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References A new Branch-and-Price Algorithm for the Traveling Tournament Problem (TTP) Column Generation 2008, Aussois, France Stefan Irnich 1 sirnich@or.rwth-aachen.de RWTH Aachen University Deutsche Post Endowed Chair of Optimization of

  1. References Round Robin Sports League Scheduling If an away game is followed by another away game or a home game is followed by another home game, this is called a break. In an attractive schedules, the number of breaks is small. Distance minimization Break minimization objective: minimize distance, no additional constraints: i.e., function separable by constructive methods teams → timetables with n − 2 breaks (SRR), 3 n − 6 (MDRR) etc. constraints: number of in general: minimization of (consecutive) breaks breaks, s.t. additional approaches: heuristics, constraints combined IP and CP approaches: heuristics, IP or (Easton et al. , 2003) combined IP/CP (Nemhauser and Trick, 1998; Henz, 2001) 5 / 33

  2. References Round Robin Sports League Scheduling If an away game is followed by another away game or a home game is followed by another home game, this is called a break. In an attractive schedules, the number of breaks is small. Distance minimization Break minimization objective: minimize distance, no additional constraints: i.e., function separable by constructive methods teams → timetables with n − 2 breaks (SRR), 3 n − 6 (MDRR) etc. constraints: number of in general: minimization of (consecutive) breaks breaks, s.t. additional approaches: heuristics, constraints combined IP and CP approaches: heuristics, IP or (Easton et al. , 2003) combined IP/CP (Nemhauser and Trick, 1998; Henz, 2001) 5 / 33

  3. References What is the TTP? The Traveling Tournament Problem (TTP) is as follows: Input: n the number of teams D = ( d ij ) an n × n integer distance matrix L and U integer parameters Output: A double round robin tournament on the n teams such that the number of consecutive home games and consecutive away games are between L and U inclusive (typically L = 1 non-restrictive), and the total distance travelled by the teams is minimized (optional) no-repeater constraints: game A-B not followed by game B-A 6 / 33

  4. References What is the TTP? The Traveling Tournament Problem (TTP) is as follows: Input: n the number of teams D = ( d ij ) an n × n integer distance matrix L and U integer parameters Output: A double round robin tournament on the n teams such that the number of consecutive home games and consecutive away games are between L and U inclusive (typically L = 1 non-restrictive), and the total distance travelled by the teams is minimized (optional) no-repeater constraints: game A-B not followed by game B-A 6 / 33

  5. References What is the TTP? The Traveling Tournament Problem (TTP) is as follows: Input: n the number of teams D = ( d ij ) an n × n integer distance matrix L and U integer parameters Output: A double round robin tournament on the n teams such that the number of consecutive home games and consecutive away games are between L and U inclusive (typically L = 1 non-restrictive), and the total distance travelled by the teams is minimized (optional) no-repeater constraints: game A-B not followed by game B-A 6 / 33

  6. References What is the TTP? The Traveling Tournament Problem (TTP) is as follows: Input: n the number of teams D = ( d ij ) an n × n integer distance matrix L and U integer parameters Output: A double round robin tournament on the n teams such that the number of consecutive home games and consecutive away games are between L and U inclusive (typically L = 1 non-restrictive), and the total distance travelled by the teams is minimized (optional) no-repeater constraints: game A-B not followed by game B-A 6 / 33

  7. References What is the TTP? The Traveling Tournament Problem (TTP) is as follows: Input: n the number of teams D = ( d ij ) an n × n integer distance matrix L and U integer parameters Output: A double round robin tournament on the n teams such that the number of consecutive home games and consecutive away games are between L and U inclusive (typically L = 1 non-restrictive), and the total distance travelled by the teams is minimized (optional) no-repeater constraints: game A-B not followed by game B-A 6 / 33

  8. References Extensive Formulation Easton et al. (2003) proposed the following extensive (CG) formulation based on the tour variables λ t p : P t set of tours for team t tour p ∈ P t has cost c p s , t ′ = { tour of team t with away game against t ′ in slot s } A t � � c p λ t min p p ∈ P t t ∈ T � � � ∀ t ∈ T λ t ′ λ t s.t. + = 1 (1) p p ∀ s ∈ { 1 , . . . , 2 ¯ n } p ∈ Pt : t ′∈ T : p ∈ Pt ′ : t ′� = t ∃ t ′ : p ∈A t p ∈A t ′ s , t ′ st � �� � � �� � team t ′ � = t plays team t plays away away against t � λ t p = 1 ∀ t ∈ T (2) p ∈ P t λ t ∀ t ∈ T , p ∈ P t p ∈ { 0 , 1 } (3) 7 / 33

  9. References Extensive Formulation Easton et al. (2003) proposed the following extensive (CG) formulation based on the tour variables λ t p : P t set of tours for team t tour p ∈ P t has cost c p s , t ′ = { tour of team t with away game against t ′ in slot s } A t � � c p λ t min p p ∈ P t t ∈ T � � � ∀ t ∈ T λ t ′ λ t s.t. + = 1 (1) p p ∀ s ∈ { 1 , . . . , 2 ¯ n } p ∈ Pt : t ′∈ T : p ∈ Pt ′ : t ′� = t ∃ t ′ : p ∈A t p ∈A t ′ s , t ′ st � �� � � �� � team t ′ � = t plays team t plays away away against t � λ t p = 1 ∀ t ∈ T (2) p ∈ P t λ t ∀ t ∈ T , p ∈ P t p ∈ { 0 , 1 } (3) 7 / 33

  10. References Hardness of the TTP Empirically, a problem very hard to solve ! (Easton et al. , 2003): NL4: Instance with n = 4 teams Solved to optimality in ≈ 30 seconds. NL6: Instance with n = 6 teams Solved to optimality in 20 ∗ · 900 seconds. NL8: Instance with n = 8 teams (relaxed) Solved to optimality in > 20 ∗ · 4 days. ∗ using 20 workstations in parallel No (theoretical) complexity results, but suspected to be NP -hard! Design of good heuristic methods is a non-trivial task, see, e.g., (Van Hentenryck and Vergados, 2006) 8 / 33

  11. References Hardness of the TTP Empirically, a problem very hard to solve ! (Easton et al. , 2003): NL4: Instance with n = 4 teams Solved to optimality in ≈ 30 seconds. NL6: Instance with n = 6 teams Solved to optimality in 20 ∗ · 900 seconds. NL8: Instance with n = 8 teams (relaxed) Solved to optimality in > 20 ∗ · 4 days. ∗ using 20 workstations in parallel No (theoretical) complexity results, but suspected to be NP -hard! Design of good heuristic methods is a non-trivial task, see, e.g., (Van Hentenryck and Vergados, 2006) 8 / 33

  12. References Hardness of the TTP Empirically, a problem very hard to solve ! (Easton et al. , 2003): NL4: Instance with n = 4 teams Solved to optimality in ≈ 30 seconds. NL6: Instance with n = 6 teams Solved to optimality in 20 ∗ · 900 seconds. NL8: Instance with n = 8 teams (relaxed) Solved to optimality in > 20 ∗ · 4 days. ∗ using 20 workstations in parallel No (theoretical) complexity results, but suspected to be NP -hard! Design of good heuristic methods is a non-trivial task, see, e.g., (Van Hentenryck and Vergados, 2006) 8 / 33

  13. References Compact Formulation Easton et al. (2003) did not provide a compact formulation Compact formulation with binary variables x t ijs , where x t = ( x t ijs ) represents the movement of team t x t ijs = 1 if team t moves at time slot s from the venue of team i to the venue of team j (playing there at time slot s + 1) Remarks: teams are numbered by t ∈ T := { 1 , 2 , . . . , n } venues are numbered as their corresponding teams, i.e., i , j ∈ T = { 1 , 2 , . . . , n } time slots are numbered s ∈ { 1 , 2 , . . . , 2 ¯ n } with two artificial time slots 0 and 2 ¯ n + 1 9 / 33

  14. References Compact Formulation Easton et al. (2003) did not provide a compact formulation Compact formulation with binary variables x t ijs , where x t = ( x t ijs ) represents the movement of team t x t ijs = 1 if team t moves at time slot s from the venue of team i to the venue of team j (playing there at time slot s + 1) Remarks: teams are numbered by t ∈ T := { 1 , 2 , . . . , n } venues are numbered as their corresponding teams, i.e., i , j ∈ T = { 1 , 2 , . . . , n } time slots are numbered s ∈ { 1 , 2 , . . . , 2 ¯ n } with two artificial time slots 0 and 2 ¯ n + 1 9 / 33

  15. References Compact Formulation Easton et al. (2003) did not provide a compact formulation Compact formulation with binary variables x t ijs , where x t = ( x t ijs ) represents the movement of team t x t ijs = 1 if team t moves at time slot s from the venue of team i to the venue of team j (playing there at time slot s + 1) Remarks: teams are numbered by t ∈ T := { 1 , 2 , . . . , n } venues are numbered as their corresponding teams, i.e., i , j ∈ T = { 1 , 2 , . . . , n } time slots are numbered s ∈ { 1 , 2 , . . . , 2 ¯ n } with two artificial time slots 0 and 2 ¯ n + 1 9 / 33

  16. References Compact Formulation Easton et al. (2003) did not provide a compact formulation Compact formulation with binary variables x t ijs , where x t = ( x t ijs ) represents the movement of team t x t ijs = 1 if team t moves at time slot s from the venue of team i to the venue of team j (playing there at time slot s + 1) Remarks: teams are numbered by t ∈ T := { 1 , 2 , . . . , n } venues are numbered as their corresponding teams, i.e., i , j ∈ T = { 1 , 2 , . . . , n } time slots are numbered s ∈ { 1 , 2 , . . . , 2 ¯ n } with two artificial time slots 0 and 2 ¯ n + 1 9 / 33

  17. References Compact Formulation Easton et al. (2003) did not provide a compact formulation Compact formulation with binary variables x t ijs , where x t = ( x t ijs ) represents the movement of team t x t ijs = 1 if team t moves at time slot s from the venue of team i to the venue of team j (playing there at time slot s + 1) Remarks: teams are numbered by t ∈ T := { 1 , 2 , . . . , n } venues are numbered as their corresponding teams, i.e., i , j ∈ T = { 1 , 2 , . . . , n } time slots are numbered s ∈ { 1 , 2 , . . . , 2 ¯ n } with two artificial time slots 0 and 2 ¯ n + 1 9 / 33

  18. References Compact Formulation Easton et al. (2003) did not provide a compact formulation Compact formulation with binary variables x t ijs , where x t = ( x t ijs ) represents the movement of team t x t ijs = 1 if team t moves at time slot s from the venue of team i to the venue of team j (playing there at time slot s + 1) Remarks: teams are numbered by t ∈ T := { 1 , 2 , . . . , n } venues are numbered as their corresponding teams, i.e., i , j ∈ T = { 1 , 2 , . . . , n } time slots are numbered s ∈ { 1 , 2 , . . . , 2 ¯ n } with two artificial time slots 0 and 2 ¯ n + 1 9 / 33

  19. References Compact Formulation Easton et al. (2003) did not provide a compact formulation Compact formulation with binary variables x t ijs , where x t = ( x t ijs ) represents the movement of team t x t ijs = 1 if team t moves at time slot s from the venue of team i to the venue of team j (playing there at time slot s + 1) Remarks: teams are numbered by t ∈ T := { 1 , 2 , . . . , n } venues are numbered as their corresponding teams, i.e., i , j ∈ T = { 1 , 2 , . . . , n } time slots are numbered s ∈ { 1 , 2 , . . . , 2 ¯ n } with two artificial time slots 0 and 2 ¯ n + 1 9 / 33

  20. References Compact Formulation Easton et al. (2003) did not provide a compact formulation Compact formulation with binary variables x t ijs , where x t = ( x t ijs ) represents the movement of team t x t ijs = 1 if team t moves at time slot s from the venue of team i to the venue of team j (playing there at time slot s + 1) Remarks: teams are numbered by t ∈ T := { 1 , 2 , . . . , n } venues are numbered as their corresponding teams, i.e., i , j ∈ T = { 1 , 2 , . . . , n } time slots are numbered s ∈ { 1 , 2 , . . . , 2 ¯ n } with two artificial time slots 0 and 2 ¯ n + 1 9 / 33

  21. References Compact Formulation Interpretation of decision variables as arcs of a layered network ( V , A ) nodes v t is ∈ V “team t is at venue i at time s ” ∼ i-s artificial node v t h 0 (“source”) and v t n + 1 (“sink”) h , 2 ¯ arcs ( v t is , v t j , s + 1 ) ∈ A corresponding to decision variables x t ijs Network for team t = 4 : 10 / 33

  22. References 2 ¯ n � � � � d ij x t min ijs t ∈ T i ∈ T j ∈ T s = 0 n 2 ¯ � � x t s.t. ijs = 1 ∀ t , i ∈ T : i � = t (1) s = 1 j ∈ T � � x t x t ijs − ji , s + 1 = 0 ∀ t , j ∈ T , ∀ s ∈ { 0 , .., 2 ¯ n − 1 } (2) i ∈ T i ∈ T U − 1 U − 1 � � � � x t x t tt , s + u ≤ U − 1 and ij , s + u ≤ U − 1 u = 0 u = 0 i � = t j � = t ∀ t ∈ T , ∀ s ∈ { 1 , .., 2 ¯ n − U } (3) ∀ t ∈ T � � � � x t ′ x t + = 1 (4) ijs tjs ∀ s ∈ { 1 , .., 2 ¯ n } t ′∈ T : i ∈ T : j ∈ T j ∈ T i � = t t ′� = t � �� � � �� � team t ′ � = t plays team t plays away away against t x t ijs ∈ { 0 , 1 } ∀ t , i , j ∈ T , ∀ s ∈ { 0 , .., 2 ¯ n } (5) Explanation: (1) each team t must play once at every opponent’s venue 11 / 33

  23. References 2 ¯ n � � � � d ij x t min ijs t ∈ T i ∈ T j ∈ T s = 0 n 2 ¯ � � x t s.t. ijs = 1 ∀ t , i ∈ T : i � = t (1) s = 1 j ∈ T � � x t x t ijs − ji , s + 1 = 0 ∀ t , j ∈ T , ∀ s ∈ { 0 , .., 2 ¯ n − 1 } (2) i ∈ T i ∈ T U − 1 U − 1 � � � � x t x t tt , s + u ≤ U − 1 and ij , s + u ≤ U − 1 u = 0 u = 0 i � = t j � = t ∀ t ∈ T , ∀ s ∈ { 1 , .., 2 ¯ n − U } (3) ∀ t ∈ T � � � � x t ′ x t + = 1 (4) ijs tjs ∀ s ∈ { 1 , .., 2 ¯ n } t ′∈ T : i ∈ T : j ∈ T j ∈ T i � = t t ′� = t � �� � � �� � team t ′ � = t plays team t plays away away against t x t ijs ∈ { 0 , 1 } ∀ t , i , j ∈ T , ∀ s ∈ { 0 , .., 2 ¯ n } (5) Explanation: (2) flow conservation 11 / 33

  24. References 2 ¯ n � � � � d ij x t min ijs t ∈ T i ∈ T j ∈ T s = 0 n 2 ¯ � � x t s.t. ijs = 1 ∀ t , i ∈ T : i � = t (1) s = 1 j ∈ T � � x t x t ijs − ji , s + 1 = 0 ∀ t , j ∈ T , ∀ s ∈ { 0 , .., 2 ¯ n − 1 } (2) i ∈ T i ∈ T U − 1 U − 1 � � � � x t x t tt , s + u ≤ U − 1 and ij , s + u ≤ U − 1 u = 0 u = 0 i � = t j � = t ∀ t ∈ T , ∀ s ∈ { 1 , .., 2 ¯ n − U } (3) ∀ t ∈ T � � � � x t ′ x t + = 1 (4) ijs tjs ∀ s ∈ { 1 , .., 2 ¯ n } t ′∈ T : i ∈ T : j ∈ T j ∈ T i � = t t ′� = t � �� � � �� � team t ′ � = t plays team t plays away away against t x t ijs ∈ { 0 , 1 } ∀ t , i , j ∈ T , ∀ s ∈ { 0 , .., 2 ¯ n } (5) Explanation: (3) upper bound on number of consecutive home/away games 11 / 33

  25. References 2 ¯ n � � � � d ij x t min ijs t ∈ T i ∈ T j ∈ T s = 0 n 2 ¯ � � x t s.t. ijs = 1 ∀ t , i ∈ T : i � = t (1) s = 1 j ∈ T � � x t x t ijs − ji , s + 1 = 0 ∀ t , j ∈ T , ∀ s ∈ { 0 , .., 2 ¯ n − 1 } (2) i ∈ T i ∈ T U − 1 U − 1 � � � � x t x t tt , s + u ≤ U − 1 and ij , s + u ≤ U − 1 u = 0 u = 0 i � = t j � = t ∀ t ∈ T , ∀ s ∈ { 1 , .., 2 ¯ n − U } (3) ∀ t ∈ T � � � � x t ′ x t + = 1 (4) ijs tjs ∀ s ∈ { 1 , .., 2 ¯ n } t ′∈ T : i ∈ T : j ∈ T j ∈ T i � = t t ′� = t � �� � � �� � team t ′ � = t plays team t plays away away against t x t ijs ∈ { 0 , 1 } ∀ t , i , j ∈ T , ∀ s ∈ { 0 , .., 2 ¯ n } (5) Explanation: (4) coupling constraints: each team t must play in every time slot s 11 / 33

  26. References Compact Formulation Dantzig-Wolfe Decomposition: decomposition by teams t constraints (1), (2), and (3) are separable by team → constraints of the CG pricing problem constraints (4) are coupling constraints between different teams → constraints of the CG master Result is identical to CG formulation given by Easton et al. (2003). Compact formulation is useful to show that branching rules are (in)complete to derive new branching rules to derive valid inequalties compatible with pricing ( → robust cuts) 12 / 33

  27. References Compact Formulation Dantzig-Wolfe Decomposition: decomposition by teams t constraints (1), (2), and (3) are separable by team → constraints of the CG pricing problem constraints (4) are coupling constraints between different teams → constraints of the CG master Result is identical to CG formulation given by Easton et al. (2003). Compact formulation is useful to show that branching rules are (in)complete to derive new branching rules to derive valid inequalties compatible with pricing ( → robust cuts) 12 / 33

  28. References Compact Formulation Optional No-Repeater Constraints (NRC) can be easily added: ∀ t , t ′ ∈ T , t � = t ′ , ∀ s ∈ { 1 , .., 2 ¯ tt ′ s + x t ′ x t tt ′ s ≤ 1 n − 1 } first variable: t plays away against t ′ in slot s + 1 second variable: t ′ plays away against t in slot s Remarks: this is a cubic number of constraints: n ( n − 1 )( 2 ¯ n − 1 ) NRC mix variables of different teams t � = t ′ ⇒ NRC cannot be modelled in the pricing problem; must remain in the CG master program ⇒ dynamic constraint generation implemented! 13 / 33

  29. References Compact Formulation Optional No-Repeater Constraints (NRC) can be easily added: ∀ t , t ′ ∈ T , t � = t ′ , ∀ s ∈ { 1 , .., 2 ¯ tt ′ s + x t ′ x t tt ′ s ≤ 1 n − 1 } first variable: t plays away against t ′ in slot s + 1 second variable: t ′ plays away against t in slot s Remarks: this is a cubic number of constraints: n ( n − 1 )( 2 ¯ n − 1 ) NRC mix variables of different teams t � = t ′ ⇒ NRC cannot be modelled in the pricing problem; must remain in the CG master program ⇒ dynamic constraint generation implemented! 13 / 33

  30. References Solution of the Pricing Problem Easton et al. (2003) suggest solving the pricing problem by constraint programming (CP): Advantages: easy to implement; easy to extend to cope with additional constraints Drawback: CP does not exploit the structure of the network The Pricing Problem of each team t is a Shortest-Path Problem with Resource Constraints (SPPRC) : 1 one resource α ∈ { 0 , 1 , . . . , U − 1 } for counting the number of consecutive home/way games 2 paths must be (task-)elementary visit every opponent’s venue at most (=exactly) once visit home venue at most (=exactly) ¯ n times 14 / 33

  31. References Solution of the Pricing Problem Easton et al. (2003) suggest solving the pricing problem by constraint programming (CP): Advantages: easy to implement; easy to extend to cope with additional constraints Drawback: CP does not exploit the structure of the network The Pricing Problem of each team t is a Shortest-Path Problem with Resource Constraints (SPPRC) : 1 one resource α ∈ { 0 , 1 , . . . , U − 1 } for counting the number of consecutive home/way games 2 paths must be (task-)elementary visit every opponent’s venue at most (=exactly) once visit home venue at most (=exactly) ¯ n times 14 / 33

  32. References Solution of the Pricing Problem Easton et al. (2003) suggest solving the pricing problem by constraint programming (CP): Advantages: easy to implement; easy to extend to cope with additional constraints Drawback: CP does not exploit the structure of the network The Pricing Problem of each team t is a Shortest-Path Problem with Resource Constraints (SPPRC) : 1 one resource α ∈ { 0 , 1 , . . . , U − 1 } for counting the number of consecutive home/way games 2 paths must be (task-)elementary visit every opponent’s venue at most (=exactly) once visit home venue at most (=exactly) ¯ n times 14 / 33

  33. References Solution of the Pricing Problem Standard labeling approach in network for team t : use labels ( i , S , α, ˜ c ) , where i is the last node (venue) S ⊆ T \ { t , i } is the set of previously visited opponent venues α is the number of consecutive home/away games ˜ c is the minimum (reduced) cost to reach state ( i , S , α ) Example with n = 4 teams, U = 3 : Tour ( 3 , H , H , H , 1 , 2 ) of team t = 4 = H creates the following sequence of labels (cost ˜ c omitted): ( s , ∅ , 0 ) (initial source node s ) 1 ( 3 , ∅ , 0 ) 2 ( H , { 3 } , 0 ) 3 ( H , { 3 } , 1 ) 4 ( H , { 3 } , 2 ) ← next game must be away 5 ( 1 , { 3 } , 0 ) 6 ( 2 , { 1 , 3 } , 1 ) 7 ( d , { 1 , 2 , 3 } , 0 ) (destination node d ) 8 15 / 33

  34. References Solution of the Pricing Problem Standard labeling approach in network for team t : use labels ( i , S , α, ˜ c ) , where i is the last node (venue) S ⊆ T \ { t , i } is the set of previously visited opponent venues α is the number of consecutive home/away games ˜ c is the minimum (reduced) cost to reach state ( i , S , α ) Example with n = 4 teams, U = 3 : Tour ( 3 , H , H , H , 1 , 2 ) of team t = 4 = H creates the following sequence of labels (cost ˜ c omitted): ( s , ∅ , 0 ) (initial source node s ) 1 ( 3 , ∅ , 0 ) 2 ( H , { 3 } , 0 ) 3 ( H , { 3 } , 1 ) 4 ( H , { 3 } , 2 ) ← next game must be away 5 ( 1 , { 3 } , 0 ) 6 ( 2 , { 1 , 3 } , 1 ) 7 ( d , { 1 , 2 , 3 } , 0 ) (destination node d ) 8 15 / 33

  35. References Solution of the Pricing Problem Standard labeling approach in network for team t : use labels ( i , S , α, ˜ c ) , where i is the last node (venue) S ⊆ T \ { t , i } is the set of previously visited opponent venues α is the number of consecutive home/away games ˜ c is the minimum (reduced) cost to reach state ( i , S , α ) Example with n = 4 teams, U = 3 : Tour ( 3 , H , H , H , 1 , 2 ) of team t = 4 = H creates the following sequence of labels (cost ˜ c omitted): ( s , ∅ , 0 ) (initial source node s ) 1 ( 3 , ∅ , 0 ) 2 ( H , { 3 } , 0 ) 3 ( H , { 3 } , 1 ) 4 ( H , { 3 } , 2 ) ← next game must be away 5 ( 1 , { 3 } , 0 ) 6 ( 2 , { 1 , 3 } , 1 ) 7 ( d , { 1 , 2 , 3 } , 0 ) (destination node d ) 8 15 / 33

  36. References Solution of the Pricing Problem Standard labeling approach in network for team t : use labels ( i , S , α, ˜ c ) , where i is the last node (venue) S ⊆ T \ { t , i } is the set of previously visited opponent venues α is the number of consecutive home/away games ˜ c is the minimum (reduced) cost to reach state ( i , S , α ) Example with n = 4 teams, U = 3 : Tour ( 3 , H , H , H , 1 , 2 ) of team t = 4 = H creates the following sequence of labels (cost ˜ c omitted): ( s , ∅ , 0 ) (initial source node s ) 1 ( 3 , ∅ , 0 ) 2 ( H , { 3 } , 0 ) 3 ( H , { 3 } , 1 ) 4 ( H , { 3 } , 2 ) ← next game must be away 5 ( 1 , { 3 } , 0 ) 6 ( 2 , { 1 , 3 } , 1 ) 7 ( d , { 1 , 2 , 3 } , 0 ) (destination node d ) 8 15 / 33

  37. References Solution of the Pricing Problem Standard labeling approach in network for team t : use labels ( i , S , α, ˜ c ) , where i is the last node (venue) S ⊆ T \ { t , i } is the set of previously visited opponent venues α is the number of consecutive home/away games ˜ c is the minimum (reduced) cost to reach state ( i , S , α ) Example with n = 4 teams, U = 3 : Tour ( 3 , H , H , H , 1 , 2 ) of team t = 4 = H creates the following sequence of labels (cost ˜ c omitted): ( s , ∅ , 0 ) (initial source node s ) 1 ( 3 , ∅ , 0 ) 2 ( H , { 3 } , 0 ) 3 ( H , { 3 } , 1 ) 4 ( H , { 3 } , 2 ) ← next game must be away 5 ( 1 , { 3 } , 0 ) 6 ( 2 , { 1 , 3 } , 1 ) 7 ( d , { 1 , 2 , 3 } , 0 ) (destination node d ) 8 15 / 33

  38. References Solution of the Pricing Problem Standard labeling approach in network for team t : use labels ( i , S , α, ˜ c ) , where i is the last node (venue) S ⊆ T \ { t , i } is the set of previously visited opponent venues α is the number of consecutive home/away games ˜ c is the minimum (reduced) cost to reach state ( i , S , α ) Example with n = 4 teams, U = 3 : Tour ( 3 , H , H , H , 1 , 2 ) of team t = 4 = H creates the following sequence of labels (cost ˜ c omitted): ( s , ∅ , 0 ) (initial source node s ) 1 ( 3 , ∅ , 0 ) 2 ( H , { 3 } , 0 ) 3 ( H , { 3 } , 1 ) 4 ( H , { 3 } , 2 ) ← next game must be away 5 ( 1 , { 3 } , 0 ) 6 ( 2 , { 1 , 3 } , 1 ) 7 ( d , { 1 , 2 , 3 } , 0 ) (destination node d ) 8 15 / 33

  39. References Solution of the Pricing Problem Standard labeling approach in network for team t : use labels ( i , S , α, ˜ c ) , where i is the last node (venue) S ⊆ T \ { t , i } is the set of previously visited opponent venues α is the number of consecutive home/away games ˜ c is the minimum (reduced) cost to reach state ( i , S , α ) Example with n = 4 teams, U = 3 : Tour ( 3 , H , H , H , 1 , 2 ) of team t = 4 = H creates the following sequence of labels (cost ˜ c omitted): ( s , ∅ , 0 ) (initial source node s ) 1 ( 3 , ∅ , 0 ) 2 ( H , { 3 } , 0 ) 3 ( H , { 3 } , 1 ) 4 ( H , { 3 } , 2 ) ← next game must be away 5 ( 1 , { 3 } , 0 ) 6 ( 2 , { 1 , 3 } , 1 ) 7 ( d , { 1 , 2 , 3 } , 0 ) (destination node d ) 8 15 / 33

  40. References Solution of the Pricing Problem Standard labeling approach in network for team t : use labels ( i , S , α, ˜ c ) , where i is the last node (venue) S ⊆ T \ { t , i } is the set of previously visited opponent venues α is the number of consecutive home/away games ˜ c is the minimum (reduced) cost to reach state ( i , S , α ) Example with n = 4 teams, U = 3 : Tour ( 3 , H , H , H , 1 , 2 ) of team t = 4 = H creates the following sequence of labels (cost ˜ c omitted): ( s , ∅ , 0 ) (initial source node s ) 1 ( 3 , ∅ , 0 ) 2 ( H , { 3 } , 0 ) 3 ( H , { 3 } , 1 ) 4 ( H , { 3 } , 2 ) ← next game must be away 5 ( 1 , { 3 } , 0 ) 6 ( 2 , { 1 , 3 } , 1 ) 7 ( d , { 1 , 2 , 3 } , 0 ) (destination node d ) 8 15 / 33

  41. References Solution of the Pricing Problem Standard labeling approach in network for team t : use labels ( i , S , α, ˜ c ) , where i is the last node (venue) S ⊆ T \ { t , i } is the set of previously visited opponent venues α is the number of consecutive home/away games ˜ c is the minimum (reduced) cost to reach state ( i , S , α ) Example with n = 4 teams, U = 3 : Tour ( 3 , H , H , H , 1 , 2 ) of team t = 4 = H creates the following sequence of labels (cost ˜ c omitted): ( s , ∅ , 0 ) (initial source node s ) 1 ( 3 , ∅ , 0 ) 2 ( H , { 3 } , 0 ) 3 ( H , { 3 } , 1 ) 4 ( H , { 3 } , 2 ) ← next game must be away 5 ( 1 , { 3 } , 0 ) 6 ( 2 , { 1 , 3 } , 1 ) 7 ( d , { 1 , 2 , 3 } , 0 ) (destination node d ) 8 15 / 33

  42. References Solution of the Pricing Problem Standard labeling approach in network for team t : use labels ( i , S , α, ˜ c ) , where i is the last node (venue) S ⊆ T \ { t , i } is the set of previously visited opponent venues α is the number of consecutive home/away games ˜ c is the minimum (reduced) cost to reach state ( i , S , α ) Example with n = 4 teams, U = 3 : Tour ( 3 , H , H , H , 1 , 2 ) of team t = 4 = H creates the following sequence of labels (cost ˜ c omitted): ( s , ∅ , 0 ) (initial source node s ) 1 ( 3 , ∅ , 0 ) 2 ( H , { 3 } , 0 ) 3 ( H , { 3 } , 1 ) 4 ( H , { 3 } , 2 ) ← next game must be away 5 ( 1 , { 3 } , 0 ) 6 ( 2 , { 1 , 3 } , 1 ) 7 ( d , { 1 , 2 , 3 } , 0 ) (destination node d ) 8 15 / 33

  43. References Solution of the Pricing Problem Standard labeling approach in network for team t : use labels ( i , S , α, ˜ c ) , where i is the last node (venue) S ⊆ T \ { t , i } is the set of previously visited opponent venues α is the number of consecutive home/away games ˜ c is the minimum (reduced) cost to reach state ( i , S , α ) Example with n = 4 teams, U = 3 : Tour ( 3 , H , H , H , 1 , 2 ) of team t = 4 = H creates the following sequence of labels (cost ˜ c omitted): ( s , ∅ , 0 ) (initial source node s ) 1 ( 3 , ∅ , 0 ) 2 ( H , { 3 } , 0 ) 3 ( H , { 3 } , 1 ) 4 ( H , { 3 } , 2 ) ← next game must be away 5 ( 1 , { 3 } , 0 ) 6 ( 2 , { 1 , 3 } , 1 ) 7 ( d , { 1 , 2 , 3 } , 0 ) (destination node d ) 8 15 / 33

  44. References Solution of the Pricing Problem Standard labeling approach in network for team t : use labels ( i , S , α, ˜ c ) , where i is the last node (venue) S ⊆ T \ { t , i } is the set of previously visited opponent venues α is the number of consecutive home/away games ˜ c is the minimum (reduced) cost to reach state ( i , S , α ) Example with n = 4 teams, U = 3 : Tour ( 3 , H , H , H , 1 , 2 ) of team t = 4 = H creates the following sequence of labels (cost ˜ c omitted): ( s , ∅ , 0 ) (initial source node s ) 1 ( 3 , ∅ , 0 ) 2 ( H , { 3 } , 0 ) 3 ( H , { 3 } , 1 ) 4 ( H , { 3 } , 2 ) ← next game must be away 5 ( 1 , { 3 } , 0 ) 6 ( 2 , { 1 , 3 } , 1 ) 7 ( d , { 1 , 2 , 3 } , 0 ) (destination node d ) 8 15 / 33

  45. References Solution of the Pricing Problem Standard labeling approach in network for team t : use labels ( i , S , α, ˜ c ) , where i is the last node (venue) S ⊆ T \ { t , i } is the set of previously visited opponent venues α is the number of consecutive home/away games ˜ c is the minimum (reduced) cost to reach state ( i , S , α ) Example with n = 4 teams, U = 3 : Tour ( 3 , H , H , H , 1 , 2 ) of team t = 4 = H creates the following sequence of labels (cost ˜ c omitted): ( s , ∅ , 0 ) (initial source node s ) 1 ( 3 , ∅ , 0 ) 2 ( H , { 3 } , 0 ) 3 ( H , { 3 } , 1 ) 4 ( H , { 3 } , 2 ) ← next game must be away 5 ( 1 , { 3 } , 0 ) 6 ( 2 , { 1 , 3 } , 1 ) 7 ( d , { 1 , 2 , 3 } , 0 ) (destination node d ) 8 15 / 33

  46. References Solution of the Pricing Problem Standard labeling approach in network for team t : use labels ( i , S , α, ˜ c ) , where i is the last node (venue) S ⊆ T \ { t , i } is the set of previously visited opponent venues α is the number of consecutive home/away games ˜ c is the minimum (reduced) cost to reach state ( i , S , α ) Example with n = 4 teams, U = 3 : Tour ( 3 , H , H , H , 1 , 2 ) of team t = 4 = H creates the following sequence of labels (cost ˜ c omitted): ( s , ∅ , 0 ) (initial source node s ) 1 ( 3 , ∅ , 0 ) 2 ( H , { 3 } , 0 ) 3 ( H , { 3 } , 1 ) 4 ( H , { 3 } , 2 ) ← next game must be away 5 ( 1 , { 3 } , 0 ) 6 ( 2 , { 1 , 3 } , 1 ) 7 ( d , { 1 , 2 , 3 } , 0 ) (destination node d ) 8 15 / 33

  47. References Solution of the Pricing Problem Non-standard approach to cope with this kind of SPPRC required! Observations: 1 there is dominance only between labels with identical last node i and subset S 2 in case of a lower bound L > 1 on consecutive home/away games, there is no dominance (at all) between different labels 3 if L = 1 (i.e. no lower bound), a label with α = 0 can only dominate U − 1 labels with α = 1 , 2 , . . . , U − 1 Consequences: we ignore dominance between labels with different α ( → factor U − 1 is small, typically U = 3) we use an expanded network where nodes are states ( i , S , α ) no dominance check required; solution with pulling algorithm for ordinary SPP 16 / 33

  48. References Solution of the Pricing Problem Non-standard approach to cope with this kind of SPPRC required! Observations: 1 there is dominance only between labels with identical last node i and subset S 2 in case of a lower bound L > 1 on consecutive home/away games, there is no dominance (at all) between different labels 3 if L = 1 (i.e. no lower bound), a label with α = 0 can only dominate U − 1 labels with α = 1 , 2 , . . . , U − 1 Consequences: we ignore dominance between labels with different α ( → factor U − 1 is small, typically U = 3) we use an expanded network where nodes are states ( i , S , α ) no dominance check required; solution with pulling algorithm for ordinary SPP 16 / 33

  49. References Solution of the Pricing Problem Non-standard approach to cope with this kind of SPPRC required! Observations: 1 there is dominance only between labels with identical last node i and subset S 2 in case of a lower bound L > 1 on consecutive home/away games, there is no dominance (at all) between different labels 3 if L = 1 (i.e. no lower bound), a label with α = 0 can only dominate U − 1 labels with α = 1 , 2 , . . . , U − 1 Consequences: we ignore dominance between labels with different α ( → factor U − 1 is small, typically U = 3) we use an expanded network where nodes are states ( i , S , α ) no dominance check required; solution with pulling algorithm for ordinary SPP 16 / 33

  50. References Solution of the Pricing Problem Non-standard approach to cope with this kind of SPPRC required! Observations: 1 there is dominance only between labels with identical last node i and subset S 2 in case of a lower bound L > 1 on consecutive home/away games, there is no dominance (at all) between different labels 3 if L = 1 (i.e. no lower bound), a label with α = 0 can only dominate U − 1 labels with α = 1 , 2 , . . . , U − 1 Consequences: we ignore dominance between labels with different α ( → factor U − 1 is small, typically U = 3) we use an expanded network where nodes are states ( i , S , α ) no dominance check required; solution with pulling algorithm for ordinary SPP 16 / 33

  51. References Solution of the Pricing Problem Non-standard approach to cope with this kind of SPPRC required! Observations: 1 there is dominance only between labels with identical last node i and subset S 2 in case of a lower bound L > 1 on consecutive home/away games, there is no dominance (at all) between different labels 3 if L = 1 (i.e. no lower bound), a label with α = 0 can only dominate U − 1 labels with α = 1 , 2 , . . . , U − 1 Consequences: we ignore dominance between labels with different α ( → factor U − 1 is small, typically U = 3) we use an expanded network where nodes are states ( i , S , α ) no dominance check required; solution with pulling algorithm for ordinary SPP 16 / 33

  52. References Solution of the Pricing Problem Non-standard approach to cope with this kind of SPPRC required! Observations: 1 there is dominance only between labels with identical last node i and subset S 2 in case of a lower bound L > 1 on consecutive home/away games, there is no dominance (at all) between different labels 3 if L = 1 (i.e. no lower bound), a label with α = 0 can only dominate U − 1 labels with α = 1 , 2 , . . . , U − 1 Consequences: we ignore dominance between labels with different α ( → factor U − 1 is small, typically U = 3) we use an expanded network where nodes are states ( i , S , α ) no dominance check required; solution with pulling algorithm for ordinary SPP 16 / 33

  53. References Solution of the Pricing Problem Non-standard approach to cope with this kind of SPPRC required! Observations: 1 there is dominance only between labels with identical last node i and subset S 2 in case of a lower bound L > 1 on consecutive home/away games, there is no dominance (at all) between different labels 3 if L = 1 (i.e. no lower bound), a label with α = 0 can only dominate U − 1 labels with α = 1 , 2 , . . . , U − 1 Consequences: we ignore dominance between labels with different α ( → factor U − 1 is small, typically U = 3) we use an expanded network where nodes are states ( i , S , α ) no dominance check required; solution with pulling algorithm for ordinary SPP 16 / 33

  54. References Solution of the Pricing Problem Expanded Network for n = 4 : 17 / 33

  55. References Solution of the Pricing Problem Size of the expanded network ( V ex , A ex ) : Here for U = 3! #teams #nodes #arcs #tours Factor | V ex | | A ex | #tours/#arcs n 4 92 198 120 ≈ 0 . 6 6 1138 3596 23280 ≈ 6 . 5 9.93888 · 10 6 8 8548 32108 ≈ 310 7.43831 · 10 9 10 55448 245910 ≈ 30 , 000 8.63608 · 10 12 12 320416 1621450 ≈ 5 , 300 , 000 Note: Pulling algorithm for SPP has time complexity O ( | A ex | ) Factor #tours/#arcs shows combinatorial leverage effect (Glover and Punnen, 1994; Ahuja et al. , 1999) 18 / 33

  56. References Solution of the Pricing Problem Size of the expanded network ( V ex , A ex ) : Here for U = 3! #teams #nodes #arcs #tours Factor | V ex | | A ex | #tours/#arcs n 4 92 198 120 ≈ 0 . 6 6 1138 3596 23280 ≈ 6 . 5 9.93888 · 10 6 8 8548 32108 ≈ 310 7.43831 · 10 9 10 55448 245910 ≈ 30 , 000 8.63608 · 10 12 12 320416 1621450 ≈ 5 , 300 , 000 Note: Pulling algorithm for SPP has time complexity O ( | A ex | ) Factor #tours/#arcs shows combinatorial leverage effect (Glover and Punnen, 1994; Ahuja et al. , 1999) 18 / 33

  57. References Solution of the Pricing Problem Size of the expanded network ( V ex , A ex ) : Here for U = 3! #teams #nodes #arcs #tours Factor | V ex | | A ex | #tours/#arcs n 4 92 198 120 ≈ 0 . 6 6 1138 3596 23280 ≈ 6 . 5 9.93888 · 10 6 8 8548 32108 ≈ 310 7.43831 · 10 9 10 55448 245910 ≈ 30 , 000 8.63608 · 10 12 12 320416 1621450 ≈ 5 , 300 , 000 Note: Pulling algorithm for SPP has time complexity O ( | A ex | ) Factor #tours/#arcs shows combinatorial leverage effect (Glover and Punnen, 1994; Ahuja et al. , 1999) 18 / 33

  58. References Solution of the Pricing Problem Observations: networks are symmetric: slots 1 , . . . , ¯ n are mirror images of slots ¯ n + 1 , . . . , 2 ¯ n only rdc on arcs differ Consequences: build and store only the “first half” of the expanded network corresponding to slots 1 , . . . , ¯ n solve SPP bidirectionally: SPP forward to create rdc labels at slots 1 , . . . , ¯ n 1 SPP backward to create rdc labels at slot 2 ¯ n , . . . , ¯ n + 1 2 merge step: combine fw and bw labels from slots ¯ n and ¯ n + 1 3 i.e. ℓ fw = ( i , S , α ) and ℓ bw = ( i ′ , S ′ , α ′ ) with S ∪ S ′ ∪ { i , i ′ } = T \ { t } and α + α ′ < U − 1 19 / 33

  59. References Solution of the Pricing Problem Observations: networks are symmetric: slots 1 , . . . , ¯ n are mirror images of slots ¯ n + 1 , . . . , 2 ¯ n only rdc on arcs differ Consequences: build and store only the “first half” of the expanded network corresponding to slots 1 , . . . , ¯ n solve SPP bidirectionally: SPP forward to create rdc labels at slots 1 , . . . , ¯ n 1 SPP backward to create rdc labels at slot 2 ¯ n , . . . , ¯ n + 1 2 merge step: combine fw and bw labels from slots ¯ n and ¯ n + 1 3 i.e. ℓ fw = ( i , S , α ) and ℓ bw = ( i ′ , S ′ , α ′ ) with S ∪ S ′ ∪ { i , i ′ } = T \ { t } and α + α ′ < U − 1 19 / 33

  60. References Solution of the Pricing Problem Observations: networks are symmetric: slots 1 , . . . , ¯ n are mirror images of slots ¯ n + 1 , . . . , 2 ¯ n only rdc on arcs differ Consequences: build and store only the “first half” of the expanded network corresponding to slots 1 , . . . , ¯ n solve SPP bidirectionally: SPP forward to create rdc labels at slots 1 , . . . , ¯ n 1 SPP backward to create rdc labels at slot 2 ¯ n , . . . , ¯ n + 1 2 merge step: combine fw and bw labels from slots ¯ n and ¯ n + 1 3 i.e. ℓ fw = ( i , S , α ) and ℓ bw = ( i ′ , S ′ , α ′ ) with S ∪ S ′ ∪ { i , i ′ } = T \ { t } and α + α ′ < U − 1 19 / 33

  61. References Solution of the Pricing Problem Observations: networks are symmetric: slots 1 , . . . , ¯ n are mirror images of slots ¯ n + 1 , . . . , 2 ¯ n only rdc on arcs differ Consequences: build and store only the “first half” of the expanded network corresponding to slots 1 , . . . , ¯ n solve SPP bidirectionally: SPP forward to create rdc labels at slots 1 , . . . , ¯ n 1 SPP backward to create rdc labels at slot 2 ¯ n , . . . , ¯ n + 1 2 merge step: combine fw and bw labels from slots ¯ n and ¯ n + 1 3 i.e. ℓ fw = ( i , S , α ) and ℓ bw = ( i ′ , S ′ , α ′ ) with S ∪ S ′ ∪ { i , i ′ } = T \ { t } and α + α ′ < U − 1 19 / 33

  62. References Solution of the Pricing Problem Observations: networks are symmetric: slots 1 , . . . , ¯ n are mirror images of slots ¯ n + 1 , . . . , 2 ¯ n only rdc on arcs differ Consequences: build and store only the “first half” of the expanded network corresponding to slots 1 , . . . , ¯ n solve SPP bidirectionally: SPP forward to create rdc labels at slots 1 , . . . , ¯ n 1 SPP backward to create rdc labels at slot 2 ¯ n , . . . , ¯ n + 1 2 merge step: combine fw and bw labels from slots ¯ n and ¯ n + 1 3 i.e. ℓ fw = ( i , S , α ) and ℓ bw = ( i ′ , S ′ , α ′ ) with S ∪ S ′ ∪ { i , i ′ } = T \ { t } and α + α ′ < U − 1 19 / 33

  63. References Solution of the Pricing Problem Observations: networks are symmetric: slots 1 , . . . , ¯ n are mirror images of slots ¯ n + 1 , . . . , 2 ¯ n only rdc on arcs differ Consequences: build and store only the “first half” of the expanded network corresponding to slots 1 , . . . , ¯ n solve SPP bidirectionally: SPP forward to create rdc labels at slots 1 , . . . , ¯ n 1 SPP backward to create rdc labels at slot 2 ¯ n , . . . , ¯ n + 1 2 merge step: combine fw and bw labels from slots ¯ n and ¯ n + 1 3 i.e. ℓ fw = ( i , S , α ) and ℓ bw = ( i ′ , S ′ , α ′ ) with S ∪ S ′ ∪ { i , i ′ } = T \ { t } and α + α ′ < U − 1 19 / 33

  64. References Solution of the Pricing Problem Observations: networks are symmetric: slots 1 , . . . , ¯ n are mirror images of slots ¯ n + 1 , . . . , 2 ¯ n only rdc on arcs differ Consequences: build and store only the “first half” of the expanded network corresponding to slots 1 , . . . , ¯ n solve SPP bidirectionally: SPP forward to create rdc labels at slots 1 , . . . , ¯ n 1 SPP backward to create rdc labels at slot 2 ¯ n , . . . , ¯ n + 1 2 merge step: combine fw and bw labels from slots ¯ n and ¯ n + 1 3 i.e. ℓ fw = ( i , S , α ) and ℓ bw = ( i ′ , S ′ , α ′ ) with S ∪ S ′ ∪ { i , i ′ } = T \ { t } and α + α ′ < U − 1 19 / 33

  65. References Solution of the Pricing Problem Variable Elimination ( → Network Reduction): Results of (Irnich et al. , 2007) can be used to eliminate arcs from the (individual!) expanded networks of each team. Given 1 a dual feasible solution π to the LP-relaxation of the CG master program 2 corresponding lower bound z LB = 1 1 ⊤ π for TTP 3 and an upper bound UB on TTP c t p ( π ) of a tour variable λ t Proposition A: If the rdc ˜ p exceeds the optimality gap UB − z LB , then λ t p = 0 in any optimal solution. Proposition B: If a ∈ A t ex is an arc of the expanded network of team t and every tour through a has rdc greater than UB − z LB , then no optimal tour p ∈ P t contains a . 20 / 33

  66. References Solution of the Pricing Problem Variable Elimination ( → Network Reduction): Results of (Irnich et al. , 2007) can be used to eliminate arcs from the (individual!) expanded networks of each team. Given 1 a dual feasible solution π to the LP-relaxation of the CG master program 2 corresponding lower bound z LB = 1 1 ⊤ π for TTP 3 and an upper bound UB on TTP c t p ( π ) of a tour variable λ t Proposition A: If the rdc ˜ p exceeds the optimality gap UB − z LB , then λ t p = 0 in any optimal solution. Proposition B: If a ∈ A t ex is an arc of the expanded network of team t and every tour through a has rdc greater than UB − z LB , then no optimal tour p ∈ P t contains a . 20 / 33

  67. References Solution of the Pricing Problem Variable Elimination ( → Network Reduction): Results of (Irnich et al. , 2007) can be used to eliminate arcs from the (individual!) expanded networks of each team. Given 1 a dual feasible solution π to the LP-relaxation of the CG master program 2 corresponding lower bound z LB = 1 1 ⊤ π for TTP 3 and an upper bound UB on TTP c t p ( π ) of a tour variable λ t Proposition A: If the rdc ˜ p exceeds the optimality gap UB − z LB , then λ t p = 0 in any optimal solution. Proposition B: If a ∈ A t ex is an arc of the expanded network of team t and every tour through a has rdc greater than UB − z LB , then no optimal tour p ∈ P t contains a . 20 / 33

  68. References Solution of the Pricing Problem Variable Elimination ( → Network Reduction): Results of (Irnich et al. , 2007) can be used to eliminate arcs from the (individual!) expanded networks of each team. Given 1 a dual feasible solution π to the LP-relaxation of the CG master program 2 corresponding lower bound z LB = 1 1 ⊤ π for TTP 3 and an upper bound UB on TTP c t p ( π ) of a tour variable λ t Proposition A: If the rdc ˜ p exceeds the optimality gap UB − z LB , then λ t p = 0 in any optimal solution. Proposition B: If a ∈ A t ex is an arc of the expanded network of team t and every tour through a has rdc greater than UB − z LB , then no optimal tour p ∈ P t contains a . 20 / 33

  69. References Solution of the Pricing Problem Variable Elimination ( → Network Reduction): Results of (Irnich et al. , 2007) can be used to eliminate arcs from the (individual!) expanded networks of each team. Given 1 a dual feasible solution π to the LP-relaxation of the CG master program 2 corresponding lower bound z LB = 1 1 ⊤ π for TTP 3 and an upper bound UB on TTP c t p ( π ) of a tour variable λ t Proposition A: If the rdc ˜ p exceeds the optimality gap UB − z LB , then λ t p = 0 in any optimal solution. Proposition B: If a ∈ A t ex is an arc of the expanded network of team t and every tour through a has rdc greater than UB − z LB , then no optimal tour p ∈ P t contains a . 20 / 33

  70. References Solution of the Pricing Problem Variable Elimination ( → Network Reduction): Results of (Irnich et al. , 2007) can be used to eliminate arcs from the (individual!) expanded networks of each team. Given 1 a dual feasible solution π to the LP-relaxation of the CG master program 2 corresponding lower bound z LB = 1 1 ⊤ π for TTP 3 and an upper bound UB on TTP c t p ( π ) of a tour variable λ t Proposition A: If the rdc ˜ p exceeds the optimality gap UB − z LB , then λ t p = 0 in any optimal solution. Proposition B: If a ∈ A t ex is an arc of the expanded network of team t and every tour through a has rdc greater than UB − z LB , then no optimal tour p ∈ P t contains a . 20 / 33

  71. References Solution of the Pricing Problem Variable Elimination ( → Network Reduction) (Cont.): Smallest rdc of tours through an arbitrary arc a = ( v , w ) ∈ A t ex can be computed with a bidirectional SPP approach: 1 solve full fw SPP; cost labels ˜ c fw ( v ) at nodes v 2 solve full bw SPP; cost labels ˜ c bw ( v ) at nodes v 3 value rdc ( v , w ) := ˜ c fw ( v ) + ˜ c bw ( w ) is minimum rdc of c vw + ˜ paths through a = ( v , w ) 4 Rule: Eliminate a = ( v , w ) if rdc ( v , w ) > UB − z LB Effort: O ( 3 · | A t ex | ) , i.e., same as pricing 21 / 33

  72. References Solution of the Pricing Problem Variable Elimination ( → Network Reduction) (Cont.): Smallest rdc of tours through an arbitrary arc a = ( v , w ) ∈ A t ex can be computed with a bidirectional SPP approach: 1 solve full fw SPP; cost labels ˜ c fw ( v ) at nodes v 2 solve full bw SPP; cost labels ˜ c bw ( v ) at nodes v 3 value rdc ( v , w ) := ˜ c fw ( v ) + ˜ c bw ( w ) is minimum rdc of c vw + ˜ paths through a = ( v , w ) 4 Rule: Eliminate a = ( v , w ) if rdc ( v , w ) > UB − z LB Effort: O ( 3 · | A t ex | ) , i.e., same as pricing 21 / 33

  73. References Solution of the Pricing Problem Variable Elimination ( → Network Reduction) (Cont.): Smallest rdc of tours through an arbitrary arc a = ( v , w ) ∈ A t ex can be computed with a bidirectional SPP approach: 1 solve full fw SPP; cost labels ˜ c fw ( v ) at nodes v 2 solve full bw SPP; cost labels ˜ c bw ( v ) at nodes v 3 value rdc ( v , w ) := ˜ c fw ( v ) + ˜ c bw ( w ) is minimum rdc of c vw + ˜ paths through a = ( v , w ) 4 Rule: Eliminate a = ( v , w ) if rdc ( v , w ) > UB − z LB Effort: O ( 3 · | A t ex | ) , i.e., same as pricing 21 / 33

  74. References Solution of the Pricing Problem Variable Elimination ( → Network Reduction) (Cont.): Smallest rdc of tours through an arbitrary arc a = ( v , w ) ∈ A t ex can be computed with a bidirectional SPP approach: 1 solve full fw SPP; cost labels ˜ c fw ( v ) at nodes v 2 solve full bw SPP; cost labels ˜ c bw ( v ) at nodes v 3 value rdc ( v , w ) := ˜ c fw ( v ) + ˜ c bw ( w ) is minimum rdc of c vw + ˜ paths through a = ( v , w ) 4 Rule: Eliminate a = ( v , w ) if rdc ( v , w ) > UB − z LB Effort: O ( 3 · | A t ex | ) , i.e., same as pricing 21 / 33

  75. References Solution of the Pricing Problem Variable Elimination ( → Network Reduction) (Cont.): Smallest rdc of tours through an arbitrary arc a = ( v , w ) ∈ A t ex can be computed with a bidirectional SPP approach: 1 solve full fw SPP; cost labels ˜ c fw ( v ) at nodes v 2 solve full bw SPP; cost labels ˜ c bw ( v ) at nodes v 3 value rdc ( v , w ) := ˜ c fw ( v ) + ˜ c bw ( w ) is minimum rdc of c vw + ˜ paths through a = ( v , w ) 4 Rule: Eliminate a = ( v , w ) if rdc ( v , w ) > UB − z LB Effort: O ( 3 · | A t ex | ) , i.e., same as pricing 21 / 33

  76. References Solution of the Pricing Problem Variable Elimination ( → Network Reduction) (Cont.): Smallest rdc of tours through an arbitrary arc a = ( v , w ) ∈ A t ex can be computed with a bidirectional SPP approach: 1 solve full fw SPP; cost labels ˜ c fw ( v ) at nodes v 2 solve full bw SPP; cost labels ˜ c bw ( v ) at nodes v 3 value rdc ( v , w ) := ˜ c fw ( v ) + ˜ c bw ( w ) is minimum rdc of c vw + ˜ paths through a = ( v , w ) 4 Rule: Eliminate a = ( v , w ) if rdc ( v , w ) > UB − z LB Effort: O ( 3 · | A t ex | ) , i.e., same as pricing 21 / 33

  77. References Solution of the Pricing Problem Variable Elimination ( → Network Reduction): Instance Bounds before reduction avg. after reduction LB UB | V ex | | A ex | | V ex | | A ex | % | A ex | remaining NL4[-NRC] 8160 8276 92 198 72 119 60% NL6 22557 23552 1138 3596 763 1727 48% NL6-NRC 22557 23916 1138 3596 1056 2821 78% NL8 38670 39480 8548 32108 5618 13200 41% NL8-NRC 38670 39721 8548 32108 7470 20016 62% NL10[-NRC] 56659 59436 55448 245910 55442 245561 99% 22 / 33

  78. References Branching Easton et al. (2003) suggest branching on home/away decisions for team t and time slot s : team plays home in slot 3 team plays away in slot 3 Remarks: branching rule is incomplete! must be supplemented with complete branching rule however: most effective branching rule in practice 23 / 33

  79. References Branching Easton et al. (2003) suggest branching on home/away decisions for team t and time slot s : team plays home in slot 3 team plays away in slot 3 Remarks: branching rule is incomplete! must be supplemented with complete branching rule however: most effective branching rule in practice 23 / 33

  80. References Branching Easton et al. (2003) suggest branching on home/away decisions for team t and time slot s : team plays home in slot 3 team plays away in slot 3 Remarks: branching rule is incomplete! must be supplemented with complete branching rule however: most effective branching rule in practice 23 / 33

  81. References Branching Easton et al. (2003) suggest branching on home/away decisions for team t and time slot s : team plays home in slot 3 team plays away in slot 3 Remarks: branching rule is incomplete! must be supplemented with complete branching rule however: most effective branching rule in practice 23 / 33

  82. References Branching Branching on nodes of the (non-expanded) network arcs of the (non-expanded) network is complete, i.e., guarantees integer solutions! Strategy: Branching on home/away decisions first, branching on nodes second (if all home/away decision are made) Strong branching is essential, since B&B-tree get extremly large: consider a number of candidate branching decisions completely evaluate corresponding son nodes ( → full CG) choose branching variable, where weaker son gives largest LB How many candidates? The deeper a node in the tree, the less candidates should be considered! � UB − z LB ( node ) � γ #Candidates := ⌊ 1 + MaxCand · ⌋ UB − z LB ( root ) good results for γ ∈ [ 1 . 5 ; 2 ] and MaxCand ∈ [ 10 , 200 ] 24 / 33

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