May 2004 Minimizing travels by maximizing breaks1/32 Minimizing Travels by Maximizing Breaks in Round Robin Tournament Schedules Celso RIBEIRO UFF and PUC-Rio, Brazil Sebastián URRUTIA PUC-Rio, Brazil

May 2004 Minimizing travels by maximizing breaks2/32 Summary Motivation Tournament schedules and the traveling tournament problem Connecting breaks with distances Maximum number of breaks for SRR tournaments Polygon method Maximum number of breaks for TTP- constrained MDRR tournaments Numerical results Concluding remarks

May 2004 Minimizing travels by maximizing breaks3/32 Motivation Motivation for this work: –Context: research group on applications of OR techniques to problems in sports management and scheduling – Effective algorithms for the Traveling Tournament Problem: the total distance traveled is an important variable to be minimized in tournament scheduling, to reduce traveling costs and to give more time to the players for resting and training. – Real life application: finding a good schedule to the Brazilian national soccer championship (26 teams)

May 2004 Minimizing travels by maximizing breaks4/32 Tournament schedules Conditions: –n (even) teams take part in a tournament. –Each team has its own stadium at its home city. –Each team is located at its home city in the beginning, to where it returns at the end. –Distances between the stadiums are known. –A team playing two consecutive away games goes directly from one city to the other, without returning to its home city.

May 2004 Minimizing travels by maximizing breaks5/32 Tournament schedules Conditions (cont’d): –Single round-robin tournament (SRR): Each team plays every other team exactly once in n-1 prescheduled rounds. –Double round-robin tournament (DRR): Each team plays every other team exactly twice in 2(n-1) prescheduled rounds (each of them with exactly n/2 games), once at home and once away.

May 2004 Minimizing travels by maximizing breaks6/32 Tournament schedules Conditions (cont’d): –Mirrored double round-robin tournament (MDRR): Each team plays every other team exactly twice in 2(n-1) prescheduled rounds (each of them with exactly n/2 games), once at home and once away. MDRR is a SRR tournament in the first (n-1) rounds, followed by the same SRR tournament with reversed venues in the last (n-1) rounds. –A tournament schedule determines at which round and in which stadium each game takes place.

May 2004 Minimizing travels by maximizing breaks7/32 Tournament schedules Home-away pattern (HAP): –Matrix with as many rows as teams (n) and as many columns as rounds in the tournament. –Each row of a HAP is a sequence of H’s and A’s. –An H (resp. A) in position r of row t means that team t has a home (resp. an away) game in round r. –A team has a break in round r if it has two consecutive home (or away) games in rounds r-1 and r.

May 2004 Minimizing travels by maximizing breaks8/32 Tournament schedules Schedule S: –B(S) = total number of breaks (sum of the number of breaks over all teams in the tournament) –There are no two equal rows in a HAP (every two teams have to play against each other at some round) –Number of home breaks = number of away breaks = B(S)/2 –D(S) = total distance traveled (sum of the distances traveled by all teams in the tournament) –T(S) = total number of travels (number of times any team must travel from one stadium to another)

May 2004 Minimizing travels by maximizing breaks9/32 Tournament schedules Breaks minimization problems: –Schedules with a minimum number of breaks De Werra (1981,1988): constraints on geographical locations (complementary HAPs for teams in the same location, e.g. Mets and Yankees in NY), teams organized in divisions (weekday vs. weekend games), minimize the number of rounds with breaks –Minimize breaks when the order of games is fixed Elf, Junger & Rinaldi (2003)

May 2004 Minimizing travels by maximizing breaks10/32 – Hard problem: previous largest instance exactly solved to date had only n=6 teams! (n=8 with 20 processors in 4 days CPU time) Tournament schedules Distance minimization problems: –NHL schedule: minimize the total distance traveled (evolutionary tabu search) - Costa (1995) –Traveling tournament problem: minimize the total distance traveled, such that no team plays more than three consecutive away games or three consecutive home games - Easton, Nemhauser & Trick (2001,2004) –Mirrored TTP: Ribeiro & Urrutia (2004) complexity? Open!

May 2004 Minimizing travels by maximizing breaks11/32 Tournament schedules In this work: –Connection between breaks and distance problems –New class of instances for which distance minimization is equivalent to breaks maximization –Construction of schedules with maximum number of breaks and minimum distance traveled –Mirrored DRR schedules satisfying TTP contraints –Solution of larger TTP instances

May 2004 Minimizing travels by maximizing breaks12/32 Tournament schedules Variants: –no-repeaters –no synchronized rounds –multiple games (more than two, variable) –teams with complementary patterns in the same city –pre-scheduled games and TV constraints –stadium availability –minimize airfare and hotel costs, etc.

May 2004 Minimizing travels by maximizing breaks13/32 Connecting breaks with distances Benchmark instances for distance minimization problems: –Structured circular instances with n = 4 to 20 teams –MLB instances with n = 4 to 16 teams –All available from Michael Trick’s web page –2003 edition of the Brazilian national soccer championship with 24 teams

May 2004 Minimizing travels by maximizing breaks14/32 discounted by the number of teams that do not travel (home breaks) Connecting breaks with distances New uniform instances: all distances equal to one D(S) = T(S) R = number of rounds T(S) = n/2 + n(R-1) – B(S)/2 + n/2 = nR – B(S)/2 travels to play in intermediary rounds if all teams were to travel, travels after playing the last gametravels to play the first game

May 2004 Minimizing travels by maximizing breaks15/32 Connecting breaks with distances In the particular case of a uniform instance: D(S) = T(S) Then, D(S) = nR – B(S)/2 maximize breaks => minimize travels => => minimize distance traveled for uniform instances Motivation: UB to breaks gives LB to distance Consequence: implications in the solution of the TTP

May 2004 Minimizing travels by maximizing breaks16/32 Max breaks for SRR tournaments SRR tournaments: maximum number of breaks for any team is (n-2): all home games or all away games Only two teams may have (n-2) breaks: all games away and all games at home Remaining (n-2) teams: at most (n-3) breaks each Upper bound to the number of breaks: UB SRR = 2(n-2) + (n-2)(n-3) = n 2 – 3n + 2

May 2004 Minimizing travels by maximizing breaks17/32 Polygon method Upper bound to the number of breaks: UB SRR = 2(n-2) + (n-2)(n-3) = n 2 – 3n + 2 UB SRR bound is tight. We use the polygon (or circle) method to build a schedule with exactly UB SRR breaks. Phase 1: assign games to rounds –Graph with one edge for each game at each round

May 2004 Minimizing travels by maximizing breaks18/32 Polygon method Example: “polygon method” for n=6 1 st round Phase 1: game assignment

May 2004 Minimizing travels by maximizing breaks19/32 Polygon method Example: “polygon method” for n=6 2 nd round Phase 1: game assignment

May 2004 Minimizing travels by maximizing breaks20/32 Polygon method Example: “polygon method” for n=6 3 rd round Phase 1: game assignment

May 2004 Minimizing travels by maximizing breaks21/32 Polygon method Example: “polygon method” for n=6 4 th round Phase 1: game assignment

May 2004 Minimizing travels by maximizing breaks22/32 Polygon method Example: “polygon method” for n=6 5 th round Phase 1: game assignment

May 2004 Minimizing travels by maximizing breaks23/32 Polygon method Phase 2: extension of the polygon method an orientation to each edge (oriented edge coloring) Edge connecting nodes 1 and n is always oriented from 1 to n (in every round) k=2,...,n/2: the edge connecting nodes k and n+1-k is oriented from the even (resp. odd) numbered node to the odd (resp. even) numbered node in odd (resp. even) rounds Final extremity of each arc is the home team.

May 2004 Minimizing travels by maximizing breaks24/32 Polygon method Phase 2: stadium assignment

May 2004 Minimizing travels by maximizing breaks25/32 Max breaks for TTP- constrained MDRR tournaments Similar tight bounds can also be obtained for equilibrated SRR, DRR, and MDRR tournaments. Mirrored DRR tournaments in which each schedule must follow the same constraints of the traveling tournament problem: –No team can play more than three consecutive home games or more than three consecutive away games.

May 2004 Minimizing travels by maximizing breaks26/32 Upper bounds to the number of breaks can be derived using similar (although much more elaborated) counting arguments: Max breaks for TTP- constrained MDRR tournaments

May 2004 Minimizing travels by maximizing breaks27/32 Since T(S) = 2n(n-1) – B(S)/2, the upper bound UB TTP can be used in the computation of lower bounds to T(S) and, for the uniform instances, also to D(S) = T(S). Contrarily to the previous problems, a construction method to build schedules for TTP-constrained MDRR tournaments with exactly UB TTP breaks does not seem to exist to date. Max breaks for TTP- constrained MDRR tournaments

May 2004 Minimizing travels by maximizing breaks28/32 Use an effective TTP heuristic to find good approximate solutions (10 minutes): –Ribeiro & Urrutia (2004): better solutions in 10 minutes of CPU for benchmark instances than Anagnostopoulos, Michel, Van Hentenryck & Vergados (2003) in 5 days (similar machine); also best known solutions to circ18 and circ GHz Pentium IV with 512 Mb RAM memory Uniform instances with n = 4, 6, 8,..., 18, 20 Max breaks for TTP- constrained MDRR tournaments

May 2004 Minimizing travels by maximizing breaks29/32 Max breaks for TTP- constrained MDRR tournaments nD(S)LBgapB(S)

May 2004 Minimizing travels by maximizing breaks30/32 Concluding remarks New class of uniform instances Connection between breaks maximization and distance minimization problems This connection is used to prove the optimality of approximate solutions found by an effective heuristic for the TTP. New largest TTP instance exactly solved to date: n=16

May 2004 Minimizing travels by maximizing breaks31/32 Concluding remarks In spite of being easier than other classes of TTP instances, uniform instances could not be exactly solved for n > 16. Complexity results for this new class will possibly shed some light on the complexity of the traveling tournament problem.

May 2004 Minimizing travels by maximizing breaks32/32 Concluding remarks Total distance traveled for the 2003 edition of the Brazilian soccer championship with 24 teams (instance br24) in 12 hours (Pentium IV 2.0 MHz): Realized (official draw): 1,048,134 kms Our solution: 506,433 kms (52% reduction) Approximate corresponding potential savings in airfares: US$ 1,700,000