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CP and MIP approaches for soccer analysis

  • * Corresponding author: Robinson Duque

    * Corresponding author: Robinson Duque 
Abstract Full Text(HTML) Figure(5) / Table(11) Related Papers Cited by
  • 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.

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

    Citation:

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

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

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

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

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

    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 $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$
     | Show Table
    DownLoad: CSV

    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 $\checkmark$
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV
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