A new Branch-and-Price Algorithm for the Traveling Tournament Problem (TTP) Column Generation 2008, Aussois, France

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1 A new Branch-and-Price Algorithm for the Traveling Tournament Problem (TTP) Column Generation 2008, Aussois, France Stefan Irnich 1 RWTH Aachen University Deutsche Post Endowed Chair of Optimization of Distribution Networks June 18th, joint work with Ulrich Schrempp 1 / 33

2 Round Robin Sports League Scheduling Sports League Scheduling: in many sports leagues the teams play a double round robin tournament: even number n of teams every team plays against each other team twice; once home and once away each team has n := n 1 opponent teams, i.e., each team plays 2 n games schedule is tight: 2 n time slots, n 2 games in each slot 2 / 33

3 Round Robin Sports League Scheduling Sports League Scheduling: in many sports leagues the teams play a double round robin tournament: even number n of teams every team plays against each other team twice; once home and once away each team has n := n 1 opponent teams, i.e., each team plays 2 n games schedule is tight: 2 n time slots, n 2 games in each slot 2 / 33

4 Round Robin Sports League Scheduling Sports League Scheduling: in many sports leagues the teams play a double round robin tournament: even number n of teams every team plays against each other team twice; once home and once away each team has n := n 1 opponent teams, i.e., each team plays 2 n games schedule is tight: 2 n time slots, n 2 games in each slot 2 / 33

5 Round Robin Sports League Scheduling Sports League Scheduling: in many sports leagues the teams play a double round robin tournament: even number n of teams every team plays against each other team twice; once home and once away each team has n := n 1 opponent teams, i.e., each team plays 2 n games schedule is tight: 2 n time slots, n 2 games in each slot 2 / 33

6 Round Robin Sports League Scheduling Sports League Scheduling: in many sports leagues the teams play a double round robin tournament: even number n of teams every team plays against each other team twice; once home and once away each team has n := n 1 opponent teams, i.e., each team plays 2 n games schedule is tight: 2 n time slots, n 2 games in each slot 2 / 33

7 Round Robin Sports League Scheduling Example: n = 4 teams n = 3 opponents 2 n = 6 time slots, each with n 2 = 2 games A feasible solution: time slot team 1: team 2: team 3: team 4: home (H) away (A) 3 / 33

8 Round Robin Sports League Scheduling Example: n = 4 teams n = 3 opponents 2 n = 6 time slots, each with n 2 = 2 games A feasible solution: time slot team 1: team 2: team 3: team 4: home (H) away (A) 3 / 33

9 Round Robin Sports League Scheduling In some sports leagues teams travel directly from one away game to the next away game! In North-America: baseball, Basketball, Ice-Hockey etc. Example (Cont.): Teams always start at home and return home at the end! Distances traveled: team 1 : d 12 + d 23 + d 34 + d 41 team 2 : d 21 + d 13 + d 32 + d 24 + d 42 team 3 : d 31 + d 13 + d 34 + d 43 + d 32 + d 23 team 4 : d 42 + d 23 + d 31 + d 14 4 / 33

10 Round Robin Sports League Scheduling In some sports leagues teams travel directly from one away game to the next away game! In North-America: baseball, Basketball, Ice-Hockey etc. Example (Cont.): Teams always start at home and return home at the end! Distances traveled: team 1 : d 12 + d 23 + d 34 + d 41 team 2 : d 21 + d 13 + d 32 + d 24 + d 42 team 3 : d 31 + d 13 + d 34 + d 43 + d 32 + d 23 team 4 : d 42 + d 23 + d 31 + d 14 4 / 33

11 Round Robin Sports League Scheduling In some sports leagues teams travel directly from one away game to the next away game! In North-America: baseball, Basketball, Ice-Hockey etc. Example (Cont.): time slot team 1: team 2: team 3: team 4: home (H) away (A) Teams always start at home and return home at the end! Distances traveled: team 1 : d 12 + d 23 + d 34 + d 41 team 2 : d 21 + d 13 + d 32 + d 24 + d 42 team 3 : d 31 + d 13 + d 34 + d 43 + d 32 + d 23 team 4 : d 42 + d 23 + d 31 + d 14 4 / 33

12 Round Robin Sports League Scheduling In some sports leagues teams travel directly from one away game to the next away game! In North-America: baseball, Basketball, Ice-Hockey etc. Example (Cont.): time slot team 1: H H H team 2: H 1 3 H 4 H team 3: 1 H H 4 H 2 team 4: H H H + home (H) away (A) Teams always start at home and return home at the end! Distances traveled: team 1 : d 12 + d 23 + d 34 + d 41 team 2 : d 21 + d 13 + d 32 + d 24 + d 42 team 3 : d 31 + d 13 + d 34 + d 43 + d 32 + d 23 team 4 : d 42 + d 23 + d 31 + d 14 4 / 33

13 Round Robin Sports League Scheduling In some sports leagues teams travel directly from one away game to the next away game! In North-America: baseball, Basketball, Ice-Hockey etc. Example (Cont.): time slot team 1: H H H team 2: H 1 3 H 4 H team 3: 1 H H 4 H 2 team 4: H H H + home (H) away (A) Teams always start at home and return home at the end! Distances traveled: team 1 : d 12 + d 23 + d 34 + d 41 team 2 : d 21 + d 13 + d 32 + d 24 + d 42 team 3 : d 31 + d 13 + d 34 + d 43 + d 32 + d 23 team 4 : d 42 + d 23 + d 31 + d 14 4 / 33

14 Round Robin Sports League Scheduling In some sports leagues teams travel directly from one away game to the next away game! In North-America: baseball, Basketball, Ice-Hockey etc. Example (Cont.): time slot team 1: H H H team 2: H 1 3 H 4 H team 3: 1 H H 4 H 2 team 4: H H H + home (H) away (A) Teams always start at home and return home at the end! Distances traveled: team 1 : d 12 + d 23 + d 34 + d 41 team 2 : d 21 + d 13 + d 32 + d 24 + d 42 team 3 : d 31 + d 13 + d 34 + d 43 + d 32 + d 23 team 4 : d 42 + d 23 + d 31 + d 14 4 / 33

15 Round Robin Sports League Scheduling In some sports leagues teams travel directly from one away game to the next away game! In North-America: baseball, Basketball, Ice-Hockey etc. Example (Cont.): time slot team 1: H H H team 2: H 1 3 H 4 H team 3: 1 H H 4 H 2 team 4: H H H + home (H) away (A) Teams always start at home and return home at the end! Distances traveled: team 1 : d 12 + d 23 + d 34 + d 41 team 2 : d 21 + d 13 + d 32 + d 24 + d 42 team 3 : d 31 + d 13 + d 34 + d 43 + d 32 + d 23 team 4 : d 42 + d 23 + d 31 + d 14 4 / 33

16 Round Robin Sports League Scheduling In some sports leagues teams travel directly from one away game to the next away game! In North-America: baseball, Basketball, Ice-Hockey etc. Example (Cont.): time slot team 1: H H H team 2: H 1 3 H 4 H team 3: 1 H H 4 H 2 team 4: H H H + home (H) away (A) Teams always start at home and return home at the end! Distances traveled: team 1 : d 12 + d 23 + d 34 + d 41 team 2 : d 21 + d 13 + d 32 + d 24 + d 42 team 3 : d 31 + d 13 + d 34 + d 43 + d 32 + d 23 team 4 : d 42 + d 23 + d 31 + d 14 4 / 33

17 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. Break minimization no additional constraints: constructive methods timetables with n 2 breaks (SRR), 3n 6 (MDRR) etc. in general: minimization of breaks, s.t. additional constraints approaches: heuristics, IP or combined IP/CP (Nemhauser and Trick, 1998; Henz, 2001) Distance minimization objective: minimize distance, i.e., function separable by teams constraints: number of (consecutive) breaks approaches: heuristics, combined IP and CP (Easton et al., 2003) 5 / 33

18 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. Break minimization no additional constraints: constructive methods timetables with n 2 breaks (SRR), 3n 6 (MDRR) etc. in general: minimization of breaks, s.t. additional constraints approaches: heuristics, IP or combined IP/CP (Nemhauser and Trick, 1998; Henz, 2001) Distance minimization objective: minimize distance, i.e., function separable by teams constraints: number of (consecutive) breaks approaches: heuristics, combined IP and CP (Easton et al., 2003) 5 / 33

19 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. Break minimization no additional constraints: constructive methods timetables with n 2 breaks (SRR), 3n 6 (MDRR) etc. in general: minimization of breaks, s.t. additional constraints approaches: heuristics, IP or combined IP/CP (Nemhauser and Trick, 1998; Henz, 2001) Distance minimization objective: minimize distance, i.e., function separable by teams constraints: number of (consecutive) breaks approaches: heuristics, combined IP and CP (Easton et al., 2003) 5 / 33

20 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. Break minimization no additional constraints: constructive methods timetables with n 2 breaks (SRR), 3n 6 (MDRR) etc. in general: minimization of breaks, s.t. additional constraints approaches: heuristics, IP or combined IP/CP (Nemhauser and Trick, 1998; Henz, 2001) Distance minimization objective: minimize distance, i.e., function separable by teams constraints: number of (consecutive) breaks approaches: heuristics, combined IP and CP (Easton et al., 2003) 5 / 33

21 What is the TTP? References 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

22 What is the TTP? References 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

23 What is the TTP? References 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

24 What is the TTP? References 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

25 What is the TTP? References 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

26 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 A t s,t = {tour of team t with away game against t in slot s} min s.t. t T c pλ t p p P t + t p λ p P t : t :p A t s,t } {{ } team t plays away t T : t t p P t : p A t st λ t p } {{ } team t t plays away against t = 1 t T s {1,..., 2 n} p P t λ t p = 1 t T (2) λ t p {0, 1} t T, p P t (3) (1) 7 / 33

27 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 A t s,t = {tour of team t with away game against t in slot s} min s.t. t T c pλ t p p P t + t p λ p P t : t :p A t s,t } {{ } team t plays away t T : t t p P t : p A t st λ t p } {{ } team t t plays away against t = 1 t T s {1,..., 2 n} p P t λ t p = 1 t T (2) λ t p {0, 1} t T, p P t (3) (1) 7 / 33

28 Hardness of the TTP References 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 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 N P -hard! Design of good heuristic methods is a non-trivial task, see, e.g., (Van Hentenryck and Vergados, 2006) 8 / 33

29 Hardness of the TTP References 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 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 N P -hard! Design of good heuristic methods is a non-trivial task, see, e.g., (Van Hentenryck and Vergados, 2006) 8 / 33

30 Hardness of the TTP References 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 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 N P -hard! Design of good heuristic methods is a non-trivial task, see, e.g., (Van Hentenryck and Vergados, 2006) 8 / 33

31 Compact Formulation References Easton et al. (2003) did not provide a compact formulation Compact formulation with binary variables x t ijs, where x t = (xijs t ) represents the movement of team t xijs t = 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 / 33

32 Compact Formulation References Easton et al. (2003) did not provide a compact formulation Compact formulation with binary variables x t ijs, where x t = (xijs t ) represents the movement of team t xijs t = 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 / 33

33 Compact Formulation References Easton et al. (2003) did not provide a compact formulation Compact formulation with binary variables x t ijs, where x t = (xijs t ) represents the movement of team t xijs t = 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 / 33

34 Compact Formulation References Easton et al. (2003) did not provide a compact formulation Compact formulation with binary variables x t ijs, where x t = (xijs t ) represents the movement of team t xijs t = 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 / 33

35 Compact Formulation References Easton et al. (2003) did not provide a compact formulation Compact formulation with binary variables x t ijs, where x t = (xijs t ) represents the movement of team t xijs t = 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 / 33

36 Compact Formulation References Easton et al. (2003) did not provide a compact formulation Compact formulation with binary variables x t ijs, where x t = (xijs t ) represents the movement of team t xijs t = 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 / 33

37 Compact Formulation References Easton et al. (2003) did not provide a compact formulation Compact formulation with binary variables x t ijs, where x t = (xijs t ) represents the movement of team t xijs t = 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 / 33

38 Compact Formulation References Easton et al. (2003) did not provide a compact formulation Compact formulation with binary variables x t ijs, where x t = (xijs t ) represents the movement of team t xijs t = 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 / 33

39 Compact Formulation References Interpretation of decision variables as arcs of a layered network (V, A) nodes vis t V team t is at venue i at time s i-s artificial node vh0 t ( source ) and vh,2 n+1 t ( sink ) arcs (vis, t vj,s+1) t A corresponding to decision variables xijs t Network for team t = 4: 10 / 33

40 min s.t. Explanation: 2 n t T i T j T s=0 2 n s=1 j T d ij x t ijs xijs t = 1 t, i T : i t (1) xijs t xji,s+1 t = 0 t, j T, s {0,.., 2 n 1} (2) i T i T U 1 xtt,s+u t U 1 u=0 i T : j T i t x t ijs } {{ } team t plays away + and t T : j T t t U 1 xij,s+u t U 1 u=0 i t j t t T, s {1,.., 2 n U} (3) t T xtjs t = 1 s {1,.., 2 n} (4) } {{ } team t t plays away against t x t ijs {0, 1} t, i, j T, s {0,.., 2 n} (5) (1) each team t must play once at every opponent s venue 11 / 33

41 min s.t. Explanation: 2 n t T i T j T s=0 2 n s=1 j T d ij x t ijs xijs t = 1 t, i T : i t (1) xijs t xji,s+1 t = 0 t, j T, s {0,.., 2 n 1} (2) i T i T U 1 xtt,s+u t U 1 u=0 i T : j T i t x t ijs } {{ } team t plays away (2) flow conservation + and t T : j T t t U 1 xij,s+u t U 1 u=0 i t j t t T, s {1,.., 2 n U} (3) t T xtjs t = 1 s {1,.., 2 n} (4) } {{ } team t t plays away against t x t ijs {0, 1} t, i, j T, s {0,.., 2 n} (5) 11 / 33

42 min s.t. Explanation: 2 n t T i T j T s=0 2 n s=1 j T d ij x t ijs xijs t = 1 t, i T : i t (1) xijs t xji,s+1 t = 0 t, j T, s {0,.., 2 n 1} (2) i T i T U 1 xtt,s+u t U 1 u=0 i T : j T i t x t ijs } {{ } team t plays away + and t T : j T t t U 1 xij,s+u t U 1 u=0 i t j t t T, s {1,.., 2 n U} (3) t T xtjs t = 1 s {1,.., 2 n} (4) } {{ } team t t plays away against t x t ijs {0, 1} t, i, j T, s {0,.., 2 n} (5) (3) upper bound on number of consecutive home/away games 11 / 33

43 min s.t. Explanation: 2 n t T i T j T s=0 2 n s=1 j T d ij x t ijs xijs t = 1 t, i T : i t (1) xijs t xji,s+1 t = 0 t, j T, s {0,.., 2 n 1} (2) i T i T U 1 xtt,s+u t U 1 u=0 i T : j T i t x t ijs } {{ } team t plays away + and t T : j T t t U 1 xij,s+u t U 1 u=0 i t j t t T, s {1,.., 2 n U} (3) t T xtjs t = 1 s {1,.., 2 n} (4) } {{ } team t t plays away against t x t ijs {0, 1} t, i, j T, s {0,.., 2 n} (5) (4) coupling constraints: each team t must play in every time slot s 11 / 33

44 Compact Formulation References 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

45 Compact Formulation References 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

46 Compact Formulation References Optional No-Repeater Constraints (NRC) can be easily added: x t tt s + x t tt s 1 t, t T, t t, s {1,.., 2 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

47 Compact Formulation References Optional No-Repeater Constraints (NRC) can be easily added: x t tt s + x t tt s 1 t, t T, t t, s {1,.., 2 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

48 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

49 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

50 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

51 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): 1 (s,, 0) (initial source node s) 2 (3,, 0) 3 (H, {3}, 0) 4 (H, {3}, 1) 5 (H, {3}, 2) next game must be away 6 (1, {3}, 0) 7 (2, {1, 3}, 1) 8 (d, {1, 2, 3}, 0) (destination node d) 15 / 33

52 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): 1 (s,, 0) (initial source node s) 2 (3,, 0) 3 (H, {3}, 0) 4 (H, {3}, 1) 5 (H, {3}, 2) next game must be away 6 (1, {3}, 0) 7 (2, {1, 3}, 1) 8 (d, {1, 2, 3}, 0) (destination node d) 15 / 33

53 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): 1 (s,, 0) (initial source node s) 2 (3,, 0) 3 (H, {3}, 0) 4 (H, {3}, 1) 5 (H, {3}, 2) next game must be away 6 (1, {3}, 0) 7 (2, {1, 3}, 1) 8 (d, {1, 2, 3}, 0) (destination node d) 15 / 33

54 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): 1 (s,, 0) (initial source node s) 2 (3,, 0) 3 (H, {3}, 0) 4 (H, {3}, 1) 5 (H, {3}, 2) next game must be away 6 (1, {3}, 0) 7 (2, {1, 3}, 1) 8 (d, {1, 2, 3}, 0) (destination node d) 15 / 33

55 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): 1 (s,, 0) (initial source node s) 2 (3,, 0) 3 (H, {3}, 0) 4 (H, {3}, 1) 5 (H, {3}, 2) next game must be away 6 (1, {3}, 0) 7 (2, {1, 3}, 1) 8 (d, {1, 2, 3}, 0) (destination node d) 15 / 33

56 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): 1 (s,, 0) (initial source node s) 2 (3,, 0) 3 (H, {3}, 0) 4 (H, {3}, 1) 5 (H, {3}, 2) next game must be away 6 (1, {3}, 0) 7 (2, {1, 3}, 1) 8 (d, {1, 2, 3}, 0) (destination node d) 15 / 33

57 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): 1 (s,, 0) (initial source node s) 2 (3,, 0) 3 (H, {3}, 0) 4 (H, {3}, 1) 5 (H, {3}, 2) next game must be away 6 (1, {3}, 0) 7 (2, {1, 3}, 1) 8 (d, {1, 2, 3}, 0) (destination node d) 15 / 33

58 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): 1 (s,, 0) (initial source node s) 2 (3,, 0) 3 (H, {3}, 0) 4 (H, {3}, 1) 5 (H, {3}, 2) next game must be away 6 (1, {3}, 0) 7 (2, {1, 3}, 1) 8 (d, {1, 2, 3}, 0) (destination node d) 15 / 33

59 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): 1 (s,, 0) (initial source node s) 2 (3,, 0) 3 (H, {3}, 0) 4 (H, {3}, 1) 5 (H, {3}, 2) next game must be away 6 (1, {3}, 0) 7 (2, {1, 3}, 1) 8 (d, {1, 2, 3}, 0) (destination node d) 15 / 33

60 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): 1 (s,, 0) (initial source node s) 2 (3,, 0) 3 (H, {3}, 0) 4 (H, {3}, 1) 5 (H, {3}, 2) next game must be away 6 (1, {3}, 0) 7 (2, {1, 3}, 1) 8 (d, {1, 2, 3}, 0) (destination node d) 15 / 33

61 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): 1 (s,, 0) (initial source node s) 2 (3,, 0) 3 (H, {3}, 0) 4 (H, {3}, 1) 5 (H, {3}, 2) next game must be away 6 (1, {3}, 0) 7 (2, {1, 3}, 1) 8 (d, {1, 2, 3}, 0) (destination node d) 15 / 33

62 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): 1 (s,, 0) (initial source node s) 2 (3,, 0) 3 (H, {3}, 0) 4 (H, {3}, 1) 5 (H, {3}, 2) next game must be away 6 (1, {3}, 0) 7 (2, {1, 3}, 1) 8 (d, {1, 2, 3}, 0) (destination node d) 15 / 33

63 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): 1 (s,, 0) (initial source node s) 2 (3,, 0) 3 (H, {3}, 0) 4 (H, {3}, 1) 5 (H, {3}, 2) next game must be away 6 (1, {3}, 0) 7 (2, {1, 3}, 1) 8 (d, {1, 2, 3}, 0) (destination node d) 15 / 33

64 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): 1 (s,, 0) (initial source node s) 2 (3,, 0) 3 (H, {3}, 0) 4 (H, {3}, 1) 5 (H, {3}, 2) next game must be away 6 (1, {3}, 0) 7 (2, {1, 3}, 1) 8 (d, {1, 2, 3}, 0) (destination node d) 15 / 33

65 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

66 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

67 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

68 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

69 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

70 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

71 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

72 Solution of the Pricing Problem Expanded Network for n = 4: 17 / 33

73 Solution of the Pricing Problem Size of the expanded network (V ex, A ex ): Here for U = 3! #teams #nodes #arcs #tours Factor n V ex A ex #tours/#arcs , ,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

74 Solution of the Pricing Problem Size of the expanded network (V ex, A ex ): Here for U = 3! #teams #nodes #arcs #tours Factor n V ex A ex #tours/#arcs , ,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

75 Solution of the Pricing Problem Size of the expanded network (V ex, A ex ): Here for U = 3! #teams #nodes #arcs #tours Factor n V ex A ex #tours/#arcs , ,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

76 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: 1 SPP forward to create rdc labels at slots 1,..., n 2 SPP backward to create rdc labels at slot 2 n,..., n merge step: combine fw and bw labels from slots n and n + 1 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

77 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: 1 SPP forward to create rdc labels at slots 1,..., n 2 SPP backward to create rdc labels at slot 2 n,..., n merge step: combine fw and bw labels from slots n and n + 1 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

78 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: 1 SPP forward to create rdc labels at slots 1,..., n 2 SPP backward to create rdc labels at slot 2 n,..., n merge step: combine fw and bw labels from slots n and n + 1 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

79 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: 1 SPP forward to create rdc labels at slots 1,..., n 2 SPP backward to create rdc labels at slot 2 n,..., n merge step: combine fw and bw labels from slots n and n + 1 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

80 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: 1 SPP forward to create rdc labels at slots 1,..., n 2 SPP backward to create rdc labels at slot 2 n,..., n merge step: combine fw and bw labels from slots n and n + 1 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

81 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: 1 SPP forward to create rdc labels at slots 1,..., n 2 SPP backward to create rdc labels at slot 2 n,..., n merge step: combine fw and bw labels from slots n and n + 1 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

82 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: 1 SPP forward to create rdc labels at slots 1,..., n 2 SPP backward to create rdc labels at slot 2 n,..., n merge step: combine fw and bw labels from slots n and n + 1 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

83 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 π for TTP 3 and an upper bound UB on TTP Proposition A: If the rdc c t p(π) of a tour variable λ t 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

84 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 π for TTP 3 and an upper bound UB on TTP Proposition A: If the rdc c t p(π) of a tour variable λ t 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

85 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 π for TTP 3 and an upper bound UB on TTP Proposition A: If the rdc c t p(π) of a tour variable λ t 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

86 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 π for TTP 3 and an upper bound UB on TTP Proposition A: If the rdc c t p(π) of a tour variable λ t 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

87 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 π for TTP 3 and an upper bound UB on TTP Proposition A: If the rdc c t p(π) of a tour variable λ t 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

88 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 π for TTP 3 and an upper bound UB on TTP Proposition A: If the rdc c t p(π) of a tour variable λ t 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

89 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 vw + c bw (w) is minimum rdc of 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

90 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 vw + c bw (w) is minimum rdc of 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

91 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 vw + c bw (w) is minimum rdc of 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

92 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 vw + c bw (w) is minimum rdc of 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

93 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 vw + c bw (w) is minimum rdc of 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

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