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1. Assignments are symmetric assignedGame T2 T1 D L2 assignedGame T1 T2 D L1 isTeam T1 isTeam T2 day D location L1 location L2 L1 12 hasGame T1 D is true when T1 plays a game on day D hasGame T1i D game T1 T2 D isTeam T1 isTeam T2 day D hasGameAt T1 D L is true when team T1 plays a game at L on day D hasGameAt T1 D L 13 assignedGame T1 T2 D L isTeam T1 isTeam T2 T1 T2 day D location L hasPlayedNInARowAtOneLoc T1 N D L is true when at the end of day D the previous N games played by T1 were all at location L Initially 0 games have been played putting home here is arbitrary hasPlayedNInARowAtOneLoc T1 0 O home isTeam T1 Still no game has been played hasPlayedNInARowAtOneLoc T1 0 D home isTeam T1 day D not hasGame T1i D hasPlayedNInARowAtOneLoc T1 0 D 1 home If no game was played on day D then simply use the truth value at day D 1 hasPlayedNInARowAtOneLoc T1 N D L isTeam T1 consecLocCounter N day D location L not hasGame T1 D hasPlayedNInARowAtOneLoc T1 N D 1 L If a game was played on day D at the same location as the previous game then we must increment the number of consecutive games played at the appropriate location hasPlayedNInARowAtOneLoc T1 N D L isTeam T1 consecLocCounter N day D 14 location L
2. In the real NHL n 30 teams are separated into two fifteen team conferences where each conference contains three divisions of five teams each In the regular season each team plays x 6 games against divisional opponents y 4 games against other teams in the same conference and z 1 game against inter conference opponents for a total of 82 games each team plays an additional game against three inter conference rival teams Not surprisingly each team plays 41 home games and 41 away games These games are scheduled between early October and early to mid April and so if we account for holidays and breaks such as Christmas and the NHL All Star weekend where no games are scheduled the number of days d is roughly twice the number of games each individual team plays We will not consider creating schedules with more than 30 teams or 82 games per team as our goal is really to allow video games to have smaller schedules than real NHL schedules The NHL considers other issues besides optimizing travel distances and fan attendance when creating a regular season schedule For instance teams are not scheduled to play too many games in consecutive days so that players can rest and a team s home games and away games are typically interleaved to some degree Sports video game schedules should resemble their corresponding professional league schedules to give the user the feeling of realism Therefore we consider similar constraints when creating an NHL
3. E E 30 20 10 F 0 RHS LHS RHS LHS a Example 1 b Example 2 3 base Clasp Smodls EEE 25 F 2 oa oD Bol Bl 8 8 3 3 15 E E E 1 05 0 RHS LHS RHS LHS c Example 3 d Example 4 12 r r lpase EE Clasp E Smodls EEE 10 4 Time seconds O RHS LHS e Example 5 Figure 3 1 The computation times required by lparse to ground our program and by Clasp and Smodels to find a minimum schedule in each of our five examples Note that in Example 3 Smodels does not find a minimum schedule in less than five minutes location L Then given any window of consecutive days of length M we can simply use a cardinality constraint to ensure that it is impossible for a team to play more than G games in the window G 1 hasGame T1 D day D Dmin lt D D lt Dmax isTeam T1 day Dmin day Dmax Dmin lt Dmax Dmax Dmin M 1 atMost G M atMost 2 3 atMost 3 5 atMost 5 8 If we ever want to change the domain of G M in our definition of this constraint our encoding can easily be modified by adding or removing the atMost predicates Note that we could have introduced a LHS version of this instead similar to what was done in Section 3 1 however RHS versions were found to be faster for this constraint and the below constraints as well We omit the details of these experiments here No team plays less than 1 game per 7 consecutive days We can encode this constra
4. Smodels see speed ups in computation time when using the RHS definition and lparse is generally a bit faster at grounding the program with the RHS definition We suspect that this is partially due to the sizes of the grounded programs as the RHS grounding is smaller about 31MB compared to the LHS grounded program which is about 37MB for Example 3 Graphs of these results are displayed in Figure 3 1 below For all future experiments we use the RHS definition and Clasp as this combination performs the best 3 2 Proper Schedules Being able to find minimum schedules in just seconds is a good step towards our true goal as mentioned earlier of finding proper schedules in a short amount of time Since a proper schedule is also a minimum schedule we reuse all of the code in MinScheduleBuilder 1p with the RHS definition from Section 3 1 We then extend our encoding to address each of the six additional constraints presented earlier in Section 2 which we detail below No team plays more than G games per M consecutive days where G M 2 3 3 5 5 8 To simplify this and future constraint encodings we introduce a predicate hasGame T1 D which is true when team T1 plays a game on day D hasGame T1 D game T1i T2 D L isTeam T1 isTeam T2 Ti T2 day D 1 r r 90 r r lase EE i parse EE Clasp E 80 smodels mumm 70 60 a a c c 50 8 8 3 3 40 4
5. T2 day D Facts day 1 d isTeam T team T Conf Div location home location away consecLoc 5 consecLocCounter 0 6 16 References Cla http www cs uni potsdam de clasp IRT78 A Itai M Rodeh and S L Tanimoto Some matching problems for bipartite graphs Journal of the Association for Computing Machinery 25 1978 517 525 Sch99 A Schaerf Scheduling sport tournaments using constraint logic programming Con straints An International Journal 4 1999 43 65 Smo http www tcs hut fi Software smodels SMYMO06 A Suzuka R Miyashiro A Yoshise and T Matsui Dependent randomized round ing to the home away assignment problem in sports scheduling IEICE Trans Fun damentals E89 A 2006 1407 1416 Syr T Syrj nen Lparse 1 0 user s manual http www tcs hut fi Software smodels lparse ps gz Tri02 M A Trick Integer and constraint programming approaches for round robin tourna ment scheduling PATAT 2002 63 77 17
6. hasGameAt T1 D L hasPlayedNInARowAtOneLoc T1 N 1 D 1 L If a game was played on day D at a different location to the previous game then we have only played 1 consecutive game at the new location hasPlayedNInARowAtOneLoc T1 1 D L1 isTeam T1 day D location L1 hasGameAt T1 D L1 hasPlayedNInARowAtOneLoc T1 N D 1 L2 consecLocCounter N location L2 L1 12 No team can play more than N consecutive games at one location where consecLoc N isTeam T1 consecLoc N location L hasPlayedNInARowAtOneLoc T1 N 1 D L day D homeAndHome T1 T2 is true if Ti and T2 are in the same division x gt 2 and T1 and T2 play a home and home series against each other homeAndHome T1 T2 team T1 Conf Div team T2 Conf Div Ti T2 assignedGame T1 T2 D1 L1 assignedGame T1 T2 D2 L2 day D1 day D2 abs D2 D1 1 location L1 location L2 15 Li t L2 x gt 2 If two teams are in the same division and play each other more than twice they play at least one home and home series team T1 Conf Div team T2 Conf Div Ti T2 not homeAndHome T1 T2 x gt 2 matchup T1 T2 D says T1 plays at T2 on day D this is just to simplify output matchup T1 T2 D assignedGame T1 T2 D away isTeam T1 isTeam T2 day D matchup T2 T1 D assignedGame T1 T2 D home isTeam T1 isTeam
7. real NHL scheduling formats In Section 3 we present and explain our implementation and show that it is efficient even for schedules the size of the real life NHL counterparts Finally in Section 4 we conclude our findings and propose some further work 2 League Scheduling A league is made up of n teams where each team belongs to one of c conferences Each conference of teams is further partitioned into v divisions Teams in the same division play x games against each other teams in the same conference but different divisions play y games against each other and teams in different conferences play z games against each other typically x gt y gt z though we do not enforce this For simplicity we assume that each of the c conferences are of equal size and that each of the v divisions are also of equal size otherwise we would need x y and z to be dependent on each division so that each team plays an equal number of games in total In addition for each game one team is denoted as the home team and the other as the away team We require that each team plays an equal number of home games as away games Again for simplicity we will restrict x y and z to be even This way we can enforce that for two teams T and S T is the home team for exactly half of the games against S and vice versa Finally each game of the form S at T thus denoting S as the away team and T as the home team is assigned to one of d days so that no team plays more t
8. such a program is certainly of interest for future work In addition to MLB scheduling there are a number of issues worth further investigation Firstly we expect that our program with a few appropriate changes and additions should be capable of removing the simplifications we imposed in Section 2 In particular we should be able to address the cases where conferences and divisions are unevenly sized and the values of x y and z are not necessarily even This would mean defining x y and z values specific to each division and enforcing teams to play an equal number of total away games and home games plus or minus 1 rather than just for games between each pair of teams Next we could include some special cases to remove or adjust certain constraints when searching for schedules in extreme 11 cases such as the 4 team 82 game case These special rules would look similar to the home and home series condition where we assert that x gt 2 and may allow these extreme cases to be solved in a reasonable amount of time Finally we note here that it seems extremely unlikely but perhaps possible that the stable model found by PSBP1 1p would not lend itself to a proper schedule after reassigning home and away team distinctions In this scenario PSBP2 1p would be unable to find a solution so what should we do We could simply resort to the schedule corresponding to the output of PSBP1 1p as it is nearly a proper schedule or request a new s
9. PSBP1 1p in a format that lparse can understand Thus we write a small simple parser in Java called ScheduleParse java which simply takes the Clasp output from PSBP1 1p assumed to be in a file called games 1p and writes the game T1 T2 D L predicates found in the stable model to a file called parsedGames 1p Thus to run our program for instance on Example 3 we require three commands where again Teams 1p contains the correct number and arrangement of teams lparse c d 164 c x 8 c y 2 c z 2 ProperScheduleBuilderPart1 1lp Teams 1lp clasp 1 1 3 gt games 1p java ScheduleParse lparse c d 164 c x 8 c y 2 c z 2 ProperScheduleBuilderPart2 1p Teams 1p parsedGames 1lp clasp 1 1 3 Then each predicate of the form matchup T1 T2 D see Appendix A in the stable model of PSBP2 1p represents the game T1 at T2 on day D The code for ScheduleParse java is straightforward and is omitted here We run our program on each of our example instances in Table 3 1 and the results are shown in Table 3 2 below To our delight a proper schedule can be found in any example in under three and a half minutes As expected Example 3 requires the most computation time as it is the largest instance However note that Example 4 requires more time than Example 3 lparse plus Clasp to find a solution for PSBP1 1p This is likely because the constraint no two teams play each other more than H times per N consecutive days is more difficult to satisfy
10. Sports Scheduling in Video Games Richard Gibson April 24 2009 Abstract This paper investigates the problem of creating schedules on line for sports video games where the user specifies the number of teams and games played We detail and implement a program using answer set programming which can find such schedules efficiently for a hypothetical NHL video game while maintaining the general format of real NHL schedules 1 Introduction Today there are many sports video games which allow a human player which we call the user throughout this paper to take control of a professional team and play through an entire or abbreviated season of games against computer controlled teams However the options available to the user are often limited and do not allow for fully personalized season schedules For instance there are baseball games where the user may only choose from a few set numbers of games between 14 and 162 In these games there is no ability to choose for instance how many times each divisional opponent is faced or how many inter league games are played let alone the number of teams to use in the season In certain hockey games the situation is even worse in season mode the user may only play the full 30 team 82 game i e each team plays 82 games hard coded schedule that was set out by the real NHL before the video game was shipped This results in long tedious seasons that the user may never finish With today s techn
11. attempt to increase efficiency even further For instance we tried simpler encodings for PSBP1 1p that did not incorporate home or away team distinctions since PSBP2 1p ignore these anyways Unfortunately the results of these efforts were unexpectedly less efficient and thus were abandoned However we do not doubt that there is room for improvement in the efficiency of our program While we only focused on finding schedules for NHL video games we can also address scheduling for video games of other professional sports leagues The NBA basketball schedule is very similar to the NHL schedule the 30 team 82 game NBA regular season schedule takes place between early November and mid to late April where like the NHL teams are split among two conferences of three divisions each This suggests that our program would also be appropriate to use in an NBA video game for on line customizable scheduling However the MLB baseball regular season schedule is a 30 team 162 game schedule from early April to late September This means that the MLB has a higher games per team to days ratio than the NHL and NBA days off for a team are much more rare in the MLB which would make the search for a schedule harder if we adjust the value of d in our program accordingly Also MLB games are usually grouped into three or four game series between the same two teams at the same venue and any scheduling program for an MLB video game should mimic this type of format Creating
12. han one game per day An assignment of all the games in this manner is called a minimum schedule Intuitively the problem of finding a minimum schedule is difficult We can think of this as a restricted matching problem of a bipartite graph here the problem is to find a complete matching ie a matching which covers one of the two vertex partitions M of a bipartite graph B V E such that given a collection of subsets Fj Ep C E and positive integers T1 5 Tk M satisfies M N E lt r for all j 1 k For us we can denote B as the complete bipartite graph where one partition of V is the set of games of the form Game Tj at Tj2 to be played and the other partition is 1 d x 1 n n copies of each day Then for each day 1 d and each pair of games Game and Game i j that have at least one team in common i e Tj1 Ti2 O Tj1 Tj2 gt 1 we create the restriction Eije Game l m Game p m p 1 n with rije 1 Assuming that d is large enough for a minimum schedule to exist this ensures that a solution M to this restricted matching problem directly corresponds to a minimum schedule by assigning Game to day where Game m M for some m 1 n As the restricted matching problem is NP complete IRT78 we can think of finding a minimum schedule in a sense as hard as well Unfortunately this is not enough to assert NP completeness and we have yet to find a proof
13. home series between appropriate teams team T1 Conf Div team T2 Conf Div Ti T2 not homeAndHome T1 T2 x gt 2 Our encoding to this point which we call ProperScheduleBuilderPart1 lp PSBP1 1p is fairly efficient at finding schedules subject to the constraints included The first two columns of Table 3 2 below show the computation times for both lparse and Clasp for each of our five example instances from Table 3 1 The most difficult instance for PSBP1 1lp was Example 4 but lparse and Clasp can still find a schedule in under two and a half minutes However these schedules may not be proper schedules as we have yet to address the constraint that no team plays more than 5 consecutive home games or 5 consecutive away games Unfortunately we could not find any way of extending PSBP1 1p to incorporate this constraint without dramatically increasing computation time for larger instances particularly for Example 3 To ensure that we eventually find a proper schedule we create a second Iparse encoding named ProperScheduleBuilderPart2 1lp PSBP2 1p This encoding takes as input the sched ule found by PSBP1 1p and reconfigures the home and away team assignments for each game until we have a proper schedule The mechanics of this are a little tricky and the full encoding of PSBP2 1p with comments is produced in Appendix A We now have a program capable of finding proper schedules Note that PSBP2 1p requires the output of
14. int simply by adjusting the cardinalities and window sizes of the previous contraint hasGame T1 D day D Dmin lt D D lt Dmax G 1 isTeam T1 day Dmin day Dmax Dmin lt Dmax Dmax Dmin M 1 atLeast G M atLeast 1 7 No two teams play each other more than H times per N consecutive days where H N 2 7 3 14 This constraint is encoded similar to the previous two except that we cannot use the convenient hasGame predicate here Gti game T1 T2 D L location L day D Dmin lt D D lt Dmax isTeam T1 isTeam T2 Ti T2 day Dmin day Dmax Dmin lt Dmax Dmax Dmin M 1 diverse G M diverse 2 7 diverse 3 14 At least one game is played each day Again we keep the left hand side empty hasGame T1 D isTeam T1 0 day D If x gt 2 then teams play at least one home and home series against each team in their division We find it convenient to split this constraint into two parts The first defines a predicate homeAndHome T1 T2 which is true when x gt 2 teams T1 and T2 are in the same division and they play a home and home series against each other homeAndHome T1 T2 team Ti Conf Div team T2 Conf Div Ti T2 game T1 T2 D1 L1 game T1 T2 D2 L2 day D1 day D2 abs D2 D1 1 location L1 location L2 Li 12 x gt 2 We then reject any schedules missing home and
15. largest we wish our program to solve Examples 4 and 5 test cases where we have few teams but many games and many teams but few games respectively Example 4 is a 6 team 62 game schedule and Example 5 is a 24 team 20 game schedule The parameters used for each example are listed in Table 3 1 To run the program for say Example 1 with Smodels we enter lparse c x 2 c y 0 c z 0 c d 12 MinScheduleBuilder 1lp Teams 1lp smodels at the command line n clviz yl z d Example1 4 1 1 2100 12 Example 2 16 2 2 4 2 2 72 Example 3 302 3 8 2 2 164 Example 4 6 1 2 16 100 124 Example 5 24 3 12 4 2 0 40 Table 3 1 The parameters used for each example of our experiments In every example Clasp outperforms Smodels at finding a minimum schedule This is not too surprising since Clasp has special functionality for dealing with cardinality constraints which are included in four definitions of our encoding However it is surprising at how much more efficient Clasp is compared to Smodels for larger cases In Example 3 it takes Smodels about 150 minutes to find a minimum schedule using the RHS definition whereas Clasp using the RHS definition finds one in under two seconds We are not aware of other problems where Clasp outperforms Smodels by this magnitude In general the RHS definition appears to be more efficient than the LHS definition Both Clasp and in general
16. nScheduleBuiler 1p which we use to find minimum sched ules First teams are denoted by predicates of the form team teamName conferenceName divisionName and we list the teams in a separate file Teams 1p Our remaining base predi cates define the allowed days to schedule over which team names are valid and the home away team designations day 1 d isTeam T team T Conf Div location home location away Each possible game in our schedule is represented by a predicate game T1 T2 D L which is true when team T1 plays team T2 on day D with T1 at home if and only if L home We ensure that teams in the same division play each other exactly x times with each team being at home for half of the games with the following definition x 2 game T1 T2 D L day D x 2 team Ti Conf Div team T2 Conf Div location L T1 lt T2 The definitions for intra conference inter division games and for inter conference games using y and z respectively are similar Since a game between two teams is symmetric game T1 T2 D away is equivalent to game T2 T1 D home we find it useful to include the definition below game T2 T1 D L1 game T1 T2 D L2 isTeam T1 isTeam T2 location L1 location L2 L1 12 day D Before our program can find a minimum schedule we need to enforce that no team has more than one game scheduled per day We experiment with two similar definitions f
17. ologies sports video games should allow the user to tweak the schedule specifications to his liking For instance the user should be able to specify the number of teams to use in the season as well as the number of games each team plays In this paper we focus on creating schedules for a hypothetical NHL hockey video game on line in a short period of time We implement a program using answer set programming which finds schedules that both adhere to the user s specifications as well as resemble the format of real NHL schedules The efficiency of our implementation is important if starting a new season in the video game means hours of loading time the user may often be too impatient to wait and thus never even play the season mode We consider about five minutes to be about as long as the average video gamer would be willing to wait for a season to load While other sports scheduling problems have been studied we believe that our particular work here is original Firstly there are studies focusing on scheduling for round robin leagues where all teams play each other an equal number of times and each team plays every round the NHL s schedule is more general as each team does not play every other team an equal number of times In the round robin case one can partition the problem by first finding a graph theoretical arrangement of matches which can be done in polynomial time and then assign teams to the different placeholders so that constraint
18. or doing this The first which we denote as the LHS definition defines a cardinality constraint that ensures that each team has either 0 or 1 game each day O game T1 T2 D L isTeam T2 T1 T2 location L 1 isTeam T1 day D The second denoted the RHS definition simply moves the cardinality constraint in the LHS definition to the right hand side and adjusts the cardinality appropriately 2 game T1 T2 D L isTeam T2 T1 T2 location L isTeam T1 day D Including either of these two definitions in the code above is sufficient for finding a minimum schedule Note that the user specifies the teams to use in the schedule as well as the values of x y and z However the value of d the number of days to schedule over is up to us to determine So what should d be As mentioned in Section 2 the real NHL uses a value of d that is about twice the number of games each individual team plays Since our video game schedules should look similar in format to the real NHL schedules we set d to exactly twice the number of such games in all of our experiments We compare the efficiencies of Smodels and Clasp at finding minimum schedules for five different examples Example 1 is a small 4 team 6 game schedule which is the smallest sized schedule we wish our program to find see Section 3 Example 2 is a medium sized 16 team 36 game schedule and Example 3 is a large 30 team 82 game schedule the
19. s are satisfied and objectives are maximized which is NP hard Sch99 Second we are unaware of any work regarding scheduling for video games Typically in professional sports leagues like the NHL as well as in sports video games each game has a home team and an away team where the game is played at the home team s venue Because of this in the real world there are travel costs to minimize and fan attendance to maximize as well as many other additional constraints Thus the real life scheduling problem is truly an optimization problem which often takes several months of work just to find an acceptable sub optimal solution Some work in this area includes SMYM06 where the cost problem is studied by separating the assignment of matches from the home and away assignments for those matches For video games we are simply looking for a feasible solution rather than an optimal solution Since issues such as travel costs do not exist in the virtual world we do not have to worry about optimizing an objective function but rather just that the arrangement of matches be valid and interesting Finally while integer programming Tri02 and other constraint programming approaches Sch99 have been considered we are not aware of any uses of answer set programming for sports scheduling The remainder of this paper is organized as follows Section 2 introduces the parameters of the schedule that the user may change and the constraints we apply to mimic
20. table model from PSBP1 1p or even increase the number of allowed consecutive home or away games for one team However this scenario appears extremely unprobable as it was never encountered in any of our experiments and since we already have some strategies for dealing with this unlikely event it should be considered low priority Appendix A Encoding of ProperScheduleBuilderPart2 1p Below is the encoding used for ProperScheduleBuilderPart2 1p see Section 3 2 Remove the old location assignment game T1 T2 D game T1 T2 D L isTeam T1 isTeam T2 day D location L The assignedGame predicate has the same meaning as the game predicate from the first part of this program assignedGame T1 T2 D L is true only when game T1 T2 D is true and when T1 is assigned to be the home away team where L home away x 2 assignedGame T1 T2 D L game T1 T2 D x 2 team Ti Conf Div team T2 Conf Div location L T1 lt T2 y 2 assignedGame T1 T2 D L game T1 T2 D y 2 team T1 Conf Div1 team T2 Conf Div2 location L 12 Ti lt T2 Divi Div2 z 2 assignedGame T1 T2 D L game T1 T2 D z 2 team T1 Conf1 Divi team T2 Conf2 Div2 location L T1 lt T2 Conf1 Conf2 Each game must be assigned to exactly one location 1 assignedGame T1 T2 D L location L 1 game T1 T2 D isTeam T1 isTeam T2 day D T1 lt T2
21. ur goal in this paper is to find a proper schedule for a league in a short length of time where the size and length of the league is determined by the user In particular the user specifies the values of n c v x y and z The user also determines which n teams to include in the league as well as their arrangement into c conferences and v divisions We make one more simplification by assuming that n gt 4 and that each team plays at least 6 games This is done since proper schedules of unusually small size may not exist for example there is no proper schedule for the values n 2 c v 1 x 4 and d 8 because the two teams cannot play each other more than 3 times per 14 consecutive days 3 Implementation and Experimental Results We use answer set programming to find NHL video game schedules All of our implementations use lparse version 1 0 5 Syr as a front end for the answer set solvers used In Section 3 1 we compare Smodels version 2 26 Smo and Clasp version 1 1 3 Cla at finding minimum schedules of various sizes using two different encodings before tackling proper schedules in Section 3 2 All of our tests were run on a 2 00 GHz processor with 4 GB of memory We note here that our implementations were tested with the newer versions of lparse 1 1 1 and Smodels 2 33 on another machine and were still found to give correct solutions 3 1 Minimum Schedules Smodels versus Clasp Here we describe our lparse code Mi
22. video game schedule e No team plays more than G games per M consecutive days where G M 2 3 3 5 5 8 This ensures that a team s games are not too bunched together as player fatigue may have some effect in the video game e No team plays less than 1 game per 7 consecutive days This constraint helps to spread out a team s games across all of the days in the schedule e At least one game is played each day This further helps to spread out the games across all of the schedule s days e No two teams play each other more than H times per N consecutive days where H N 2 7 3 14 It is often less fun to play the same team repeatedly and the fun factor is certainly important for us e No team plays more than 5 consecutive home games or 5 consecutive away games Similar to the previous constraint it is more fun to alternate between home games and away games to some degree rather than playing for instance all the home games before the away games e If x gt 2 then teams play at least one home and home series against each team in their division A home and home series is a pair of games between two teams over 2 consecutive days where each team is at home for one of the two games Real NHL schedules often include home and home series between rival teams so we do the same for our video game schedules Any minimum schedule that also satisfies the listed constraints above is called a proper schedule O
23. when the number of teams is small and the number of games is large Some additional experiments have shown that the obscure 4 team 82 game case has unacceptably long computation times for PSBP1 1p over five minutes For now we recommend that a video game not allow the option of a league with such a large number of games when the number of teams is very small We address this further in Section 4 10 Iparse Clasp lIparse Clasp Total PSBP1 lp PSBP1 lp PSBP2 1p PSBP2 1p Example 1 0 062 0 031 0 093 0 0 326 Example 2 5 413 1 904 2 792 0 375 18 455 Example 3 62 259 43 057 44 007 60 451 209 774 Example 4 1 747 135 814 0 936 0 187 138 684 Example 5 5 366 60 825 2 964 0 250 69 405 Table 3 2 The computation times in seconds required for lparse to ground our encodings and for Clasp to find a stable model of the grounded instances The stable model found by ProperScheduleBuilderPart2 1p PSBP2 1p is a proper schedule and the total time required to find a proper schedule for each example is given 4 Conclusion We have successfully shown that NHL video game schedules with various numbers of teams and games can efficiently be found on line In addition answer set programming using Iparse and Clasp were sufficient for this task and we argue that future NHL video games should incorporate schedule customization While our encodings presented require reasonably short computation times we did

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