Noname manuscript No.(will be inserted by the editor)

An ILS heuristic for the traveling tournament problem withpredefined venues

Fabrıcio N. Costa · Sebastian Urrutia ·Celso C. Ribeiro

Received: February 29, 2008

Abstract The Traveling Tournament Problem with Predefined Venues (TTPPV) is a singleround robin variant of the Traveling Tournament Problem, in which the venue of each gameto be played is known beforehand. We propose an Iterated Local Search (ILS) heuristic forsolving real-size instances of the TTPPV, based on two types of moves. Initial solutions arederived from canonical 1-factorizations of the tournament graph or of its subgraphs. Compu-tational results show that the new ILS heuristic performs much better than heuristics basedon integer programming and improves the best known solutions for benchmark instances.We also show that commonly used neighborhoods in the sport scheduling literature are notconnected in the case of single round robin tournaments.

Keywords Traveling tournament problem · Sports scheduling · Iterated local search ·Metaheuristics · Neighborhoods

1 Introduction and motivation

A round robin tournament is one in which each team plays against every other a fixed num-ber of times in a given number of rounds. A team faces every other team exactly once (resp.twice) in a single (resp. double) round robin (SRR) (resp. DRR) tournament. A tournamentis compact if the number of rounds is minimum and every team plays a game in everyround. Every game is played in the venue of one of the opponent teams. Scheduling an SRRtournament consists in determining in which round and in which venue each game will beplayed.

The solution of the problem of scheduling a round robin tournament is usually di-vided into two stages Trick (2000). The construction of the timetable determines the round

F. N. Costa · S. UrrutiaDepartment of Computer Science, Universidade Federal de Minas Gerais, Av. Antonio Carlos 6627,Belo Horizonte, MG 31270-010, BrazilE-mail: {fabrimac,surrutia}@dcc.ufmg.br

C. C. RibeiroDepartment of Computer Science, Universidade Federal Fluminense, Rua Passo da Patria 156,Niteroi, RJ 24210-240, BrazilE-mail: celso@ic.uff.br

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in which each game is played. The Home-Away Pattern (HAP) set determines in whichcondition (home or away) each team plays in each round. Together, the timetable and thehome-away pattern set determine the tournament schedule. Literature reviews may be foundin Kendall et al (2008); Rasmussen and Trick (2008).

Some problems in the literature consider the construction of both the timetable and theHAP set. The Traveling Tournament Problem (TTP) introduced by Easton et al (2001) maybe described as follows. A double round robin tournament is played by an even number nof teams indexed by 1, . . . ,n. Each team has its own venue at its home city. All teams areinitially at their home cities, to where they return after their last away game. The distancedi j ≥ 0 from the home city of team i to that of team j is known beforehand. Whenever a teamplays two consecutive away games, it travels directly from the venue of the first opponentto that of the second. The problem calls for a DRR tournament schedule such that no teamplays more than three consecutive home games or more than three consecutive away games,there are no consecutive games involving the same pair of teams, and the total distancetraveled by the teams during the tournament is minimized.

In other problems, the timetable or HAP may be fixed and known beforehand. TheBreaks Minimization Problem (Regin (2001)) and the Timetable Constrained Distance Min-imization Problem (Rasmussen and Trick (2006)) are examples where the timetable is fixedand the objective is to find a HAP set that optimizes a certain objective function. The prob-lem of finding a timetable compatible with a given HAP set appears as a subproblem inseveral approaches to tackle real-life league scheduling problems (Nemhauser and Trick(1998); Ribeiro and Urrutia (2007b)).

The problem of scheduling a single round robin tournament may be decomposed using adifferent strategy. Instead of considering a HAP set which determines the playing conditionof each team in each round, a Home-Away Assignment (HAA) is considered. The latterdetermines the venue in which the game between each pair of teams takes place. If thetimetable is fixed, the HAP set and HAA provide the same information and determine theschedule. In the following, we consider that the HAA is fixed and known beforehand.

Melo et al (2008) introduced the Traveling Tournament Problem with Predefined Venues(TTPPV). This variant of the TTP considers a single round robin tournament, in which thevenues where the games take place are known beforehand. The set of games to be playedis represented by ordered pairs of teams determined by the HAA. The game between teamsi and j is represented either by (i, j) or by ( j, i). In the first case, the game between i andj takes place at the venue of team i; otherwise, at that of team j. Therefore, for every twoteams i and j, either one of the ordered pairs (i, j) or ( j, i) belongs to the set of games to beplayed. The TTPPV consists in finding a compact single round robin schedule compatiblewith the predefined HAA, such that the total distance traveled by the teams is minimized andno team plays more than three consecutive home games or three consecutive away games.This problem is a natural extension of the Traveling Tournament Problem to the case ofsingle round robin tournaments.

Variants of this problem find interesting applications in real-life leagues whose DRRtournaments are divided into two SRR phases. Games in the second phase are exactly thesame as those in the first phase, except for the inversion of their venues. Therefore, thevenues of the games in the second phase are known beforehand and constrained by thoseof the games in the first phase. This is the case e.g. of the Chilean soccer professionalleague (see Duran et al (2007)) and of the German table tennis federation of Lower Saxony(see Knust (2007)).

Instances of the TTPPV with up to eight teams were solved to optimality by the integerprogramming formulations presented in Melo et al (2008). Since feasible solutions have not

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been found by a commercial solver within two hours of running time for instances with 18or more teams, four heuristics based on the integer programming formulations were alsodeveloped. In this paper, we propose a local search based heuristic for the TTPPV to findgood quality solutions for realistic size instances. Section 2 describes the proposed heuristic,based on the Iterated Local Search (ILS) metaheuristic. Section 3 reports the computationalexperiments. Concluding remarks are made in the last section.

2 Heuristic approach

In this section we describe a local search based heuristic to the TTPPV. First, we show howto construct initial solutions for the problem. Next, we describe the neighborhoods that areused in a local search procedure. Finally, we outline the whole heuristic algorithm.

2.1 Initial solutions

A factor of a graph G = (V,E) is a subgraph G′ = (V,E ′) of G, with E ′ ⊆ E. G′ is said tobe a 1-factor of G if all its nodes have their degrees equal to one. A factorization of G isa set of edge-disjoint factors {G1 = (V,E1), . . . ,Gp = (V,E p)}, with E1 ∪ . . .∪E p = E. A1-factorization of G is one in which all factors are 1-factors. In an ordered 1-factorization ofG, the 1-factors are taken in a fixed order.

Schedules of an SRR tournament with n (even) teams and fixed home away assign-ments may be represented by ordered 1-factorizations of the complete undirected graph Kn(see Ribeiro and Urrutia (2007a)). Each node of this graph represents a team. An edge {i, j}in the k-th 1-factor of an ordered 1-factorization implies that the game between teams i andj is played in the k-th round.

The so called canonical 1-factorization for Kn (see de Werra (1981)) has been widelyused to construct initial solutions for round robin tournaments. We describe a method togeneralize the construction of this 1-factorization, in order to obtain a greater variety ofinitial solutions. This generalized method consists in separating Kn into three subgraphs:two complete subgraphs and one complete bipartite graph. A 1-factorization of Kn may beobtained by a combination of 1-factorizations of each subgraph of Kn. The 1-factorization ofeach subgraph of Kn may be canonical or built by the recursive application of this method.

In the canonical 1-factorization of Kn (see de Werra (1981)), the edge set of each factorGi = (V,E i) is defined as follows, for i = 1, . . . ,n−1 and V = {1, . . . ,n}:

E i = {(n, i)}∪{( f1(i,k), f2(i,k)) : k = 1, ...,n/2−1},

with

f1(i,k) =

{i+ k, if i+ k < n,

i+ k−n+1, if i+ k ≥ n,

and

f2(i,k) =

{i− k, if i− k > 0,i− k +n−1, if i− k ≤ 0.

Figure 1 shows the canonical 1-factorization of K6.The above generalized method can be directly applied if n is a multiple of four. Nodes

of V are separated into two sets V1 and V2 with n/2 nodes each. A 1-factorization is built for

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Fig. 1 Canonical 1-factorization of K6

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Fig. 2 Figures (a) to (c) show the three first 1-factors obtained by the generalized method for K8. The vertexset was partitioned into V1 = {1,2,3,4} and V2 = {5,6,7,8}. The canonical 1-factorization was built for eachsubset with four nodes.

the complete graphs defined by each set V1 and V2. To build the first n/2− 1 factors of the1-factorization of Kn, we make the union of any factor of the 1-factorization associated withV1 with any factor of the 1-factorization associated with V2. The routine proceeds as before,using pairs of unused factors of the 1-factorizations associated with V1 and V2 until n/2−1factors of the 1-factorization of Kn are obtained.

The remaining n/2 factors of the 1-factorization of Kn (corresponding to the final n/2rounds of the tournament schedule) contain exclusively edges with one extremity in V1 andthe other in V2. These remaining factors form a 1-factorization of the complete bipartitegraph defined by the partition {V1,V2}.

Figure 2 shows the three first 1-factors of K8 obtained by the generalized method.The vertex set was divided into V1 = {1,2,3,4} and V2 = {5,6,7,8}. The canonical 1-factorization was built for each complete subgraph with four nodes.

Whenever n is not a multiple of four, nodes of V are separated into two sets V1 and V2with n/2 nodes each. Two artificial nodes x and y are included in V1 and V2, respectivelyto form V ′1 = V1 ∪{x} and V ′2 = V2 ∪{y}, each of them with an even number of nodes. A1-factorization is built for the complete graphs defined by each set V ′1 and V ′2. To build thefirst n/2 factors of the 1-factorization of Kn, we combine any factor of the 1-factorization as-

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Fig. 3 Figures (a) to (c) show the three first 1-factorizations obtained by the generalized method for K6. Thecanonical 1-factorization was built for each subset V ′1 and V ′2 with four nodes.

sociated with V ′1 with any factor of the 1-factorization associated with V ′2. This combinationconsiders all edges of the factors that are not incident to any artificial nodes and the edges(v1,v2) formed by node v1 incident to x in V ′1 and node v2 incident to y in V ′2. The routineproceeds as before, using pairs of unused factors of the 1-factorizations associated with V ′1and V ′2 until n/2 factors of the 1-factorization of Kn are obtained.

The remaining n/2− 1 factors of the 1-factorization of Kn can be obtained from a1-factorization of the complete bipartite graph defined by the partition {V1,V2}. This 1-factorization contains a 1-factor that includes all edges with one endpoint in V1 and theother endpoint in V2 already in the first n/2 factors. If the latter does not contain such a fac-tor, the 1-factorization of this bipartite graph can be easily modified by a relabeling functionso as to contain it. This particular 1-factor is discarded and the others form the remainingn/2−1 factors of the 1-factorization of Kn.

Figure 3 shows the three first 1-factors of K6 built by the generalized method. The vertexset was divided into V1 = {1,2,3} and V2 = {4,5,6}. The canonical 1-factorization was builtfor each auxiliary set V ′1 = V1 ∪{x} and V ′2 = V2 ∪{y} with artificial vertices. Doted edgesrepresent those that are incident to artificial nodes in 1-factorizations of complete graphsdefined by V ′1 and V ′2. They do not appear in the 1-factorization of K6.

The bipartite canonical 1-factorization for any complete bipartite graph Kn/2,n/2 withan even number n of nodes is defined by the partition {{a1, . . . ,an/2},{b1, . . . ,bn/2}}. Theedge set of each 1-factor Gr of this 1-factorization is defined as follows, for r = 1, . . . ,n/2:

Er = {(ai,b f3(i,r)) : i = 1, ...,n/2},

with

f3(i,r) =

{i+ r−1, if i+ r−1≤ n/2;i+ r−1−n/2, otherwise.

Figure 4 shows the canonical 1-factorization of the complete bipartite graph K4,4 definedby the partition {{1,2,3,4},{5,6,7,8}}.

The construction of 1-factorizations for Kn/2,n/2 can also be generalized. We supposethat n/2 is even. The method consists in separating V1 into V1a and V1b, and V2 into V2a andV2b, with |V1a| = |V1b| = |V2a| = |V2b| = n/4. A 1-factorization is built for each completebipartite graph defined by partitions {V1a,V2a}, {V1b,V2b}, {V1a,V2b}, and {V1b,V2a}. Thefirst n/4 factors are built by a combination of the 1-factorizations of the two first graphs. Thelast n/4 factors are built by a combination of the 1-factorizations for the last two graphs.

Figure 5 shows a 1-factorization of K4,4 defined by partition {V1 = {1,2,3,4},V2 ={5,6,7,8}} constructed by the generalized method. The first set of nodes was separated into

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Fig. 4 Canonical 1-factorization of K4,4

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Fig. 5 A 1-factorization of K4,4 obtained by the generalized method.

V1a = {1,2} and V1b = {5,6} and the second into V2a = {5,6} and V2b = {7,8}. A canonical1-factorization is used for each complete bipartite graph.

Whenever n/2 is odd, V1 is divided into V1a and V1b, and V2 into V2a and V2b, with|V1a|= dn/4e, |V1b|= bn/4c, |V2a|= bn/4c and |V2b|= dn/4e. Auxiliary sets V ′2a =V2a∪{x}and V ′1b = V1b∪{y} are created adding artificial nodes x and y to the corresponding sets withodd number of nodes. A 1-factorization is built for each complete bipartite graph defined bypartitions {V1a,V ′2a}, {V ′1b,V2b}, {V1a,V2b}, and {V1b,V2a}. The first dn/4e factors are builtby combination of the 1-factorizations of the two first graphs. This combination considersall edges of the factors that are not incident to any artificial nodes and the edges (v1,v2)formed by nodes v1 incident to x and v2 incident to y.

The remaining bn/4c factors are built by combination of the two factorizations of thelast two graphs. The 1-factorization of the bipartite graph defined by the partition {V1a,V2b}has to contain a 1-factor that includes all edges with one endpoint in V1a and the otherendpoint in V2b already present in the first dn/4e factors. If it does not contain a such factor,the 1-factorization of this bipartite graph can be appropriately modified by node relabeling.This particular 1-factor is discarded and the others form the remaining bn/4c factors of the1-factorization of Kn/2,n/2.

Figure 6 shows a 1-factorization of K5,5 defined by partition {V1 = {1,2,3,4,5},V2 ={6,7,8,9,10}} built by the generalized method. The sets are divided into V1a = {1,2,3},V1b = {4,5}, V2a = {6,7}, and V2b = {8,9,10}. Auxiliary sets V ′2a = V2a ∪{x} and V ′1b =V1b ∪{y} were created adding artificial nodes to the sets with odd number of nodes. Thecanonical 1-factorization is applied to each bipartite graph defined by {V1a,V ′2a}, {V ′1b,V2b},{V1a,V2b}, and {V1b,V2a}. Factors (a)-(c) arise from the combination of the two first 1-factorizations. Edges represented by dotted lines are incident to artificial nodes and do notappear in the resulting 1-factorization of K5,5. Factors (d) and (e) are built by combination ofthe two last 1-factorizations. The 1-factorization for {V1a,V2b} was modified by a relabeling

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Fig. 6 1-factorization built for K5,5 using the generalized method.

function to contain the factor with edges (3,10), (2,8), and (1,9) already present in the firstthree factors. This 1-factor was discarded.

A particular case of the generalized method builds a 1-factorization using the canonical1-factorization of two complete graphs with n/2 nodes each and the bipartite canonical1-factorization as the remainder 1-factors. This particular 1-factorization is known in theliterature as the binary 1-factorization (de Werra (1980)).

We notice that any of the construction procedures may build ordered 1-factorizationsthat violate the limits on the maximum number of consecutive home or away games.

2.2 Neighborhoods

Let V = {1, ...,n} be the set of teams and R = {1, ...,n−1} the set of rounds of a scheduleX . We assume that this schedule is represented by a matrix with n rows and n−1 columns,where X(t,r) denotes the opponent of team t ∈V in round r ∈ R. A negative sign indicatesthat team t is playing an away game in round r. Figure 7 shows a schedule for a tournamentwith eight teams.

Teams Rounds1 2 3 4 5 6 7

1 -4 5 6 -2 8 7 -32 -6 -8 4 1 3 -5 -73 5 -4 -7 -6 -2 8 14 1 3 -2 -8 7 -6 -55 -3 -1 -8 -7 6 2 46 2 -7 -1 3 -5 4 87 -8 6 3 5 -4 -1 28 7 2 5 4 -1 -3 -6

Fig. 7 Schedule for a tournament with eight teams.

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Different neighborhood structures have been used in local search procedures for schedul-ing round robin tournaments, see e.g. Anagnostopoulos et al (2006); Gaspero and Schaerf(2007); Ribeiro and Urrutia (2007a). The basic home-away swap neighborhood used byRibeiro and Urrutia (2007a) is not considered in this study, since the home-away assign-ments are fixed beforehand. We consider four neighborhoods in the context of the TTPPV.

The first neighborhood is team swap (N1). For any two teams t1 ∈ V and t2 ∈ V , witht1 6= t2, the schedule obtained by swapping the opponents of teams t1 and t2 in all rounds ofthe schedule X is a neighbor of the latter in N1.

The second neighborhood is round swap (N2). For any two rounds r1 ∈ R and r2 ∈ R,with r1 6= r2, the schedule obtained by swapping the games of schedule X in rounds r1 andr2 is a neighbor of X in N2.

Neighborhoods N1 and N2 are not used by the heuristic proposed in this work and haveonly theoretical relevance for this paper.

The third neighborhood is partial team swap (N3). For any round r ∈ R and for any twoteams t1 ∈ V and t2 ∈ V , with t1 6= t2 and |X(t1,r)| 6= t2, let S be a minimum cardinalitysubset of rounds including round r in which the opponents of teams t1 and t2 are the same,i.e. S = {r1, ...,rk}⊆R is minimal and such that r∈ S and {t ∈V : ∃ j∈ S such that |X(t, j)|=t1} = {t ∈ V : ∃ j ∈ S such that |X(t, j)| = t2}. Given a schedule X , a round r, and teams t1and t2 defined as above, the schedule obtained by swapping the opponents of teams t1 and t2in all rounds in S is a neighbor of X in N3.

Figure 8 illustrates a move in neighborhood N3 for a tournament with eight teams andr = 2, t1 = 1, and t2 = 2. In this case, S = {2,5,6,7}. Teams 3, 5, 7, and 8 are the opponentsof teams 1 and 2 in the rounds in S. We notice that the partial team swap neighborhood N3is a generalization of the team swap neighborhood N1.

The last neighborhood is partial round swap (N4). For any team t ∈ V and for any tworounds r1 ∈ R and r2 ∈ R, with r1 6= r2, let U be a minimum cardinality subset of teams in-cluding team t in which the opponents of the teams in U in rounds r1 and r2 are the same, i.e.U = {t1, ..., tk} ⊆V is minimal and such that t ∈U and {i∈V : ∃u∈U such that |X(i,r1)|=u}= {i ∈V : ∃u ∈U such that |X(i,r2)|= u}. Given a schedule X , a team t, and rounds r1and r2 defined as above, the schedule obtained by swapping the opponents of each team inU in rounds r1 and r2 is a neighbor of X in N4.

Figure 9 shows a move in neighborhood N4 for a tournament with ten teams and t = 1,r1 = 1, and r2 = 4. In this case, U = {1,6,7}. Teams 3, 4, and 8 are the opponents ofteams in U in rounds 1 and 4. We notice that the partial round swap neighborhood N4 is ageneralization of the round swap neighborhood N2.

Moves in neighbors N1 and N2 do not change the current 1-factorization. In some situa-tions, it may happen that every move in neighborhood N3 be equivalent to a move in neigh-borhood N1. This was experimentally observed e.g. for the canonical 1-factorization forn∈ {12,14,20}. In these cases, for any round r ∈ R and for any two teams t1 ∈V and t2 ∈V ,the minimum cardinality subset S differs from the complete set R of rounds exclusively bythe round in which t1 plays against t2. Similarly, moves in neighborhood N4 may be equiva-lent to moves in neighborhood N2. Again, it is the case for the canonical 1-factorization forn ∈ {12,14,20}. Therefore, if the canonical 1-factorization is used as the initial solution inthese situations, any search method using only the neighborhoods N1, N2, N3, and N4 willnot be able to escape from the local optimum associated with the canonical 1-factorization.However, the same does not hold for other 1-factorizations built by the generalized method,in which neighborhoods N3 and N4 are different from N1 and N2, respectively.

In the following, we show that any move in neighbor N2 is equivalent to a sequence ofmoves in neighborhood N4.

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Teams Rounds1 r = 2 3 4 5 6 7

t1 = 1 -4 8 6 -2 -3 5 7

t2 = 2 -6 -5 4 1 -8 -7 3

3 5 -4 -7 -6 1 8 -24 1 3 -2 -8 7 -6 -55 -3 2 -8 -7 6 -1 46 2 -7 -1 3 -5 4 87 -8 6 3 5 -4 2 -1

8 7 -1 5 4 2 -3 -6(a) Original schedule X

Teams Rounds1 r = 2 3 4 5 6 7

t1 = 1 -4 5 6 -2 8 7 -3

t2 = 2 -6 -8 4 1 3 -5 -7

3 5 -4 -7 -6 -2 8 14 1 3 -2 -8 7 -6 -55 -3 -1 -8 -7 6 2 46 2 -7 -1 3 -5 4 87 -8 6 3 5 -4 -1 2

8 7 2 5 4 -1 -3 -6(b) New schedule after move in neighborhood N3

Fig. 8 Move in neighborhood N3 for a tournament with eight teams and r = 2, t1 = 1, and t2 = 2 (highlightedentries in (a) appear modified in (b) after the move).

We first observe that each move in N4 with parameters t, r1, and r2 involves a subset ofthe games affected by a move in N2 using parameters r1 and r2. In fact, the teams involvedin the move in N4 are those in the subset U and their opponents in rounds r1 and r2, whichare computed during the move (see above). We also remark that if, instead of using team tas a move parameter, we use another team t1 ∈U , the subset U1 of teams computed duringthe move turns out to be equal to U and, in consequence, it does not change the move. If,instead of using team t, we use another team t2 that is an opponent of a team in U in roundr1 or r2, the subset U2 of teams computed during the move turns out to be equal to the set ofopponents of teams in U in rounds r1 and r2 and the set of opponents of teams in U2 turnsout to be equal to U . Therefore, once again the move is the same. Finally, if instead of usingteam t as a move parameter, we use another team t3 that does not belong to U neither is anopponent of a team in U in round r1 or r2, the subset U3 of teams computed during the moveturns out to be disjoint to U and to the set of opponents of teams in U in rounds r1 and r2.Hence, in this case the move involves different games between rounds r1 and r2. Therefore,the set of teams may be partitioned into subsets P1, . . . ,Pp, so as that for any i = 1, . . . , p themoves in neighborhood N4 with parameters t, r1, and r2 are the same for every t ∈ Pi. Toconclude, we remark that any sequence of p moves in N4 involving rounds r1 and r2 andone team from each subset Pi, i = 1, . . . , p, is equivalent to a move in N2 using r1 and r2 asparameters.

A similar argument shows that a sequence of moves in N3 is equivalent to a move in N1.

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Teams Roundsr1 = 1 2 3 r2 = 4 5 6 7 8 9

t = 1 -3 10 7 -8 -5 2 4 9 -62 9 8 6 -5 -3 -1 10 -4 -73 1 -4 9 -6 2 -10 -7 -8 5

4 -6 3 -10 -7 8 5 -1 2 95 10 7 8 2 1 -4 -9 -6 -36 4 -9 -2 3 -10 -7 -8 5 1

7 -8 -5 -1 4 -9 6 3 -10 2

8 7 -2 -5 1 -4 -9 6 3 -109 -2 6 -3 10 7 8 5 -1 -4

10 -5 -1 4 -9 6 3 -2 7 8(a) Original schedule X

RoundsTeams r1 = 1 2 3 r2 = 4 5 6 7 8 9

t = 1 -8 10 7 -3 -5 2 4 9 -62 9 8 6 -5 -3 -1 10 -4 -73 -6 -4 9 1 2 -10 -7 -8 5

4 -7 3 -10 -6 8 5 -1 2 95 10 7 8 2 1 -4 -9 -6 -36 3 -9 -2 4 -10 -7 -8 5 1

7 4 -5 -1 -8 -9 6 3 -10 2

8 1 -2 -5 7 -4 -9 6 3 -109 -2 6 -3 10 7 8 5 -1 -4

10 -5 -1 4 -9 6 3 -2 7 8(b) New schedule after move in neighborhood N4

Fig. 9 Move in neighborhood N4 for a tournament with ten teams and t = 1, r1 = 1, and r2 = 4 (highlightedentries in (a) appear modified in (b) after the move).

2.3 Local search

We have shown in the last section that any solution that can be reached by a move in N1(resp. N2) can also be reached by a sequence of one or more moves in N3 (resp. N4). There-fore, the set of solutions connected through neighborhoods N1 and N2 is included in thatconnected through neighborhoods N3 and N4. Since neighborhoods N3 and N4 generalizeneighborhoods N1 and N2 and solutions in the two former neighborhoods can be computedas quickly as those in the two latter, we propose a local search procedure exploring onlyneighborhoods N3 and N4.

Let v(X) and d(X) be, respectively, the number of constraint violations and the total trav-eled distance in the current solution X . All moves in neighborhoods N3 and N4 are evaluatedat each local search iteration, each of them in time O(n). The number v(X ′) of constraintviolations and the total traveled distance d(X ′) are computed for each neighbor X ′ of thecurrent solution X .

If there is at least one neighbor solution X ′ such that v(X ′) ≤ v(X) and d(X ′) < d(X),then the current solution X is replaced by its neighbor with minimum traveled distanceamong all those satisfying the above condition (i.e., the current solution is replaced by a leastcost neighbor which does not deteriorate the number of constraint violations). Otherwise, ifno move is able to improve the traveled distance of the current solution X without increasing

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the number of constraint violations in the latter, then the current solution is replaced by itsneighbor decreasing the most the number of constraint violations. If no such a move exists,then the local search procedure stops and the current locally optimal solution is returned.

2.4 ILS heuristic

The Iterated Local Search (ILS) metaheuristic (see Lourenco et al (2003)) proposes the useof perturbations to escape from locally optimal solutions. The method starts by constructingan initial solution and by applying a local search procedure to it. The current solution isperturbed at each iteration and local search is applied to the perturbed solution. Next, thesolution resulting from perturbation followed by local search is compared with the currentsolution. The former is accepted as the new current solution if some predefined acceptancecriterion is met. Otherwise, a new iteration formed by perturbation followed by local searchis performed. The procedure stops when some stopping criterion is reached. Algorithm 1depicts the pseudo-code with the main steps of the ILS metaheuristic.

Algorithm 1: Pseudo-code of the ILS metaheuristic.d∗← ∞ ;1X ← BuildInitialSolution ;2X ← LocalSearch(X) ;3repeat4

if v(X) = 0 and d(X) < d∗ then5X∗← X ;6d∗← d(X) ;7

end8X ′← Perturbation(X) ;9X ′← LocalSearch(X ′) ;10X ← AcceptanceCriterion(X ,X ′) ;11

until stopping condition ;12

The traveled distance associated with the best feasible solution found is initialized in line1. The constructive procedures presented in Section 2.1 are used to build initial solutions inline 2. The local search procedure applied in lines 3 and 10 follows the strategy describedin Section 2.3. The perturbation in line 9 consists in a randomly generated sequence of two,three or four movements in N3 ∪N4. The solution X ′ obtained after the application of localsearch to the perturbed solution is accepted as the new current solution X in line 11 if andonly if it satisfies one of the conditions below:

1. v(X ′) < v(X) (the new solution X ′ has fewer constraint violations than the current solu-tion X); or

2. v(X ′) = v(X) and d(X ′) < d(X) (the new solution X ′ has the same number of constraintviolations as the current solution X , but the traveled distance according to schedule X ′

is smaller than that associated with X); or3. if at least 100 iterations have been performed since the last update of the current so-

lution X , v(X ′) ≤ v(X) (the number of constraint violations in the new solution X ′ isnot greater than that in the current solution X), and d(X ′) ≤ 1.01 · d(X) (the traveleddistance associated with the new schedule X ′ deteriorates by at most 1% the traveleddistance according with the current schedule X).

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The acceptance criterion is primarily driven to finding solutions reducing the number ofconstraint violations and, secondly, to finding improving solutions which do not deterioratethe number of constraint violations. The best feasible solution found during the search isupdated in lines 5 to 8 and returned when the stopping condition in line 12 is met.

3 Experimental results

In this section, we report on the computational experiments performed to evaluate the pro-posed ILS heuristic, which was coded in C++ and compiled with version 4.1.2 of g++ withthe optimization flag -O3. The experiments have been performed on an Intel Xeon CPU witha 3.00 GHz processor and 2 Gbytes of RAM, running under the operating system DebianGNU/Linux 4.0.

We used in the computational experiments the same instances proposed and used by Meloet al (2008). There are a total of 40 instances, 20 of them with 18 teams and the other 20with 20 teams. Distances are the same as in instances circ18 and circ20 of the TTP, both ofthem available from Trick (2007). For each instance size (i.e., the number n of teams), 20distinct home-away assignments were created: ten out of the 20 assignments are balanced(i.e., each team plays at least n/2−1 games at home and at least n/2−1 away games), whilethe remaining ten assignments are unbalanced. Two instances of each size were shown to beinfeasible by Melo et al (2008).

In the first experiment, we compare the performance of the heuristic when using thecanonical and the binary 1-factorizations to build the initial solution. Tables 1 and 2 showthe numerical results obtained after a execution of ILS heuristic with a time limit of fiveminutes using each 1-factorization to build the initial solution for instances with 18 and 20teams, respectively.

Instance Canonical Binary Improvement(%)circ18abal 844 824 2,37circ18bbal 834 814 2,40circ18cbal 802 816 -1,75circ18dbal 828 818 1,21circ18ebal 820 826 -0,73circ18fbal 804 834 -3,73circ18gbal 828 804 2,90circ18hbal 832 830 0,24circ18ibal 814 830 -1,97circ18jbal 804 812 -1,00

circ18anonbal 832 826 0,72circ18dnonbal 842 816 3,09circ18enonbal 844 832 1,42circ18fnonbal 838 834 0,48circ18gnonbal 842 834 0,95circ18hnonbal 872 844 3,21circ18inonbal 860 866 -0,70circ18jnonbal 828 832 -0,48

Average 831,56 827,33 0,51

Table 1 Numerical results for the ILS heuristic using canonical and binary 1-factorizations as initial solution(18 teams).

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Instance Canonical Binary Improvement(%)circ20abal 1378 1148 16,69circ20bbal 1400 1140 18,57circ20cbal 1390 1134 18,42circ20dbal 1380 1150 16,67circ20ebal 1370 1136 17,08circ20fbal 1380 1142 17,25circ20gbal 1372 1124 18,08circ20hbal 1382 1134 17,95circ20ibal 1374 1138 17,18circ20jbal 1376 1128 18,02

circ20anonbal 1502 1174 21,84circ20bnonbal 1420 1142 19,58circ20cnonbal 1438 1140 20,72circ20dnonbal 1422 1166 18,00circ20enonbal 1452 1170 19,42circ20gnonbal 1470 1150 21,77circ20inonbal 1408 1144 18,75circ20jnonbal 1382 1118 19,10

Average 1405,33 1143,22 18,62

Table 2 Numerical results for the ILS heuristic using canonical and binary 1-factorizations as initial solution(20 teams).

The first column shows the instance name. The second and third column give the bestsolution values obtained using each of the 1-factorizations. The last column shows the im-provement (i.e., the reduction) in percent in the value of the solution obtained using thecanonical 1-factorization when the binary 1-factorization is used.

The results obtained by the ILS heuristic with the two 1-factorizations are quite sim-ilar for the instances with 18 teams. For instances with 20 teams, the use of the binary1-factorization improved the solution value by at lest 16.67%, with an average improvementof 18,62%. This is due to the fact that for n = 20, neighborhoods N3 and N4 are equivalent toN1 and N2, respectively, if the canonical 1-factorization is used. Since moves in N1 and N2do not change the 1-factorization associated with the current schedule, in this case movesin N3 and N4 do not change it either. In consequence, the ILS heuristic gets stuck by thestructure of the canonical 1-factorization and explores just a small portion of the solutionspace. This phenomenon does not happen when the binary 1-factorization is used.

Considering that neighborhoods N1 and N2 do not connect the search space and that insome situations they are equivalent to N3 and N4, respectively (e.g. for n = 20 when thecanonical 1-factorization is used), we conclude that the solution space is not connected byneighborhoods N3 and N4 for n = 20. In this case, at least the canonical 1-factorizationis not connected to any other factorization by moves in these neighborhoods (it remainsunknown whether the search space is connected or not for n = 18). We propose the use of thegeneralized method introduced in Section 2.1 to construct a large variety of 1-factorizationsto be used as initial solutions, which will make possible for the heuristic to visit a largerportion of the solution space.

In the next experiment, we run the heuristic for a total of two hours (the same timegiven to the heuristics in Melo et al (2008)) to obtain better solutions for each instance.Since there is a variety of 1-factorizations to use for building the initial solution, we proposea method for choosing a promising 1-factorization to be used as the initial solution. Thismethod is able to build 79 non-isomorphic 1-factorizations for K18 and 43 for K20. It hastwo parameters: the number s of stages and the time t spent in each stage. In the first stage,

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each 1-factorization is used to construct an initial solution and the ILS heuristic is run fromeach initial solution for a total time of t seconds of execution. The top z/ s

√f best solutions

found in this stage go to the next one, where z is the number of solutions in the currentstage and f is the number of 1-factorizations initially built. The fraction of solutions thatsurvive and go to the next stage is such that only one solution remains after the last stage.The process is repeated until s stages are completed and the best solution found is returned.Algorithm 2 depicts the pseudo-code of the method. The generalized method is used in line1 to generate a set F of 1-factorizations. Each of the initial solutions considered in a givenstage j = 1, . . . ,s are investigated in lines 6 to 8 and replaced by the solution obtained bythe application of the ILS heuristic using it as initial solution for t/|F | time units. The bestsolutions are selected in lines 10 and 11 to survive to the next stage, while the others arediscarded.

Algorithm 2: Pseudo-code for choosing a promising initial solution.Use the generalized method to obtain the complete set F of 1-factorizations it can build;1f ← |F | ;2for j = 1, . . . ,s do3

F ′← /0;4for e ∈ F do5

Select a solution e and remove it from F : F ← F−{e} ;6Obtain a new solution e′ by applying the ILS heuristic to e for t/|F | time units;7Consider the new solution e′ as a candidate for the next stage: F ′← F ′ ∪{e′} ;8

end9Keep in F ′ the top best d|F |/ s

√f e 1-factorizations and discard the others;10

F ← F ′;11end12Return the best 1-factorization in F ;13

A robust heuristic that uses this strategy to choose the initial solution was compared toa heuristic that randomly selects the initial solution among the 1-factorizations that can bebuilt with the generalized method (with the exception of the canonical 1-factorization forn = 20). The robust heuristic spends three minutes choosing the initial solution, distributedover three stages. Next, the ILS heuristic is executed from the selected initial solution for117 minutes. Tables 3 and 4 show the numerical results obtained by each algorithm. The firstcolumn shows the instance name, while the second and third give the best solution valuesobtained by each algorithm. The last column shows the improvement (i.e., the reduction) inpercent in the value of the best solution obtained by the random algorithm when the robustheuristic is applied. The results obtained by both heuristics are quite similar for n = 18, butfor n = 20 the robust strategy found better solutions in the same computation time for 61%of the feasible instances (i.e., for 11 out of 18 instances) and improved the best solutionvalues by 0,39% on average.

To conclude, we compare the results obtained by the ILS heuristic in one second ofprocessing time with those presented in Melo et al (2008) and obtained in a computationalenvironment similar to the one used in this work. Since the time limit was very small, therandom strategy to choose the initial 1-factorization was used for all instances. Table 5displays the numerical results. The first column gives the instance name. The second columnshows the best solution value among those found by the four algorithms described by Meloet al (2008) after two hours of running time. The next two columns give the traveled distancein the best solution found by a single run of the ILS heuristic for one second, together with

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Instance Robust Random Improvement(%)circ18abal 790 776 -1,80circ18bbal 810 796 -1,76circ18cbal 802 794 -1,01circ18dbal 788 794 0,76circ18ebal 790 784 -0,77circ18fbal 792 798 0,75circ18gbal 792 784 -1,02circ18hbal 780 792 1,52circ18ibal 806 778 -3,60circ18jbal 780 794 1,76

circ18anonbal 806 814 0,98circ18dnonbal 810 800 -1,25circ18enonbal 814 804 -1,24circ18fnonbal 802 826 2,91circ18gnonbal 782 802 2,49circ18hnonbal 804 822 2,19circ18inonbal 818 818 0,00circ18jnonbal 782 790 1,01

Average 797,11 798,11 0,11

Table 3 Numerical results for Robust and Random ILS heuristics (18 teams)

Instance Robust Random Improvement(%)circ20abal 1094 1098 0,36circ20bbal 1090 1082 -0,74circ20cbal 1082 1072 -0,93circ20dbal 1100 1104 0,36circ20ebal 1076 1100 2,18circ20fbal 1072 1076 0,37circ20gbal 1068 1082 1,29circ20hbal 1094 1094 0,00circ20ibal 1084 1078 -0,56circ20jbal 1086 1096 0,91

circ20anonbal 1130 1106 -2,17circ20bnonbal 1082 1096 1,28circ20cnonbal 1096 1104 0,72circ20dnonbal 1140 1136 -0,35circ20enonbal 1128 1100 -2,55circ20gnonbal 1078 1130 4,60circ20inonbal 1082 1092 0,92circ20jnonbal 1070 1084 1,29

Average 1091,78 1096,11 0,39

Table 4 Numerical results for Robust and Random ILS heuristics (20 teams).

the corresponding improvement over the best result in Melo et al (2008). The ILS heuristicclearly outperformed the algorithms in Melo et al (2008): running the ILS heuristic for onesecond improved the best known solution values by at least 11.42% for every instance andby 17.17% on average.

4 Concluding remarks

We proposed an ILS heuristic for the traveling tournament problem with predefined venues.A generalized strategy for building initial solutions was developed and evaluated. We showedthat canonical 1-factorizations should not be used as initial solutions in some situations, for

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Instance Best ILS(1s) Improvement(%)circ18abal 1106 912 17,54circ18bbal 1100 896 18,55circ18cbal 1038 892 14,07circ18dbal 1096 882 19,53circ18ebal 1074 892 16,95circ18fbal 1060 910 14,15circ18gbal 1100 894 18,73circ18hbal 1094 880 19,56circ18ibal 1102 894 18,87circ18jbal 1078 878 18,55

circ18anonbal 1124 940 16,37circ18dnonbal 1060 914 13,77circ18enonbal 1092 922 15,57circ18fnonbal 1098 956 12,93circ18gnonbal 1098 924 15,85circ18hnonbal 1110 944 14,95circ18inonbal 1104 932 15,58circ18jnonbal 1102 898 18,51

circ20abal 1520 1236 18,68circ20bbal 1530 1252 18,17circ20cbal 1470 1234 16,05circ20dbal 1464 1238 15,44circ20ebal 1526 1214 20,45circ20fbal 1546 1236 20,05circ20gbal 1536 1210 21,22circ20hbal 1516 1268 16,36circ20ibal 1544 1238 19,82circ20jbal 1484 1222 17,65

circ20anonbal 1502 1270 15,45circ20bnonbal 1522 1258 17,35circ20cnonbal 1488 1318 11,42circ20dnonbal 1510 1294 14,30circ20enonbal 1574 1250 20,58circ20gnonbal 1540 1278 17,01circ20inonbal 1516 1236 18,47circ20jnonbal 1516 1220 19,53

Average 1303,89 1078,67 17,17

Table 5 Comparative results between ILS (one second of execution) and the previous best solution.

which the 1-factorizations generated by the generalized method are much more appropriate.Two neighborhoods were investigated and explored by the ILS heuristic. These neighbor-hoods allow the heuristic to escape from locally optimal 1-factorizations.

The new ILS heuristic clearly outperformed the previous heuristics in the literature,improving the best known solution values by at least 11.42% for every benchmark problemafter one second of running time. The average reduction over all feasible instances amountedto 17.17%. Even better results can be obtained if longer running times are accepted.

Acknowledgements Fabrıcio N. Costa is supported by a FAPEMIG grant. Sebastian Urrutia is partiallysupported by FAPEMIG (Edital Universal). Celso C. Ribeiro is partially supported by CNPq research grants301694/2007-9 and 485328/2007-0, and by FAPERJ research grant E-152.522/2006.

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