Conference on Principles and Practice of Constraint Programming, (2016), 575–583. Google Scholar
[13]	G. Durán, M. Guajardo, J. Miranda, D. Sauré, S. Souyris, A. Weintraub and R. Wolf, Scheduling the Chilean soccer league by integer programming, Interfaces, 37 (2007), 539-552. Google Scholar
[14]	G. Durán, M. Guajardo and D. Sauré, Scheduling the South American Qualifiers to the 2018 FIFA World Cup by integer programming, European Journal of Operational Research, 262 (2017), 1109-1115. doi: 10.1016/j.ejor.2017.04.043. Google Scholar
[15]	G. Durán, M. Guajardo and R. Wolf-Yadlin, Operations research techniques for scheduling Chile's second division soccer league, Interfaces, 42 (2012), 273-285. Google Scholar
[16]	Gecode Team, Gecode: Generic constraint development environment, 2016. Available from http://www.gecode.org. Google Scholar
[17]	D. Goossens and F. Spieksma, Scheduling the Belgian soccer league, Interfaces, 39 (2009), 109-118. doi: 10.1287/inte.1080.0402. Google Scholar
[18]	D. R. Goossens, J. Beliën and F. C. Spieksma, Comparing league formats with respect to match importance in Belgian football, Annals of Operations Research, 194 (2012), 223-240. doi: 10.1007/s10479-010-0764-4. Google Scholar
[19]	J. C. Guedes and F. S. Machado, Changing rewards in contests: Has the three-point rule brought more offense to soccer? Empirical Economics, 27 (2002), 607–630. doi: 10.1007/s001810100106. Google Scholar
[20]	D. Gusfield and C. Martel, The structure and complexity of sports elimination numbers, Algorithmica, 32 (2002), 73-86. doi: 10.1007/s00453-001-0074-y. Google Scholar
[21]	R. M. Haralick and G. L. Elliott, Increasing tree search efficiency for constraint satisfaction problems, Artificial Intelligence, 14 (1980), 263-313. doi: 10.1016/0004-3702(80)90051-X. Google Scholar
[22]	A. Hoffman and T. Rivlin, When is a team "mathematically" eliminated?, Princeton Symposium on Mathematical Programming, (1967), 391-401. Google Scholar
[23]	H. Kellerer, U. Pferschy and D. Pisinger, The subset sum problem, in Knapsack Problems, (2004), 73–115. Google Scholar
[24]	W. Kern and D. Paulusma, The new FIFA rules are hard: Complexity aspects of sports competitions, Discrete Applied Mathematics, 108 (2001), 317-323. doi: 10.1016/S0166-218X(00)00241-9. Google Scholar
[25]	W. Kern and D. Paulusma, The computational complexity of the elimination problem in generalized sports competitions, Discrete Optimization, 1 (2004), 205-214. doi: 10.1016/j.disopt.2003.12.003. Google Scholar
[26]	J. Kyngas and K. Nurmi, Scheduling the Finnish major ice hockey league, in Computational Intelligence in Scheduling, (2009), 84–89. doi: 10.1109/SCIS.2009.4927019. Google Scholar
[27]	C. Lucena, T. Noronha, S. Urrutia and C. Ribeiro, A multi-agent framework to retrieve and publish information on qualification and elimination data in sports tournaments, Monografias em Ciênca da Computação, (2006). Google Scholar
[28]	S. T. McCormick, Fast algorithms for parametric scheduling come from extensions to parametric maximum flow, Operations Research, 47 (1999), 744-756. doi: 10.1287/opre.47.5.744. Google Scholar
[29]	R. B. Miller, Response time in man-computer conversational transactions, Joint Computer Conference, part I, (1968), 267-277. doi: 10.1145/1476589.1476628. Google Scholar
[30]	Monash University, Data61 Decision Sciences and University of Melbourne. Minizinc, 2014, Available from http://www.minizinc.org/. Google Scholar
[31]	T. F. Noronha, C. C. Ribeiro, G. Duran, S. Souyris and A. Weintraub. A branch-and-cut algorithm for scheduling the highly-constrained Chilean soccer tournament, in International Conference on the Practice and Theory of Automated Timetabling, (2006), 174–186. doi: 10.1007/978-3-540-77345-0_12. Google Scholar
[32]	D. Pálvölgyi, Deciding soccer scores and partial orientations of graphs, Acta Univ. Sapientiae, Math, 1 (2009), 51-71. Google Scholar
[33]	C. Raack, A. Raymond, T. Schlechte and A. Werner, Standings in sports competitions using integer programming, Journal of Quantitative Analysis in Sports, 10 (2014), 131-137. doi: 10.1515/jqas-2013-0111. Google Scholar
[34]	R. V. Rasmussen, Scheduling a triple round robin tournament for the best danish soccer league, European Journal of Operational Research, 185 (2008), 795-810. doi: 10.1016/j.ejor.2006.12.050. Google Scholar
[35]	R. V. Rasmussen and M. A. Trick, Round robin scheduling-a survey, European Journal of Operational Research, 188 (2008), 617-636. doi: 10.1016/j.ejor.2007.05.046. Google Scholar
[36]	P. Refalo, Impact-based search strategies for constraint programming, in International Conference on Principles and Practice of Constraint Programming, (2004), 557–571. doi: 10.1007/978-3-540-30201-8_41. Google Scholar
[37]	C. C. Ribeiro and S. Urrutia, An application of integer programming to playoff elimination in football championships, International Transactions in Operational Research, 12 (2005), 375-386. doi: 10.1111/j.1475-3995.2005.00513.x. Google Scholar
[38]	C. C. Ribeiro and S. Urrutia, Scheduling the Brazilian soccer tournament with fairness and broadcast objectives, in International Conference on the Practice and Theory of Automated Timetabling, (2006), 147–157. doi: 10.1007/978-3-540-77345-0_10. Google Scholar
[39]	C. C. Ribeiro and S. Urrutia, Scheduling the Brazilian soccer tournament: Solution approach and practice, Interfaces, 42 (2012), 260-272. doi: 10.1287/inte.1110.0566. Google Scholar
[40]	F. Rossi, P. van Beek and T. Walsh, Handbook of Constraint Programming, volume 2 of Foundations of Artificial Intelligence, Elsevier, 2006. Google Scholar
[41]	R. A. Russell and J. M. Leung, Devising a cost effective schedule for a baseball league, Operations Research, 42 (1994), 614-625. doi: 10.1287/opre.42.4.614. Google Scholar
[42]	B. L. Schwartz, Possible winners in partially completed tournaments, SIAM Review, 8 (1966), 302-308. doi: 10.1137/1008062. Google Scholar
[43]	P. J. Stuckey, M. G. de la Banda, M. Maher, K. Marriott, J. Slaney, Z. Somogyi, M. Wallace and T. Walsh, The G12 project: Mapping solver independent models to efficient solutions, in International Conference on Logic Programming, (2005), 9–13. Google Scholar
[44]	T. Van Voorhis, Highly constrained college basketball scheduling, Journal of the Operational Research Society, 53 (2002), 603-609. doi: 10.1057/palgrave.jors.2601356. Google Scholar
[45]	M. Veksler, HaifaCSP - constraints-satisfaction problem (CSP) solver, 2016. Available from http://strichman.net.technion.ac.il/haifacsp/. Google Scholar
[46]	K. D. Wayne, A new property and a faster algorithm for baseball elimination, SIAM Journal on Discrete Mathematics, 14 (2001), 223-229. doi: 10.1137/S0895480198348847. Google Scholar
[47]	I. H. Witten and E. Frank, Data mining: Practical machine learning tools and techniques, ACM SIGMOD Record, 31 (2002), 76-77. doi: 10.1145/507338.507355. Google Scholar
[48]	M. B. Wright, Scheduling fixtures for basketball New Zealand, Computers & Operations Research, 33 (2006), 1875-1893. doi: 10.1016/j.cor.2004.09.024. Google Scholar
[49]	J. T. Yang, H.-D. Huang and J.-T. Horng, Devising a cost effective baseball scheduling by evolutionary algorithms, in Evolutionary Computation, (2002), 1660–1665. Google Scholar

show all references

References:

[1]	I. Adler, A. L. Erera, D. S. Hochbaum and E. V. Olinick, Baseball, optimization, and the world wide web, Interfaces, 32 (2002), 12-22. doi: 10.1287/inte.32.2.12.67. Google Scholar
[2]	F. Alarcón, G. Durán, M. Guajardo, J. Miranda, H. Muñoz, L. Ramírez, M. Ramírez, D. Sauré, M. Siebert, S. Souyris, A. Weintraub, R. Wolf-Yadlin and G. Zamorano, Operations research transforms the scheduling of Chilean soccer leagues and South American World Cup qualifiers, Interfaces, 47 (2017), 52-69. Google Scholar
[3]	T. Bartsch, A. Drexl and S. Kröger, Scheduling the professional soccer leagues of Austria and Germany, Computers & Operations Research, 33 (2006), 1907-1937. doi: 10.1016/j.cor.2004.09.037. Google Scholar
[4]	T. Bernholt, A. Gälich, T. Hofmeister and N. Schmitt, Football elimination is hard to decide under the 3-point-rule, International Symposium on Mathematical Foundations of Computer Science, (1999), 410-418. doi: 10.1007/3-540-48340-3_37. Google Scholar
[5]	J. Borrett, E. P. Tsang and N. R. Walsh, Adaptive constraint satisfaction: The quickest first principle, in European Conference on Artificial Intelligence, (1996). Google Scholar
[6]	F. Boussemart, F. Hemery, C. Lecoutre and L. Sais, Boosting systematic search by weighting constraints, in ECAI, (2004), 146–150. Google Scholar
[7]	S. K. Card, G. G. Robertson and J. D. Mackinlay, The information visualizer, an information workspace, in SIGCHI Conference on Human Factors in Computing Systems (ACM), (1991), 181–186. Google Scholar
[8]	J. Christensen, A. N. Knudsen and K. S. Larsen, Soccer is harder than football, International Journal of Foundations of Computer Science, 26 (2015), 477-486. doi: 10.1142/S0129054115500264. Google Scholar
[9]	M. Consortium, The mozart programming system, 2013. Available from http://mozart.github.io/. Google Scholar
[10]	D. Costa, An evolutionary tabu search algorithm and the NHL scheduling problem, INFOR: Information Systems and Operational Research, 33 (1995), 161-178. doi: 10.1080/03155986.1995.11732279. Google Scholar
[11]	R. Duque, J. F. Díaz and A. Arbelaez, Constraint programming and machine learning for interactive soccer analysis, in Learning and Intelligent Optimization, (2016), 240–246. doi: 10.1007/978-3-319-50349-3_18. Google Scholar
[12]	R. Duque, J. F. Díaz and A. Arbelaez, SABIO: An implementation of MIP and CP for interactive soccer queries, in International Conference on Principles and Practice of Constraint Programming, (2016), 575–583. Google Scholar
[13]	G. Durán, M. Guajardo, J. Miranda, D. Sauré, S. Souyris, A. Weintraub and R. Wolf, Scheduling the Chilean soccer league by integer programming, Interfaces, 37 (2007), 539-552. Google Scholar
[14]	G. Durán, M. Guajardo and D. Sauré, Scheduling the South American Qualifiers to the 2018 FIFA World Cup by integer programming, European Journal of Operational Research, 262 (2017), 1109-1115. doi: 10.1016/j.ejor.2017.04.043. Google Scholar
[15]	G. Durán, M. Guajardo and R. Wolf-Yadlin, Operations research techniques for scheduling Chile's second division soccer league, Interfaces, 42 (2012), 273-285. Google Scholar
[16]	Gecode Team, Gecode: Generic constraint development environment, 2016. Available from http://www.gecode.org. Google Scholar
[17]	D. Goossens and F. Spieksma, Scheduling the Belgian soccer league, Interfaces, 39 (2009), 109-118. doi: 10.1287/inte.1080.0402. Google Scholar
[18]	D. R. Goossens, J. Beliën and F. C. Spieksma, Comparing league formats with respect to match importance in Belgian football, Annals of Operations Research, 194 (2012), 223-240. doi: 10.1007/s10479-010-0764-4. Google Scholar
[19]	J. C. Guedes and F. S. Machado, Changing rewards in contests: Has the three-point rule brought more offense to soccer? Empirical Economics, 27 (2002), 607–630. doi: 10.1007/s001810100106. Google Scholar
[20]	D. Gusfield and C. Martel, The structure and complexity of sports elimination numbers, Algorithmica, 32 (2002), 73-86. doi: 10.1007/s00453-001-0074-y. Google Scholar
[21]	R. M. Haralick and G. L. Elliott, Increasing tree search efficiency for constraint satisfaction problems, Artificial Intelligence, 14 (1980), 263-313. doi: 10.1016/0004-3702(80)90051-X. Google Scholar
[22]	A. Hoffman and T. Rivlin, When is a team "mathematically" eliminated?, Princeton Symposium on Mathematical Programming, (1967), 391-401. Google Scholar
[23]	H. Kellerer, U. Pferschy and D. Pisinger, The subset sum problem, in Knapsack Problems, (2004), 73–115. Google Scholar
[24]	W. Kern and D. Paulusma, The new FIFA rules are hard: Complexity aspects of sports competitions, Discrete Applied Mathematics, 108 (2001), 317-323. doi: 10.1016/S0166-218X(00)00241-9. Google Scholar
[25]	W. Kern and D. Paulusma, The computational complexity of the elimination problem in generalized sports competitions, Discrete Optimization, 1 (2004), 205-214. doi: 10.1016/j.disopt.2003.12.003. Google Scholar
[26]	J. Kyngas and K. Nurmi, Scheduling the Finnish major ice hockey league, in Computational Intelligence in Scheduling, (2009), 84–89. doi: 10.1109/SCIS.2009.4927019. Google Scholar
[27]	C. Lucena, T. Noronha, S. Urrutia and C. Ribeiro, A multi-agent framework to retrieve and publish information on qualification and elimination data in sports tournaments, Monografias em Ciênca da Computação, (2006). Google Scholar
[28]	S. T. McCormick, Fast algorithms for parametric scheduling come from extensions to parametric maximum flow, Operations Research, 47 (1999), 744-756. doi: 10.1287/opre.47.5.744. Google Scholar
[29]	R. B. Miller, Response time in man-computer conversational transactions, Joint Computer Conference, part I, (1968), 267-277. doi: 10.1145/1476589.1476628. Google Scholar
[30]	Monash University, Data61 Decision Sciences and University of Melbourne. Minizinc, 2014, Available from http://www.minizinc.org/. Google Scholar
[31]	T. F. Noronha, C. C. Ribeiro, G. Duran, S. Souyris and A. Weintraub. A branch-and-cut algorithm for scheduling the highly-constrained Chilean soccer tournament, in International Conference on the Practice and Theory of Automated Timetabling, (2006), 174–186. doi: 10.1007/978-3-540-77345-0_12. Google Scholar
[32]	D. Pálvölgyi, Deciding soccer scores and partial orientations of graphs, Acta Univ. Sapientiae, Math, 1 (2009), 51-71. Google Scholar
[33]	C. Raack, A. Raymond, T. Schlechte and A. Werner, Standings in sports competitions using integer programming, Journal of Quantitative Analysis in Sports, 10 (2014), 131-137. doi: 10.1515/jqas-2013-0111. Google Scholar
[34]	R. V. Rasmussen, Scheduling a triple round robin tournament for the best danish soccer league, European Journal of Operational Research, 185 (2008), 795-810. doi: 10.1016/j.ejor.2006.12.050. Google Scholar
[35]	R. V. Rasmussen and M. A. Trick, Round robin scheduling-a survey, European Journal of Operational Research, 188 (2008), 617-636. doi: 10.1016/j.ejor.2007.05.046. Google Scholar
[36]	P. Refalo, Impact-based search strategies for constraint programming, in International Conference on Principles and Practice of Constraint Programming, (2004), 557–571. doi: 10.1007/978-3-540-30201-8_41. Google Scholar
[37]	C. C. Ribeiro and S. Urrutia, An application of integer programming to playoff elimination in football championships, International Transactions in Operational Research, 12 (2005), 375-386. doi: 10.1111/j.1475-3995.2005.00513.x. Google Scholar
[38]	C. C. Ribeiro and S. Urrutia, Scheduling the Brazilian soccer tournament with fairness and broadcast objectives, in International Conference on the Practice and Theory of Automated Timetabling, (2006), 147–157. doi: 10.1007/978-3-540-77345-0_10. Google Scholar
[39]	C. C. Ribeiro and S. Urrutia, Scheduling the Brazilian soccer tournament: Solution approach and practice, Interfaces, 42 (2012), 260-272. doi: 10.1287/inte.1110.0566. Google Scholar
[40]	F. Rossi, P. van Beek and T. Walsh, Handbook of Constraint Programming, volume 2 of Foundations of Artificial Intelligence, Elsevier, 2006. Google Scholar
[41]	R. A. Russell and J. M. Leung, Devising a cost effective schedule for a baseball league, Operations Research, 42 (1994), 614-625. doi: 10.1287/opre.42.4.614. Google Scholar
[42]	B. L. Schwartz, Possible winners in partially completed tournaments, SIAM Review, 8 (1966), 302-308. doi: 10.1137/1008062. Google Scholar
[43]	P. J. Stuckey, M. G. de la Banda, M. Maher, K. Marriott, J. Slaney, Z. Somogyi, M. Wallace and T. Walsh, The G12 project: Mapping solver independent models to efficient solutions, in International Conference on Logic Programming, (2005), 9–13. Google Scholar
[44]	T. Van Voorhis, Highly constrained college basketball scheduling, Journal of the Operational Research Society, 53 (2002), 603-609. doi: 10.1057/palgrave.jors.2601356. Google Scholar
[45]	M. Veksler, HaifaCSP - constraints-satisfaction problem (CSP) solver, 2016. Available from http://strichman.net.technion.ac.il/haifacsp/. Google Scholar
[46]	K. D. Wayne, A new property and a faster algorithm for baseball elimination, SIAM Journal on Discrete Mathematics, 14 (2001), 223-229. doi: 10.1137/S0895480198348847. Google Scholar
[47]	I. H. Witten and E. Frank, Data mining: Practical machine learning tools and techniques, ACM SIGMOD Record, 31 (2002), 76-77. doi: 10.1145/507338.507355. Google Scholar
[48]	M. B. Wright, Scheduling fixtures for basketball New Zealand, Computers & Operations Research, 33 (2006), 1875-1893. doi: 10.1016/j.cor.2004.09.024. Google Scholar
[49]	J. T. Yang, H.-D. Huang and J.-T. Horng, Devising a cost effective baseball scheduling by evolutionary algorithms, in Evolutionary Computation, (2002), 1660–1665. Google Scholar

Figure 1. Data set sample with 10 records to train a machine learning model for value selection

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Figure 2. Colombian league runtime comparison (time in seconds for parallel executions)

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Figure 3. Argentine league runtime comparison (time in seconds for parallel executions)

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Figure 4. Colombian league runtime comparison for mixed solver executions (time in seconds for parallel executions)

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Figure 5. Argentine league runtime comparison for mixed solver (time in seconds for parallel executions)

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Table 1. Solvers and model configurations used in the empirical tests

Model	Short Name	Oz	CPLEX	HaifaCSP	G12/FD	Gecode
CP model	CP	$\checkmark$		$\checkmark$	$\checkmark$	$\checkmark$
CP model + Redundant Constraints	CP-R			$\checkmark$	$\checkmark$	$\checkmark$
MIP model	MIP		$\checkmark$	$\checkmark$	$\checkmark$	$\checkmark$

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Model	Short Name	Oz	CPLEX	HaifaCSP	G12/FD	Gecode
CP model	CP	$\checkmark$		$\checkmark$	$\checkmark$	$\checkmark$
CP model + Redundant Constraints	CP-R			$\checkmark$	$\checkmark$	$\checkmark$
MIP model	MIP		$\checkmark$	$\checkmark$	$\checkmark$	$\checkmark$

Table 2. Oz Solvers and model configurations using Var/Val heuristics

Model	Short Name	Oz
CP model + Var/Val Heuristics including Machine Learning	CP-ML	$\checkmark$
Extended CP (CP model + Redundant Constraints + Var/Val Heuristics including Machine Learning)	E-CP	$\checkmark$

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Model	Short Name	Oz
CP model + Var/Val Heuristics including Machine Learning	CP-ML	$\checkmark$
Extended CP (CP model + Redundant Constraints + Var/Val Heuristics including Machine Learning)	E-CP	$\checkmark$

Table 3. Unsolved instances and average running times for the Colombian league (18 teams) evaluated sequentially and in parallel with the CP, CP-ML, and E-CP models using Mozart-Oz

1 Core Experiments
Constraint Type		$P$ Queries						$R$ Queries						$S$ Queries
	Oz CP	24	28	38	46	50	52	0	0	0	0	0	0	0	2	0	0	6	3
Fixture 7	Oz CP-ML	5	9	14	21	27	35	0	0	0	0	0	0	0	0	0	0	0	0
	Oz E-CP	1	3	4	5	11	15	0	0	0	0	0	0	0	0	0	0	0	0
	Oz CP	23	28	35	46	49	44	0	0	0	0	0	0	0	0	0	0	1	1
Fixture 9	Oz CP-ML	1	4	9	18	27	31	0	0	0	0	0	0	0	0	0	0	0	1
	Oz E-CP	2	1	4	6	11	15	0	0	0	0	0	0	0	0	0	0	0	1
	Oz CP	25	27	31	44	49	47	0	0	0	0	0	0	0	0	0	0	0	0
Fixture 11	Oz CP-ML	2	7	11	18	31	34	0	0	0	0	0	0	0	0	0	0	0	0
	Oz E-CP	2	3	4	8	17	16	0	0	0	0	0	0	0	0	0	0	0	0
	Oz CP	23	33	38	41	51	45	0	0	0	0	0	0	0	0	0	1	0	0
Fixture 14	Oz CP-ML	8	22	25	34	44	42	0	0	0	0	0	0	0	0	0	0	0	0
	Oz E-CP	2	2	7	9	14	8	0	0	0	0	0	0	0	0	0	0	0	0
	Oz CP	23	31	29	31	25	13	0	0	0	0	0	0	0	0	0	0	0	0
Fixture 16	Oz CP-ML	19	25	25	33	28	17	0	0	0	0	0	0	0	0	0	0	0	0
	Oz E-CP	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
Unsolved	Oz CP	1069						0						14
Instances	Oz CP-ML	626						0						1
	Oz E-CP	170						0						1
	Oz CP	0.22	0.48	0.32	0.41	0.77	1.1	0.07	0.07	0.07	0.07	0.07	0.07	0.07	0.07	0.06	0.07	0.23	0.22
Fixture 7	Oz CP-ML	1.21	1.35	1.84	2.41	3.29	3.09	0.06	0.06	0.06	0.06	0.06	0.05	0.05	0.11	0.05	0.05	0.26	0.2
	Oz E-CP	0.89	0.94	1.05	1.06	0.95	1.19	0.06	0.06	0.07	0.07	0.06	0.06	0.07	0.12	0.06	0.06	0.27	0.22
	Oz CP	0.09	0.42	0.23	0.66	0.32	0.73	0.07	0.07	0.07	0.06	0.06	0.06	0.06	0.06	0.06	0.06	0.06	0.22
Fixture 9	Oz CP-ML	1.69	1.48	2.26	2.81	2.79	1.67	0.05	0.05	0.05	0.05	0.05	0.05	0.05	0.05	0.05	0.05	0.07	0.07
	Oz E-CP	0.81	0.56	1.16	1.16	1.9	0.46	0.07	0.06	0.06	0.06	0.07	0.06	0.06	0.06	0.05	0.05	0.08	0.08
	Oz CP	0.56	0.1	0.76	0.69	0.87	0.45	0.06	0.06	0.06	0.06	0.06	0.06	0.06	0.06	0.08	0.05	0.05	0.05
Fixture 11	Oz CP-ML	1.53	1.09	2.02	2.68	2.57	2.47	0.06	0.06	0.05	0.06	0.05	0.05	0.05	0.06	0.08	0.06	0.05	0.04
	Oz E-CP	0.4	0.4	0.75	0.64	0.78	0.23	0.06	0.05	0.05	0.05	0.05	0.05	0.05	0.05	0.07	0.05	0.05	0.04
	Oz CP	0.42	0.14	0.49	1.02	1.54	0.74	0.05	0.05	0.05	0.05	0.05	0.04	0.05	0.05	0.05	0.05	0.05	0.04
Fixture 14	Oz CP-ML	1.17	1.3	1.46	1.49	1.08	1.48	0.04	0.03	0.03	0.03	0.03	0.03	0.04	0.03	0.03	0.06	0.03	0.03
	Oz E-CP	0.07	0.37	0.11	0.16	0.44	0.54	0.05	0.05	0.04	0.04	0.05	0.04	0.05	0.04	0.04	0.07	0.04	0.04
	Oz CP	0.12	0.05	0.9	0.56	0.46	0.66	0.04	0.04	0.04	0.04	0.04	0.04	0.05	0.04	0.04	0.04	0.04	0.04
Fixture 16	Oz CP-ML	0.56	0.5	0.93	0.05	0.12	0.04	0.04	0.04	0.04	0.04	0.04	0.04	0.04	0.04	0.04	0.04	0.04	0.04
	Oz E-CP	0.05	0.04	0.03	0.07	0.04	0.05	0.04	0.04	0.04	0.04	0.04	0.04	0.04	0.04	0.04	0.04	0.04	0.04
	Oz CP	0.51						0.06						0.07
Avg. Time	Oz CP-ML	1.61						0.05						0.06
	Oz E-CP	0.57						0.05						0.07
4 Core Experiments
Constraint Type		$P$ Queries						$R$ Queries						$S$ Queries
Fixture 7	Oz CP-ML	4	8	15	18	26	37	0	0	0	0	0	0	0	0	0	0	0	0
	Oz E-CP	1	3	4	5	11	18	0	0	0	0	0	0	0	0	0	0	0	0
Fixture 9	Oz CP-ML	2	4	9	14	23	28	0	0	0	0	0	0	0	0	0	0	0	1
	Oz E-CP	0	1	3	5	10	13	0	0	0	0	0	0	0	0	0	0	0	1
Fixture 11	Oz CP-ML	2	6	11	17	29	32	0	0	0	0	0	0	0	0	0	0	0	0
	Oz E-CP	2	3	4	7	16	16	0	0	0	0	0	0	0	0	0	0	0	0
Fixture 14	Oz CP-ML	8	21	24	34	42	44	0	0	0	0	0	0	0	0	0	0	0	0
	Oz E-CP	2	2	7	9	13	9	0	0	0	0	0	0	0	0	0	0	0	0
Fixture 16	Oz CP-ML	19	25	26	33	28	17	0	0	0	0	0	0	0	0	0	0	0	0
	Oz E-CP	0	0	0	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0
Unsolved	Oz CP-ML	606						0						1
Instances	Oz E-CP	165						0						1
Fixture 7	Oz CP-ML	0.63	0.77	0.55	1.37	2.28	1.35	0.08	0.07	0.07	0.07	0.07	0.07	0.07	0.07	0.06	0.07	0.06	0.09
	Oz E-CP	0.35	0.44	0.45	0.28	0.49	0.69	0.08	0.07	0.08	0.07	0.07	0.07	0.08	0.08	0.07	0.07	0.07	0.09
Fixture 9	Oz CP-ML	0.53	0.38	0.54	1.24	2.92	1.9	0.07	0.07	0.07	0.07	0.07	0.06	0.07	0.06	0.06	0.06	0.05	0.05
	Oz E-CP	0.43	0.25	0.55	0.34	1.03	0.82	0.07	0.07	0.07	0.07	0.07	0.07	0.07	0.06	0.06	0.06	0.06	0.06
Fixture 11	Oz CP-ML	0.38	0.45	0.92	1.0	1.46	1.6	0.07	0.06	0.06	0.06	0.06	0.05	0.06	0.06	0.05	0.05	0.05	0.05
	Oz E-CP	0.16	0.15	0.31	0.35	0.69	0.32	0.07	0.06	0.06	0.06	0.07	0.06	0.06	0.06	0.06	0.05	0.05	0.05
Fixture 14	Oz CP-ML	0.38	0.88	0.78	0.68	0.54	1.23	0.05	0.04	0.05	0.04	0.04	0.04	0.05	0.04	0.04	0.04	0.04	0.04
	Oz E-CP	0.05	0.21	0.06	0.08	0.35	0.51	0.05	0.05	0.04	0.04	0.04	0.04	0.05	0.05	0.04	0.04	0.04	0.04
Fixture 16	Oz CP-ML	0.19	0.18	0.28	0.05	0.22	0.04	0.04	0.04	0.04	0.04	0.04	0.04	0.04	0.04	0.04	0.04	0.04	0.04
	Oz E-CP	0.04	0.04	0.04	0.04	0.04	0.04	0.04	0.04	0.04	0.04	0.04	0.04	0.04	0.04	0.04	0.04	0.04	0.04
Avg. Time	Oz CP-ML	0.83						0.06						0.05
	Oz E-CP	0.31						0.06						0.05

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1 Core Experiments
Constraint Type		$P$ Queries						$R$ Queries						$S$ Queries
	Oz CP	24	28	38	46	50	52	0	0	0	0	0	0	0	2	0	0	6	3
Fixture 7	Oz CP-ML	5	9	14	21	27	35	0	0	0	0	0	0	0	0	0	0	0	0
	Oz E-CP	1	3	4	5	11	15	0	0	0	0	0	0	0	0	0	0	0	0
	Oz CP	23	28	35	46	49	44	0	0	0	0	0	0	0	0	0	0	1	1
Fixture 9	Oz CP-ML	1	4	9	18	27	31	0	0	0	0	0	0	0	0	0	0	0	1
	Oz E-CP	2	1	4	6	11	15	0	0	0	0	0	0	0	0	0	0	0	1
	Oz CP	25	27	31	44	49	47	0	0	0	0	0	0	0	0	0	0	0	0
Fixture 11	Oz CP-ML	2	7	11	18	31	34	0	0	0	0	0	0	0	0	0	0	0	0
	Oz E-CP	2	3	4	8	17	16	0	0	0	0	0	0	0	0	0	0	0	0
	Oz CP	23	33	38	41	51	45	0	0	0	0	0	0	0	0	0	1	0	0
Fixture 14	Oz CP-ML	8	22	25	34	44	42	0	0	0	0	0	0	0	0	0	0	0	0
	Oz E-CP	2	2	7	9	14	8	0	0	0	0	0	0	0	0	0	0	0	0
	Oz CP	23	31	29	31	25	13	0	0	0	0	0	0	0	0	0	0	0	0
Fixture 16	Oz CP-ML	19	25	25	33	28	17	0	0	0	0	0	0	0	0	0	0	0	0
	Oz E-CP	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
Unsolved	Oz CP	1069						0						14
Instances	Oz CP-ML	626						0						1
	Oz E-CP	170						0						1
	Oz CP	0.22	0.48	0.32	0.41	0.77	1.1	0.07	0.07	0.07	0.07	0.07	0.07	0.07	0.07	0.06	0.07	0.23	0.22
Fixture 7	Oz CP-ML	1.21	1.35	1.84	2.41	3.29	3.09	0.06	0.06	0.06	0.06	0.06	0.05	0.05	0.11	0.05	0.05	0.26	0.2
	Oz E-CP	0.89	0.94	1.05	1.06	0.95	1.19	0.06	0.06	0.07	0.07	0.06	0.06	0.07	0.12	0.06	0.06	0.27	0.22
	Oz CP	0.09	0.42	0.23	0.66	0.32	0.73	0.07	0.07	0.07	0.06	0.06	0.06	0.06	0.06	0.06	0.06	0.06	0.22
Fixture 9	Oz CP-ML	1.69	1.48	2.26	2.81	2.79	1.67	0.05	0.05	0.05	0.05	0.05	0.05	0.05	0.05	0.05	0.05	0.07	0.07
	Oz E-CP	0.81	0.56	1.16	1.16	1.9	0.46	0.07	0.06	0.06	0.06	0.07	0.06	0.06	0.06	0.05	0.05	0.08	0.08
	Oz CP	0.56	0.1	0.76	0.69	0.87	0.45	0.06	0.06	0.06	0.06	0.06	0.06	0.06	0.06	0.08	0.05	0.05	0.05
Fixture 11	Oz CP-ML	1.53	1.09	2.02	2.68	2.57	2.47	0.06	0.06	0.05	0.06	0.05	0.05	0.05	0.06	0.08	0.06	0.05	0.04
	Oz E-CP	0.4	0.4	0.75	0.64	0.78	0.23	0.06	0.05	0.05	0.05	0.05	0.05	0.05	0.05	0.07	0.05	0.05	0.04
	Oz CP	0.42	0.14	0.49	1.02	1.54	0.74	0.05	0.05	0.05	0.05	0.05	0.04	0.05	0.05	0.05	0.05	0.05	0.04
Fixture 14	Oz CP-ML	1.17	1.3	1.46	1.49	1.08	1.48	0.04	0.03	0.03	0.03	0.03	0.03	0.04	0.03	0.03	0.06	0.03	0.03
	Oz E-CP	0.07	0.37	0.11	0.16	0.44	0.54	0.05	0.05	0.04	0.04	0.05	0.04	0.05	0.04	0.04	0.07	0.04	0.04
	Oz CP	0.12	0.05	0.9	0.56	0.46	0.66	0.04	0.04	0.04	0.04	0.04	0.04	0.05	0.04	0.04	0.04	0.04	0.04
Fixture 16	Oz CP-ML	0.56	0.5	0.93	0.05	0.12	0.04	0.04	0.04	0.04	0.04	0.04	0.04	0.04	0.04	0.04	0.04	0.04	0.04
	Oz E-CP	0.05	0.04	0.03	0.07	0.04	0.05	0.04	0.04	0.04	0.04	0.04	0.04	0.04	0.04	0.04	0.04	0.04	0.04
	Oz CP	0.51						0.06						0.07
Avg. Time	Oz CP-ML	1.61						0.05						0.06
	Oz E-CP	0.57						0.05						0.07
4 Core Experiments
Constraint Type		$P$ Queries						$R$ Queries						$S$ Queries
Fixture 7	Oz CP-ML	4	8	15	18	26	37	0	0	0	0	0	0	0	0	0	0	0	0
	Oz E-CP	1	3	4	5	11	18	0	0	0	0	0	0	0	0	0	0	0	0
Fixture 9	Oz CP-ML	2	4	9	14	23	28	0	0	0	0	0	0	0	0	0	0	0	1
	Oz E-CP	0	1	3	5	10	13	0	0	0	0	0	0	0	0	0	0	0	1
Fixture 11	Oz CP-ML	2	6	11	17	29	32	0	0	0	0	0	0	0	0	0	0	0	0
	Oz E-CP	2	3	4	7	16	16	0	0	0	0	0	0	0	0	0	0	0	0
Fixture 14	Oz CP-ML	8	21	24	34	42	44	0	0	0	0	0	0	0	0	0	0	0	0
	Oz E-CP	2	2	7	9	13	9	0	0	0	0	0	0	0	0	0	0	0	0
Fixture 16	Oz CP-ML	19	25	26	33	28	17	0	0	0	0	0	0	0	0	0	0	0	0
	Oz E-CP	0	0	0	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0
Unsolved	Oz CP-ML	606						0						1
Instances	Oz E-CP	165						0						1
Fixture 7	Oz CP-ML	0.63	0.77	0.55	1.37	2.28	1.35	0.08	0.07	0.07	0.07	0.07	0.07	0.07	0.07	0.06	0.07	0.06	0.09
	Oz E-CP	0.35	0.44	0.45	0.28	0.49	0.69	0.08	0.07	0.08	0.07	0.07	0.07	0.08	0.08	0.07	0.07	0.07	0.09
Fixture 9	Oz CP-ML	0.53	0.38	0.54	1.24	2.92	1.9	0.07	0.07	0.07	0.07	0.07	0.06	0.07	0.06	0.06	0.06	0.05	0.05
	Oz E-CP	0.43	0.25	0.55	0.34	1.03	0.82	0.07	0.07	0.07	0.07	0.07	0.07	0.07	0.06	0.06	0.06	0.06	0.06
Fixture 11	Oz CP-ML	0.38	0.45	0.92	1.0	1.46	1.6	0.07	0.06	0.06	0.06	0.06	0.05	0.06	0.06	0.05	0.05	0.05	0.05
	Oz E-CP	0.16	0.15	0.31	0.35	0.69	0.32	0.07	0.06	0.06	0.06	0.07	0.06	0.06	0.06	0.06	0.05	0.05	0.05
Fixture 14	Oz CP-ML	0.38	0.88	0.78	0.68	0.54	1.23	0.05	0.04	0.05	0.04	0.04	0.04	0.05	0.04	0.04	0.04	0.04	0.04
	Oz E-CP	0.05	0.21	0.06	0.08	0.35	0.51	0.05	0.05	0.04	0.04	0.04	0.04	0.05	0.05	0.04	0.04	0.04	0.04
Fixture 16	Oz CP-ML	0.19	0.18	0.28	0.05	0.22	0.04	0.04	0.04	0.04	0.04	0.04	0.04	0.04	0.04	0.04	0.04	0.04	0.04
	Oz E-CP	0.04	0.04	0.04	0.04	0.04	0.04	0.04	0.04	0.04	0.04	0.04	0.04	0.04	0.04	0.04	0.04	0.04	0.04
Avg. Time	Oz CP-ML	0.83						0.06						0.05
	Oz E-CP	0.31						0.06						0.05

Table 4. Unsolved instances and average running times for the Argentine league (30 teams) evaluated sequentially and in parallel with the CP, CP-ML, and E-CP models using Mozart-Oz

1 Core Experiments
Constraint Type		$P$ Queries						$R$ Queries						$S$ Queries
	Oz CP	31	37	44	52	68	67	0	0	0	0	0	0	0	0	0	0	0	7
Fixture 5	Oz CP-ML	11	16	24	28	47	49	0	0	0	0	0	0	0	0	0	0	0	1
	Oz E-CP	9	14	15	21	36	33	0	0	0	0	0	0	0	0	0	0	0	1
	Oz CP	28	37	45	56	69	67	0	0	0	0	0	0	0	0	0	1	4	3
Fixture 10	Oz CP-ML	6	12	18	27	42	46	0	0	0	0	0	0	0	0	0	0	0	0
	Oz E-CP	5	9	10	19	27	30	0	0	0	0	0	0	0	0	0	0	0	0
	Oz CP	29	35	46	57	65	73	0	0	0	0	0	0	0	0	1	0	2	5
Fixture 15	Oz CP-ML	5	11	14	24	31	40	0	0	0	0	0	0	0	0	0	0	0	0
	Oz E-CP	3	3	6	12	20	24	0	0	0	0	0	0	0	0	0	0	0	0
	Oz CP	33	36	43	65	67	63	0	0	0	0	0	0	0	0	0	0	1	3
Fixture 20	Oz CP-ML	8	14	17	30	41	47	0	0	0	0	0	0	0	0	0	0	0	0
	Oz E-CP	3	1	9	10	30	33	0	0	0	0	0	0	0	0	0	0	0	0
	Oz CP	29	39	35	38	37	28	0	0	0	0	0	0	0	0	0	0	1	0
Fixture 25	Oz CP-ML	23	27	28	36	34	26	0	0	0	0	0	0	0	0	0	0	0	0
	Oz E-CP	3	1	1	2	4	3	0	0	0	0	0	0	0	0	0	0	0	0
Unsolved	Oz CP	1419						0						28
Instances	Oz CP-ML	782						0						1
	Oz E-CP	396						0						1
	Oz CP	0.23	0.46	0.21	0.23	0.31	0.89	0.56	0.53	0.51	0.5	0.5	0.49	0.43	0.44	0.4	0.39	0.38	0.34
Fixture 5	Oz CP-ML	2.1	2.98	2.72	3.38	3.98	3.82	0.28	0.27	0.27	0.28	0.3	0.34	0.32	0.31	0.3	0.28	0.26	0.44
	Oz E-CP	1.82	1.88	2.54	2.26	2.79	3.84	0.49	0.49	0.49	0.49	0.47	0.46	0.44	0.41	0.41	0.38	0.37	0.51
	Oz CP	0.33	0.2	0.24	0.35	0.18	0.41	0.37	0.39	0.39	0.39	0.38	0.37	0.34	0.34	0.32	0.32	0.28	0.25
Fixture 10	Oz CP-ML	1.69	2.35	2.55	3.3	3.4	3.85	0.27	0.28	0.29	0.29	0.28	0.27	0.25	0.24	0.23	0.26	0.45	0.29
	Oz E-CP	1.35	1.51	2.26	2.17	2.69	2.4	0.38	0.37	0.38	0.36	0.37	0.34	0.32	0.32	0.3	0.32	0.51	0.33
	Oz CP	0.48	0.2	0.14	0.56	0.43	0.14	0.29	0.29	0.33	0.32	0.29	0.31	0.27	0.28	0.26	0.23	0.19	0.21
Fixture 15	Oz CP-ML	1.59	1.79	2.85	3.38	4.36	5.03	0.21	0.21	0.22	0.22	0.21	0.2	0.19	0.19	0.23	0.17	0.23	0.4
	Oz E-CP	1.27	1.26	2.04	2.39	2.9	3.12	0.28	0.28	0.28	0.28	0.27	0.25	0.23	0.2	0.19	0.15	0.19	0.38
	Oz CP	0.12	0.71	0.26	0.7	0.12	0.24	0.19	0.21	0.22	0.22	0.2	0.18	0.19	0.18	0.18	0.16	0.14	0.12
Fixture 20	Oz CP-ML	1.49	1.7	2.43	4.33	3.05	2.81	0.15	0.14	0.14	0.14	0.14	0.13	0.14	0.14	0.14	0.13	0.14	0.19
	Oz E-CP	0.8	1.28	1.41	2.33	1.95	1.49	0.14	0.14	0.14	0.13	0.12	0.12	0.11	0.11	0.12	0.11	0.13	0.18
	Oz CP	0.07	0.22	0.42	0.13	0.12	0.07	0.09	0.08	0.08	0.07	0.06	0.06	0.1	0.1	0.1	0.09	0.07	0.08
Fixture 25	Oz CP-ML	0.59	1.3	0.68	0.24	0.28	0.18	0.06	0.05	0.05	0.05	0.04	0.04	0.07	0.07	0.06	0.06	0.08	0.05
	Oz E-CP	0.17	0.06	0.21	0.21	0.44	0.09	0.09	0.08	0.08	0.07	0.06	0.06	0.09	0.09	0.09	0.08	0.11	0.07
	Oz CP	0.29						0.3						0.24
Avg. Time	Oz CP-ML	2.38						0.19						0.21
	Oz E-CP	1.6						0.26						0.24
4 Core Experiments
Constraint Type		$P$ Queries						$R$ Queries						$S$ Queries
Fixture 5	Oz CP-ML	10	17	21	26	48	49	0	0	0	0	0	0	0	0	0	0	0	1
	Oz E-CP	9	15	16	18	37	32	0	0	0	0	0	0	0	0	0	0	0	1
Fixture 10	Oz CP-ML	6	12	17	26	38	47	0	0	0	0	0	0	0	0	0	0	0	0
	Oz E-CP	4	8	11	19	25	28	0	0	0	0	0	0	0	0	0	0	0	0
Fixture 15	Oz CP-ML	5	10	12	19	33	38	0	0	0	0	0	0	0	0	0	0	0	0
	Oz E-CP	4	3	5	11	21	26	0	0	0	0	0	0	0	0	0	0	0	0
Fixture 20	Oz CP-ML	8	16	17	30	40	44	0	0	0	0	0	0	0	0	0	0	0	0
	Oz E-CP	3	3	9	10	30	32	0	0	0	0	0	0	0	0	0	0	0	0
Fixture 25	Oz CP-ML	22	27	28	36	34	26	0	0	0	0	0	0	0	0	0	0	0	0
	Oz E-CP	3	1	1	2	4	3	0	0	0	0	0	0	0	0	0	0	0	0
Unsolved	Oz CP-ML	762						0						1
Instances	Oz E-CP	393						0						1
Fixture 5	Oz CP-ML	1.02	1.22	1.35	2.15	1.72	1.91	0.46	0.45	0.47	0.45	0.45	0.44	0.4	0.4	0.39	0.37	0.36	0.32
	Oz E-CP	0.78	0.74	1.41	2.01	1.63	2.13	0.45	0.44	0.45	0.46	0.44	0.43	0.39	0.39	0.39	0.36	0.35	0.32
Fixture 10	Oz CP-ML	0.86	0.88	0.91	1.72	2.37	1.69	0.36	0.36	0.36	0.35	0.34	0.33	0.32	0.31	0.3	0.29	0.33	0.25
	Oz E-CP	0.95	0.62	0.77	0.69	0.92	1.2	0.3	0.32	0.34	0.34	0.34	0.33	0.33	0.31	0.29	0.29	0.31	0.25
Fixture 15	Oz CP-ML	0.57	0.7	1.42	2.18	1.58	3.19	0.28	0.27	0.27	0.27	0.26	0.25	0.23	0.23	0.23	0.21	0.2	0.23
	Oz E-CP	0.32	0.42	1.07	0.9	1.23	1.32	0.27	0.27	0.27	0.27	0.26	0.25	0.23	0.24	0.23	0.21	0.2	0.22
Fixture 20	Oz CP-ML	0.45	0.46	0.81	2.21	0.69	2.43	0.19	0.18	0.18	0.18	0.17	0.15	0.17	0.16	0.16	0.16	0.14	0.11
	Oz E-CP	0.25	0.62	0.54	1.21	0.82	1.59	0.18	0.17	0.17	0.17	0.15	0.15	0.15	0.14	0.15	0.14	0.12	0.11
Fixture 25	Oz CP-ML	0.49	0.51	0.25	0.16	0.32	0.13	0.08	0.07	0.07	0.07	0.05	0.05	0.08	0.09	0.08	0.08	0.08	0.07
	Oz E-CP	0.08	0.07	0.19	0.07	0.08	0.08	0.07	0.06	0.07	0.06	0.05	0.05	0.08	0.08	0.08	0.07	0.07	0.06
Avg. Time	Oz CP-ML	1.15						0.26						0.22
	Oz E-CP	0.77						0.25						0.22

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1 Core Experiments
Constraint Type		$P$ Queries						$R$ Queries						$S$ Queries
	Oz CP	31	37	44	52	68	67	0	0	0	0	0	0	0	0	0	0	0	7
Fixture 5	Oz CP-ML	11	16	24	28	47	49	0	0	0	0	0	0	0	0	0	0	0	1
	Oz E-CP	9	14	15	21	36	33	0	0	0	0	0	0	0	0	0	0	0	1
	Oz CP	28	37	45	56	69	67	0	0	0	0	0	0	0	0	0	1	4	3
Fixture 10	Oz CP-ML	6	12	18	27	42	46	0	0	0	0	0	0	0	0	0	0	0	0
	Oz E-CP	5	9	10	19	27	30	0	0	0	0	0	0	0	0	0	0	0	0
	Oz CP	29	35	46	57	65	73	0	0	0	0	0	0	0	0	1	0	2	5
Fixture 15	Oz CP-ML	5	11	14	24	31	40	0	0	0	0	0	0	0	0	0	0	0	0
	Oz E-CP	3	3	6	12	20	24	0	0	0	0	0	0	0	0	0	0	0	0
	Oz CP	33	36	43	65	67	63	0	0	0	0	0	0	0	0	0	0	1	3
Fixture 20	Oz CP-ML	8	14	17	30	41	47	0	0	0	0	0	0	0	0	0	0	0	0
	Oz E-CP	3	1	9	10	30	33	0	0	0	0	0	0	0	0	0	0	0	0
	Oz CP	29	39	35	38	37	28	0	0	0	0	0	0	0	0	0	0	1	0
Fixture 25	Oz CP-ML	23	27	28	36	34	26	0	0	0	0	0	0	0	0	0	0	0	0
	Oz E-CP	3	1	1	2	4	3	0	0	0	0	0	0	0	0	0	0	0	0
Unsolved	Oz CP	1419						0						28
Instances	Oz CP-ML	782						0						1
	Oz E-CP	396						0						1
	Oz CP	0.23	0.46	0.21	0.23	0.31	0.89	0.56	0.53	0.51	0.5	0.5	0.49	0.43	0.44	0.4	0.39	0.38	0.34
Fixture 5	Oz CP-ML	2.1	2.98	2.72	3.38	3.98	3.82	0.28	0.27	0.27	0.28	0.3	0.34	0.32	0.31	0.3	0.28	0.26	0.44
	Oz E-CP	1.82	1.88	2.54	2.26	2.79	3.84	0.49	0.49	0.49	0.49	0.47	0.46	0.44	0.41	0.41	0.38	0.37	0.51
	Oz CP	0.33	0.2	0.24	0.35	0.18	0.41	0.37	0.39	0.39	0.39	0.38	0.37	0.34	0.34	0.32	0.32	0.28	0.25
Fixture 10	Oz CP-ML	1.69	2.35	2.55	3.3	3.4	3.85	0.27	0.28	0.29	0.29	0.28	0.27	0.25	0.24	0.23	0.26	0.45	0.29
	Oz E-CP	1.35	1.51	2.26	2.17	2.69	2.4	0.38	0.37	0.38	0.36	0.37	0.34	0.32	0.32	0.3	0.32	0.51	0.33
	Oz CP	0.48	0.2	0.14	0.56	0.43	0.14	0.29	0.29	0.33	0.32	0.29	0.31	0.27	0.28	0.26	0.23	0.19	0.21
Fixture 15	Oz CP-ML	1.59	1.79	2.85	3.38	4.36	5.03	0.21	0.21	0.22	0.22	0.21	0.2	0.19	0.19	0.23	0.17	0.23	0.4
	Oz E-CP	1.27	1.26	2.04	2.39	2.9	3.12	0.28	0.28	0.28	0.28	0.27	0.25	0.23	0.2	0.19	0.15	0.19	0.38
	Oz CP	0.12	0.71	0.26	0.7	0.12	0.24	0.19	0.21	0.22	0.22	0.2	0.18	0.19	0.18	0.18	0.16	0.14	0.12
Fixture 20	Oz CP-ML	1.49	1.7	2.43	4.33	3.05	2.81	0.15	0.14	0.14	0.14	0.14	0.13	0.14	0.14	0.14	0.13	0.14	0.19
	Oz E-CP	0.8	1.28	1.41	2.33	1.95	1.49	0.14	0.14	0.14	0.13	0.12	0.12	0.11	0.11	0.12	0.11	0.13	0.18
	Oz CP	0.07	0.22	0.42	0.13	0.12	0.07	0.09	0.08	0.08	0.07	0.06	0.06	0.1	0.1	0.1	0.09	0.07	0.08
Fixture 25	Oz CP-ML	0.59	1.3	0.68	0.24	0.28	0.18	0.06	0.05	0.05	0.05	0.04	0.04	0.07	0.07	0.06	0.06	0.08	0.05
	Oz E-CP	0.17	0.06	0.21	0.21	0.44	0.09	0.09	0.08	0.08	0.07	0.06	0.06	0.09	0.09	0.09	0.08	0.11	0.07
	Oz CP	0.29						0.3						0.24
Avg. Time	Oz CP-ML	2.38						0.19						0.21
	Oz E-CP	1.6						0.26						0.24
4 Core Experiments
Constraint Type		$P$ Queries						$R$ Queries						$S$ Queries
Fixture 5	Oz CP-ML	10	17	21	26	48	49	0	0	0	0	0	0	0	0	0	0	0	1
	Oz E-CP	9	15	16	18	37	32	0	0	0	0	0	0	0	0	0	0	0	1
Fixture 10	Oz CP-ML	6	12	17	26	38	47	0	0	0	0	0	0	0	0	0	0	0	0
	Oz E-CP	4	8	11	19	25	28	0	0	0	0	0	0	0	0	0	0	0	0
Fixture 15	Oz CP-ML	5	10	12	19	33	38	0	0	0	0	0	0	0	0	0	0	0	0
	Oz E-CP	4	3	5	11	21	26	0	0	0	0	0	0	0	0	0	0	0	0
Fixture 20	Oz CP-ML	8	16	17	30	40	44	0	0	0	0	0	0	0	0	0	0	0	0
	Oz E-CP	3	3	9	10	30	32	0	0	0	0	0	0	0	0	0	0	0	0
Fixture 25	Oz CP-ML	22	27	28	36	34	26	0	0	0	0	0	0	0	0	0	0	0	0
	Oz E-CP	3	1	1	2	4	3	0	0	0	0	0	0	0	0	0	0	0	0
Unsolved	Oz CP-ML	762						0						1
Instances	Oz E-CP	393						0						1
Fixture 5	Oz CP-ML	1.02	1.22	1.35	2.15	1.72	1.91	0.46	0.45	0.47	0.45	0.45	0.44	0.4	0.4	0.39	0.37	0.36	0.32
	Oz E-CP	0.78	0.74	1.41	2.01	1.63	2.13	0.45	0.44	0.45	0.46	0.44	0.43	0.39	0.39	0.39	0.36	0.35	0.32
Fixture 10	Oz CP-ML	0.86	0.88	0.91	1.72	2.37	1.69	0.36	0.36	0.36	0.35	0.34	0.33	0.32	0.31	0.3	0.29	0.33	0.25
	Oz E-CP	0.95	0.62	0.77	0.69	0.92	1.2	0.3	0.32	0.34	0.34	0.34	0.33	0.33	0.31	0.29	0.29	0.31	0.25
Fixture 15	Oz CP-ML	0.57	0.7	1.42	2.18	1.58	3.19	0.28	0.27	0.27	0.27	0.26	0.25	0.23	0.23	0.23	0.21	0.2	0.23
	Oz E-CP	0.32	0.42	1.07	0.9	1.23	1.32	0.27	0.27	0.27	0.27	0.26	0.25	0.23	0.24	0.23	0.21	0.2	0.22
Fixture 20	Oz CP-ML	0.45	0.46	0.81	2.21	0.69	2.43	0.19	0.18	0.18	0.18	0.17	0.15	0.17	0.16	0.16	0.16	0.14	0.11
	Oz E-CP	0.25	0.62	0.54	1.21	0.82	1.59	0.18	0.17	0.17	0.17	0.15	0.15	0.15	0.14	0.15	0.14	0.12	0.11
Fixture 25	Oz CP-ML	0.49	0.51	0.25	0.16	0.32	0.13	0.08	0.07	0.07	0.07	0.05	0.05	0.08	0.09	0.08	0.08	0.08	0.07
	Oz E-CP	0.08	0.07	0.19	0.07	0.08	0.08	0.07	0.06	0.07	0.06	0.05	0.05	0.08	0.08	0.08	0.07	0.07	0.06
Avg. Time	Oz CP-ML	1.15						0.26						0.22
	Oz E-CP	0.77						0.25						0.22

Table 5. Test results using MiniZinc solvers and CPLEX for the Colombian League with 18 teams

Colombian League - 1 Core Test
Model	Solver	Solved Instances	Unsolved Instances	Avg T. Solved Instances	Std T. Solved Instances	Solved SAT Instances	Solved UNSAT Instances
	G12/FD	8592	408	0.85	2.44	5851	2741
CP	Gecode	8300	700	0.67	2.21	5584	2716
	HaifaCSP	8949	51	0.36	1.79	6178	2771
	G12/FD	8475	525	0.79	2.25	5701	2774
CP-R	Gecode	8199	801	0.82	2.56	5664	2535
	HaifaCSP	8997	3	0.15	0.75	6220	2777
	CPLEX	8995	5	0.45	0.29	6216	2779
MIP	G12/FD	8468	532	1.36	3.4	5769	2699
	Gecode	8198	802	0.77	1.93	5499	2699
	HaifaCSP	8916	84	0.61	2.07	6150	2766
Colombian League - 4 Core Test
CP	Gecode	8427	573	0.49	1.99	5704	2723
	HaifaCSP	8961	39	0.45	1.7	6202	2759
CP-R	HaifaCSP	8994	6	0.16	0.34	6220	2774
MIP	CPLEX	8999	1	0.47	0.28	6220	2779
	HaifaCSP	8960	40	0.54	1.82	6207	2753

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Colombian League - 1 Core Test
Model	Solver	Solved Instances	Unsolved Instances	Avg T. Solved Instances	Std T. Solved Instances	Solved SAT Instances	Solved UNSAT Instances
	G12/FD	8592	408	0.85	2.44	5851	2741
CP	Gecode	8300	700	0.67	2.21	5584	2716
	HaifaCSP	8949	51	0.36	1.79	6178	2771
	G12/FD	8475	525	0.79	2.25	5701	2774
CP-R	Gecode	8199	801	0.82	2.56	5664	2535
	HaifaCSP	8997	3	0.15	0.75	6220	2777
	CPLEX	8995	5	0.45	0.29	6216	2779
MIP	G12/FD	8468	532	1.36	3.4	5769	2699
	Gecode	8198	802	0.77	1.93	5499	2699
	HaifaCSP	8916	84	0.61	2.07	6150	2766
Colombian League - 4 Core Test
CP	Gecode	8427	573	0.49	1.99	5704	2723
	HaifaCSP	8961	39	0.45	1.7	6202	2759
CP-R	HaifaCSP	8994	6	0.16	0.34	6220	2774
MIP	CPLEX	8999	1	0.47	0.28	6220	2779
	HaifaCSP	8960	40	0.54	1.82	6207	2753

Table 6. Test results using MiniZinc solvers and CPLEX for the Argentine League with 30 teams

Argentine League - 1 Core Test
Model	Solver	Solved Instances	Unsolved Instances	Avg T. Solved Instances	Std T. Solved Instances	Solved SAT Instances	Solved UNSAT Instances
	G12/FD	7635	1365	2.14	2.35	6166	1469
CP	Gecode	7406	1594	1.5	1.72	5973	1433
	HaifaCSP	8363	637	1.07	3.23	6821	1542
	G12/FD	7389	1611	1.8	2.14	5839	1550
CP-R	Gecode	7752	1248	1.48	2.12	6197	1555
	HaifaCSP	8858	142	0.94	2.63	7280	1578
	CPLEX	8979	21	1.42	1.63	7411	1568
MIP	G12/FD	6401	2599	3.66	3.58	5176	1225
	Gecode	6745	2255	1.56	2.54	5428	1317
	HaifaCSP	8263	737	1.8	3.58	6742	1521
Argentine League - 4 Core Test
CP	Gecode	7537	1463	0.93	1.77	6101	1436
	HaifaCSP	8541	459	1.69	3.96	7017	1524
CP-R	HaifaCSP	8912	88	1.36	2.98	7333	1579
MIP	CPLEX	8921	79	1.12	2.19	7338	1583
	HaifaCSP	8491	509	2.56	4.38	6988	1503

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Argentine League - 1 Core Test
Model	Solver	Solved Instances	Unsolved Instances	Avg T. Solved Instances	Std T. Solved Instances	Solved SAT Instances	Solved UNSAT Instances
	G12/FD	7635	1365	2.14	2.35	6166	1469
CP	Gecode	7406	1594	1.5	1.72	5973	1433
	HaifaCSP	8363	637	1.07	3.23	6821	1542
	G12/FD	7389	1611	1.8	2.14	5839	1550
CP-R	Gecode	7752	1248	1.48	2.12	6197	1555
	HaifaCSP	8858	142	0.94	2.63	7280	1578
	CPLEX	8979	21	1.42	1.63	7411	1568
MIP	G12/FD	6401	2599	3.66	3.58	5176	1225
	Gecode	6745	2255	1.56	2.54	5428	1317
	HaifaCSP	8263	737	1.8	3.58	6742	1521
Argentine League - 4 Core Test
CP	Gecode	7537	1463	0.93	1.77	6101	1436
	HaifaCSP	8541	459	1.69	3.96	7017	1524
CP-R	HaifaCSP	8912	88	1.36	2.98	7333	1579
MIP	CPLEX	8921	79	1.12	2.19	7338	1583
	HaifaCSP	8491	509	2.56	4.38	6988	1503

Table 7. CPLEX(MIP) vs. Oz(E-CP) common solved instances comparison

Colombian League CPLEX(MIP) 4 cores vs Oz(E-CP) 4 cores
Model	Solver	Faster in	SAT	UNSAT
MIP	CPLEX	89	83	6
E-CP	Oz	8745	6091	2654
Argentine League CPLEX(MIP) 4 cores vs Oz(E-CP) 4 cores
Model	Solver	Faster in	SAT	UNSAT
MIP	CPLEX	230	226	3
E-CP	Oz	8320	6837	1481

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Colombian League CPLEX(MIP) 4 cores vs Oz(E-CP) 4 cores
Model	Solver	Faster in	SAT	UNSAT
MIP	CPLEX	89	83	6
E-CP	Oz	8745	6091	2654
Argentine League CPLEX(MIP) 4 cores vs Oz(E-CP) 4 cores
Model	Solver	Faster in	SAT	UNSAT
MIP	CPLEX	230	226	3
E-CP	Oz	8320	6837	1481

Table 8. HaifaCSP(CP-R) vs. Oz(E-CP) common solved instances comparison

Colombian League HaifaCSP(CP-R) 4 cores vs Oz(E-CP) 4 cores
Model	Solver	Faster in	SAT	UNSAT
CP-R	HaifaCSP	1171	399	772
E-CP	Oz	7223	5500	1723
Argentine League HaifaCSP(CP-R) 4 cores vs Oz(E-CP) 4 cores
Model	Solver	Faster in	SAT	UNSAT
CP-R	HaifaCSP	599	231	368
E-CP	Oz	7927	6809	1116

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Colombian League HaifaCSP(CP-R) 4 cores vs Oz(E-CP) 4 cores
Model	Solver	Faster in	SAT	UNSAT
CP-R	HaifaCSP	1171	399	772
E-CP	Oz	7223	5500	1723
Argentine League HaifaCSP(CP-R) 4 cores vs Oz(E-CP) 4 cores
Model	Solver	Faster in	SAT	UNSAT
CP-R	HaifaCSP	599	231	368
E-CP	Oz	7927	6809	1116

Table 9. Test results using MiniZinc solvers and CPLEX for the Colombian League with 18 teams

Colombian League - 4 Core Mixed Solvers Tests
Model	Solver	Solved Instances	Unsolved Instances	Avg T. Solved Instances	Std T. Solved Instances	Solved SAT Instances	Solved UNSAT Instances
E-CP & MIP	Oz & CPLEX	8999	1	0.11	0.3	6220	2779
E-CP & CP-R	Oz & HaifaCSP	8994	6	0.1	0.25	6220	2774

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Colombian League - 4 Core Mixed Solvers Tests
Model	Solver	Solved Instances	Unsolved Instances	Avg T. Solved Instances	Std T. Solved Instances	Solved SAT Instances	Solved UNSAT Instances
E-CP & MIP	Oz & CPLEX	8999	1	0.11	0.3	6220	2779
E-CP & CP-R	Oz & HaifaCSP	8994	6	0.1	0.25	6220	2774

Table 10. Test results using MiniZinc solvers and CPLEX for the Argentine League with 30 teams

Argentine League - 4 Core Mixed Solvers Tests
Model	Solver	Solved Instances	Unsolved Instances	Avg T. Solved Instances	Std T. Solved Instances	Solved SAT Instances	Solved UNSAT Instances
E-CP & MIP	Oz & CPLEX	8958	42	0.56	1.74	7376	1582
E-CP & CP-R	Oz & HaifaCSP	8940	60	0.62	2.05	7361	1579

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Argentine League - 4 Core Mixed Solvers Tests
Model	Solver	Solved Instances	Unsolved Instances	Avg T. Solved Instances	Std T. Solved Instances	Solved SAT Instances	Solved UNSAT Instances
E-CP & MIP	Oz & CPLEX	8958	42	0.56	1.74	7376	1582
E-CP & CP-R	Oz & HaifaCSP	8940	60	0.62	2.05	7361	1579

Table 11. Number of solved instances at different time limits using the best model-solver configurations

Colombian League - 4 Core Test
Model	Solver	1sec	10sec	20sec	30sec -(Total)	Unsolved Instances
CP	HaifaCSP	8279	8896	8948	8961	39
CP-R	HaifaCSP	8845	8994	8994	8994	6
E-CP	OZ	8756	8804	8832	8834	166
MIP	CPLEX	8708	8998	8999	8999	1
E-CP & MIP	OZ & CPLEX	8755	8999	8999	8999	1
Argentine League - 4 Core Test
CP	HaifaCSP	6598	8157	8422	8541	459
CP-R	HaifaCSP	6866	8673	8853	8912	88
E-CP	OZ	8399	8571	8588	8606	394
MIP	CPLEX	7481	8784	8900	8921	79
E-CP & MIP	OZ & CPLEX	8399	8877	8946	8958	42

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Colombian League - 4 Core Test
Model	Solver	1sec	10sec	20sec	30sec -(Total)	Unsolved Instances
CP	HaifaCSP	8279	8896	8948	8961	39
CP-R	HaifaCSP	8845	8994	8994	8994	6
E-CP	OZ	8756	8804	8832	8834	166
MIP	CPLEX	8708	8998	8999	8999	1
E-CP & MIP	OZ & CPLEX	8755	8999	8999	8999	1
Argentine League - 4 Core Test
CP	HaifaCSP	6598	8157	8422	8541	459
CP-R	HaifaCSP	6866	8673	8853	8912	88
E-CP	OZ	8399	8571	8588	8606	394
MIP	CPLEX	7481	8784	8900	8921	79
E-CP & MIP	OZ & CPLEX	8399	8877	8946	8958	42

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2019 Impact Factor: 1.366

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2019 Impact Factor: 1.366

American Institute of Mathematical Sciences

CP and MIP approaches for soccer analysis

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