Deermnng he Number of Games Needed o Guaranee an NHL Playoff Spo Tyrel Russell and Peer van Bee Cheron School of Compuer Scence Unversy of Waerloo {crussel,vanbee}@uwaerloo.ca Absrac. Many spors fans nves a grea deal of me no wachng and analyzng he performance of her favoure eam. However, he ools a her dsposal are prmarly heursc or based on fol wsdom. Ths paper provdes a concree mechansm for calculang he mnmum number of pons needed o guaranee a playoff spo n he Naonal Hocey League (NHL). Along wh deermnng how many games need o be won o guaranee a playoff spo comes he noon of mus wn games. Our mehod can denfy hose games where, f a eam loses, hey no longer conrol her own playoff desny. As a sde effec of hs, we can also denfy when eams ge lucy and sll mae he playoffs even hough anoher eam could have elmnaed hem. Inroducon Hocey fans are neresed n nowng when her eam clnches a playoff spo. Ths problem can be modelled as a sasfacon problem and solved usng consran programmng []. However, f he eam has no clnched a playoff spo, hs mehod provdes no nformaon abou how close a eam s o earnng a playoff poson. The problem of deermnng how close a eam s o clnchng a playoff spo can be modelled as an opmzaon problem ha deermnes he mnmum number of pons ha s necessary o guaranee a spo. Ths bound on he number of pons can also be used o deermne when a eam has no guaranee of mang he playoffs and when a eam has los a crucal game and lef desny n he hands of anoher eam. These facors are neresng o hocey fans and can be generalzed o oher spors wh playoff srucures, such as baseball and baseball. We solve he problem usng a consran model and a mxure of echnques from consran programmng and operaons research ncludng newor flows, consran propagaon and domnance consrans. We decompose he problem so ha we can use a mul-sage approach ha adds consrans a each sage f no feasble soluon s found. The NHL deermnes posonng by pons and, f ed by pons, by several e breang condons. The sages of he solver correspond o he exra consrans needed o deermne he exra e breang condons. The frs sage uses a combnaon of enumeraon and newor flows o deermne a gh lower bound on he pons needed and wheher here exss
a feasble soluon usng only pons as a creron. The second sage also uses newor flows o chec he frs e breang condon. The hrd sage uses a bacracng consran solver o deermne f here are any soluons usng he second e breang condon. Our solver can deermne he mnmum number of pons for a gven eam, a any pon n he season, whn en mnues and, for daes near he end of he season, n seconds. In spors, analyss, reporers and coaches ofen refer o mus wn games. The mehod used n hs paper can denfy games where losng ha game, he eam pus s playoff aspraons no he hands of s opponens. Whle hs does no mean he eam wll no qualfy for he playoffs, does mean ha nohng he eam does guaranees a playoff spo. We denfy nne eams n he 006-07 season ha los one of hese mus wn games and found hemselves n a poson o earn a playoff spo agan hrough he acons of her opponens. Several eams experenced hs phenomenon four mes durng he season. In he remander of he paper, we gve a bref descrpon of he mechancs of playoff qualfcaon n he NHL. Some relaed wor s presened n Secon 3. A formal defnon of he opmzaon problem for deermnng he mnmal number of pons needed o guaranee a eam maes he playoffs s presened. Aferwards, we nroduce he concep of an elmnaon se and explan how hs s used o deermne gh lower bound values on he opmal value. Snce he bound requres he relaxaon of e breang condons, we dscuss how we can rencorporae hose condons usng he same ses. Fnally, he specfc consran model s nroduced along wh some opmzaons n he form of exploed domnance and combned conssency checng. The NHL Playoff Sysem Snce he las expanson of he league n 000, he NHL has conssed of hry eams arranged no wo conferences of ffeen eams. Each conference s broen no hree dvsons wh fve eams each. Each eam n he NHL plays eghy wo games wh 4 home games and 4 away games. Egh eams mae he playoffs from each conference and o earn a playoff poson a eam mus eher be a dvson leader or one of he fve bes eams n her conference no ncludng dvson leaders, called wld card eams. Posonng n he NHL s deermned by one prmary creron and several secondary e breang crera. The prmary creron s he number of pons earned by a eam. The wo common secondary crera are oal wns and pons earned agans eams wh he same number of pons and wns. A hrd e breang crera, whch s he number of goals scored on opponens, s somemes used o brea es beween eams ed applyng he frs hree crera. Le many Norh Amercan spors, an NHL game mus end n a wn or loss. However, he NHL has a unque scorng sysem as here are pons awarded for reachng overme even f a eam does no wn he game. The game s separaed no hree weny mnue perods and, f eams are ed afer sxy mnues, a fve mnue overme perod. If eams are ed afer overme, here s a shooou
o decde he wnner. If he game ends durng regulaon me he frs sxy mnues hen he wnner of he game s awarded wo pons and he loser earns no pons. If, however, he game ends eher durng he overme perod, whch s sudden deah,.e. ends when a goal s scored, or he shooou, whch mus conclude wh a wnner, hen he wnner sll earns wo pons bu he loser earns a sngle pon n consolaon. 3 Relaed Wor Russell and van Bee [] creaed a consran programmng model for calculang wheher a gven eam had clnched a playoff spo. However, he opmzaon of he decson model requres a relaxaon of he domnance consrans. Ths relaxaon of he consrans along wh he ncreased search space leads o a sgnfcan ncrease n he execuon me of he solver such ha relavely lae season nsances could no be solved whn a day. Approaches for deermnng he mnmum number of games needed o guaranee a playoff spo have been proposed for he Brazlan fooball champonshp [] and n Major League Baseball [3]. These spors eher have a smpler scorng model or a smpler playoff qualfcaon mehod. Robnson [4] suggesed a model for he NHL bu hs model dd no allow for wld card eams or e breang condons. As well, only a heorecal model was presened whou expermenal resuls. Gusfeld and Marel [5] show ha hs problem s NP-Hard. They pu forh a mehod for calculang bounds on when a eam has been elmnaed from he playoffs bu her mehod only wors for a sngle wld card eam and a smple wn-loss scorng model. Our echnque uses a smlar mehod bu dffers n approach as we mus accoun for mulple wld card eams. Wayne [6] nroduced he concep of a consan ha could be used o deermne wheher or no a eam was elmnaed from he playoffs. Specfcally, he nroduced he concep of lower bound consan W whch denoed he mnmum number of pons needed o earn a playoff spo. Gusfeld and Marel [5] show how hs dea can be exended o nclude a sngle wld card eam and mulple dvson leaders. In hs paper, we wll also dscuss he exsence of an upper bound consan whch represens he mnmum number of pons needed o guaranee a playoff spo. 4 A Formal Problem Defnon To defne he problem formally, ceran conceps and noaons mus frs be nroduced. We denoe he se of eams n he NHL as T. We denoe he conference ha eam belongs o as C and he dvson ha eam belongs o as D. Table a shows he dfferen me varables ha we use o superscrp he oher feaure-based varables. Table b shows he dfferen feaures n he NHL and he noaon for each feaure based on he number and ype of opponen. For each nsance, here s a schedule and a dae d 0 along wh he resuls of games pror o d 0. We defne a scenaro S o be a compleon of he schedule from d 0
Dae Noaon Curren d 0 End d e Generc d Feaure Pons Wns w d j Overme Losses ol d j Games Remanng g d 0 j vs. j vs. Oppose Conference Toal p d,j ocp d p d ocw d ocol d ocg d w d ol d g d (a) (b) Table. Varable Noaon (a) The varables represenng he dfferen daes under consderaon. (b) The varables represenng he curren sae of he resuls a a gven me d. by assgnng wns, losses, and overme losses o he games scheduled afer d 0. We refer o he maxmum possble pons ha could be earned by a eam f hey won all of her remanng games from a gven dae d as mpp d. We refer o he maxmum pons over all eams T T a a gven me d as max d (T ) and he mnmum pons over all eams T T a a gven me d as mn d (T ). A eam only qualfes for a playoff spo f hey are a dvson leader or a wld card eam. We defne a dvson leader o be he eam ha has he maxmal pons a he end of he season whn her own dvson (.e. p de = max de (D )) and has beer e breaers han any eam wh equvalen pons n her dvson a me d e. We defne a wld card eam o be any eam ha s no a dvson leader bu has a p de greaer han a leas seven oher eams n s conference ha are also no dvson leaders. Gven eam, a gven dae of he season, d 0, a gven schedule of remanng games and gven resuls up o d 0 n he season, a Playoff Opmzaon Problem s o deermne he mnmal number of pons a he end of he season, p de, such ha here exss no scenaro where does no qualfy for he playoffs as eher he leader of he dvson or one of he fve wld card eams. Noe ha we refer o he gven eam as eher he elmnaon eam or smply for he remander of he documen. 5 Soluon Overvew In hs secon, we provde an overvew o he solver ha we use o solve he opmzaon problem. In order o solve all nsances, we use a mul-sage solver ha apples dfferen echnques a each sage. In he frs sage, we enumerae all of he feasble elmnaon ses of eams (see Sec. 6) and derve a gh lower bound for he number of pons needed. If p d0 s greaer han he bound hen hey have already qualfed and f mpp d0 s less han he bound hen hey can no longer guaranee. If he bound falls n beween hose values hen wh each se
ha obans our lower bound, we chec each e brea condon o deermne f hs lower bound s a feasble number of pons o guaranee a playoff spo. The second sage checs o see f he frs e brea condon, wns, s enough o elmnae. We do hs by enumerang he possble wn values and checng hem wh a feasble flow algorhm (see Sec. 7). If here s no feasble soluon usng only pons and wns as crera, he hrd sage agan uses a feasble flow algorhm o chec f here are any ses where eams are ed n boh pons and wns (see Sec. 7). If here exss feasble e breang ses, we use a bacracng consran solver o deermne f one of he ses can elmnae (see Sec. 8). If here are no soluons a hs pon hen can guaranee a playoff spo f hey earn enough pons o reach he bound. Oherwse, he soluon o he problem s one greaer han our lower bound. 6 Generang and Boundng he Elmnaon Ses In hs secon, we defne elmnaon ses and presen a mehod for deermnng a bound on he pons achevable by ha se. To calculae he lower bound, we generae all ses of egh eams ha could compose he hree dvson leaders and fve wld card eams. Each of hese ses has he poenal o elmnae a some pon bound. The larges bound over all of hese ses forms a gh lower bound on he soluon o our problem, dfferng by a mos one pon. We defne an Elmnaon Se, E, as a se of egh eams from he same conference wh a leas one eam from each dvson and does no nclude. For each eam, hey mus eher have mpp d0 from a dvson D such ha D D. > p d0 or be he only eam n he se We defne he bound of an elmnaon se, E, as he max (mn de (E)) under all scenaros S where eher p de = mn de (E) or p de = mpp d0. The maxmum bound over all elmnaon ses s a gh lower bound on he soluon o he complee problem dfferng by a mos one pon. 6. Calculang he Bound To calculae he bound for a gven elmnaon se, we adap an dea by Brown [7] usng erave max flows o solve a sharng problem. We mplemened a smlar algorhm ha shares he games beween he eams so ha he wors eam n he se has he mos pons possble. By consrucng a flow graph ha allows us o deermne a feasble share, we erae unl a vald dsrbuon of games s found. We sar ou wh a possble bound and deermne s feasbly. If he bound s no feasble, we updae he bound and chec he feasbly of he new bound. In order o fnd he bes bound, eams wn as many pons as possble. Ths means ha every loss by a eam n he elmnaon se s an overme loss and eams n he elmnaon se wn all of her games agans eams ha are no excep. We formalze he pons earned by a eam under hs suaon as,
(a) (b) (c) (d) Fg.. The relaxed bound algorhm. (a) The orgnal problem wh 6 games remanng. The sold recangles represens p d 0 and he dashed recangles represens mpp d 0. The frs sep s o sor he eams by p d 0. (b) shows he sored eams wh he mn (mpp d 0 ) shown as he sold horzonal lne. (c) We assgn games o he eams wh he leas pons. In hs case, one game o he frs eam. (d) In he nex eraon, four eams need games and we allocae four of he remanng fve games. The bound s reached and he fnal soluon has one game remanng. p = p d0 + ocg d0 + j / E {} g d0 j + j E {} Equaon () represens he sum of he pons already earned (p d0 agans eams no n he se E {} (ocg d0 + j / E {} gd0 j g d0 j. () ), he wns ) and one pon each from games agans eams n E {} ( j E {} gd0 j ). These preprocessng seps are vald domnance relaons as we are loong for he scenaro where we ge he maxmum mn de (E) and hese seps eher ncrease he pons of a eam n E or leave hem he same whle no affecng he maxmum possble pons of he eams n E. 6. The Relaxed Bound To deermne he sarng pon for he lower bound, we solve a relaxaon of he bound calculaon where we relax he consran ha a specfc number of games mus be played beween wo eams. Insead, we consder all games as a pool of unplayed games wh no assgned opponens and assgn hem o he wors eam unl he games are used or he mn E (mpp d0 ) s reached. Fgure shows an example bound calculaon. 6.3 The Flow Newor and he Bound Once we have a sarng pon calculaed by he relaxed bound algorhm, we are loong o fnd he frs feasble bound when we nclude he consrans removed
bound InfeasbleBound(E,); repea Needs CalculaeNeeds(E,,bound); need P E n; G ConsrucGraph(Needs); flow CalculaeFlow(G); f flow < need hen bound bound ; unl flow need ; reurn bound Algorhm : Ths algorhm shows he seps for calculang he bound for a gven elmnaon se, E. Frs, we generae an nfeasble bound as a sarng pon. From ha sarng pon, we generae, for each eam, he number of pons needed o reach he bound, denong he se of needs as Needs. Then we chec feasbly usng a flow algorhm. If he flow mees he needs, we reurn he bound. Oherwse, we reduce he bound and erae. durng he nfeasble calculaons. We formulae hs as a feasble flow problem [8] wh an arfcal sn and source. These graphs loo smlar o he graphs consruced by Schwarz [9] n hs paper on baseball elmnaon. Every eam n he elmnaon se and he eam needs o wn a ceran number of games o reach he bound and hs mus be ncorporaed no he graph. We defne he need of a eam, n, o be bound p (where bound s he curren lower bound on pons and p s defned n ()). The excepon o hs rule s where he bound may be greaer han mpp. In ha case, we calculae n as mpp p. We use he p values snce we are sll loong for bes case resuls for he se E {}. A bound s feasble f he maxmum flow n he graph s equal o he sum of he needs of he eams n he elmnaon se. If no, a new bound mus be red. Algorhm descrbes he process by whch he bound s calculaed. We denoe he maxmum bound calculaed by he algorhm over all elmnaon ses as p and prune any se ha does no reach ha bound. Once we now he needs for a gven elmnaon se, s relavely smple o consruc he graph. An example graph s n Fg. showng he varables and he assocaed capaces n he graph. We creae wo nodes s and o be he source and sn, respecvely. We add one node for each par of eams n he se and one node for each eam n he se. On op of hs, we add an exra node ha represens games played agans by any eam n E. Each node represenng a par of eams has hree edges where one s an ncomng edge from s wh a capacy equal o he number of games beween hose wo eams g d0 j and wo are ougong edges o he nodes for he eams wh he same capacy as he ncomng edge. There s also an edge from each node represenng a eam n he se o he sn node wh a capacy equal o he need of he node. Las, he node represenng he games agans has an edge from he source wh a capacy of g d0 n and one ln each o every eam node wh a capacy equal o he number of games played beween hem.
, w w 3, g g 3 g g s 3, 3 g3 g n, 3 w w w 3 w 3 w 3 w 3 [n, g ] [n, g ] [n 3, g 3] g g s 3, 3 g3 g n, 3 g g g 3 g 3 g 3 g 3 n 3 n n w 3 g 3 (a) (b) Fg.. (a) A varable represenaon of he values n he flow graph for a hree eam elmnaon se ({,, 3}). The mplemenaon of he graph uses he doman of he varables as he capacy bounds. (b) The capaces o deermne feasbly for he flow graph. All varables are gven her doman maxmum. Snce he flow s spl n nodes (, ), (, 3) and (, 3), a feasble flow s a vald assgnmen. If he max flow can saurae he needs of he eams hen here s a feasble soluon for hs elmnaon se. 7 Wn Values and Te Breang Ses In hs secon, we descrbe how wn values are used o deermne f an nsance has a soluon and how o generae feasble e brea ses. Once we have a pon bound and se of eams ha could poenally reach ha bound, we deermne he possble values for he secondary crera and only solve feasble nsances. Ths means ha we deermne f he elmnaon se can elmnae wh only wns or wheher some eams mus be ed. We use a modfcaon of he orgnal flow problem o deermne boh of hese quanes. Frs, observe ha f a eam earns p pons hen we have he followng consran, (p de = p) (p p d0 ) g d0 w de w d0 (p p d0 ). () Ths consran represens ha any eam wh p pons a he end of he season would have earned he fewes exra wns f every loss was an overme loss hus earnng a leas one pon per game and he mos exra wns when hey wn as many games as possble whle sll only earnng p pons. Ths consran also holds rue for so we can deermne a feasble range of wns for he elmnaon eam gven he elmnaon se, E, and he pon bound p. For each possble number of wns w for, we deermne f he se can elmnae wh ha number of wns and, f no, whch ses of eams can be ed. Boh of hese ass can be solved by checng for feasble flows on he same graph usng slghly dfferen needs n each case. We modfy he graph for calcu-
g n [0, g 3] [g, g ] s, [0, g 3] [0, g ] [0, g ] [n, g ] [n, n ] g v, n + n g g n g3 g 3 s n g w (a) (b) Fg. 3. (a) A newor flow graph wh hree eams. Team has a lower bound consran on he number of wns and s n he elmnaon se and no n he e brea se, eam s n he e brea se and has a fxed number of possble wns, and eam 3 s n neher he elmnaon se or he e brea se and has no bounds on eher pons or wns. (b) A flow graph ransformed o remove he lower bound capaces. Two addonal nodes are added v and w. A feasble flow exss n he orgnal graph f he maxmum flow s equal o he sum of he lower bounds on he orgnal graph (n + n + g ). lang pon bounds by allowng as a proper eam on he rgh hand sde and addng lns drecly o he nodes for games ousde he se (see Fg. 3a). Games agans opponens ousde E {} do no have o be won by any eam n he se so hey have no lower bound bu hose n he se mus be won by one of he eams; herefore, hey have a lower bound. We consruc he graph so ha each eam ha mus be ed wh n wns has a lower and upper bound equal o her need. The need calculaon for hs graph s dfferen han he pon graph. Snce boh pons and wns are boh fxed, we ge he followng equaon for calculang need, n = w w d0 f (p de = p) (w de = w) ) g d0 < 0. (3) ) g d0 f (p p d0 ) g d0 + w d0 > w ) g d0 + oherwse 0 f (p p d0 (p p d0 (p p d0 Each condon of Equaon (3) represens he number of wns needed o elmnae n he bes case scenaro usng as few wns as possble. The frs condon denoes ha he elmnaon eam mus have exacly w wns. The second condon denoes ha eams ha would have equal or more pons usng only exra pons from overme losses do no have o wn any more games. The hrd condon ensures ha eams ha wn he mnmal number of games o reach p have more wns han and elmnae. The fourh condon correcs he number of wns needed when he second and hrd condon do no hold by addng an addonal wn o he mnmal number of wns. For eams ed wh n wns, we nroduce a e brea se. We defne a Te Brea Se as any subse
of he eams n C where every eam can reach boh he pon bound p and he wn bound w exacly. We es all subses by seng he need of eams n he e brea equal o w w d0. Snce our graph has mnmum and maxmum capaces on he edges, we ransform he graph no a dfferen max flow problem as descrbed by Ahuja e al. [8]. The ransformaon can be seen n Fg. 3. Once we have checed he wns e breaer wh he flow graph and deermned whch ses of nodes can be ed n boh pons and wns, we deermne f any of hose e brea ses can elmnae wh pons agans eams ha are ed. We model hs problem as a sasfacon problem and solve usng bacracng search as descrbed nex. 8 The Decson Problem In hs secon, we descrbe he consran model used o deermne f he fnal e breas elmnae. Once we have fxed he elmnaon se (E), pon bound (p), wn bound (w) and e brea se (T B), we verfy hs combnaon elmnaes he eam. We examne possble scenaros of wns (w,j ) and overme losses (ol,j ) as hese are he wo facors ha affec he pons and hence he oucome of a gven scenaro. We brea he eams no four muually exclusve classes o help descrbe our model. 8. The Model A = { E / T B} C = { / E T B} B = { E T B} D = { / E / T B} There are four major consrans o hs model. Each of whch s modfed slghly dependng on whch class a gven eam belongs. Consran (4) represens he consran ha each of he eams mus eher mee or exceed he bounds dependng on her class. These rules are derved from he NHL e breang rules. Consran (5) represens he consran ha each game mus have a wnner. The excepon o boh of hese consrans are hose eams n D. These eams are no resrced by he bounds and hus we can gnore any game where hey are playng oher eams n D. We also mus consran he number of overme losses so ha he eam does no earn more overme losses han losses. Ths consran s refleced n (6). Lasly, we mus deal wh consrans on games played agans eams n he oppose conference. Teams n A can wn hese games freely, eams n D can lose hem freely and eams n B and C can wn hem dependng on he consrans appled n (4). We defne hese consrans explcly n (7). j T B w de j + j T B p de ol de j > p (p de p de > j T B p de = p w de > w) f A. = p w de = w w de j + j T B ol de j f B. = p w de = w f C. (4)
( j A w de j ocw de ocw de j w de j + wde j = g j f / D. = wd0 j wde j = wd0 j + gd0 j ( j B C w de j + wde j = wd0 j + gd0 j j j ( j D w de j = ocw d0 + ocol de = wd0 j wde j = wd0 j ) f D. (5) w de j + olde j = wd0 j + old0 j + gd0 j f A. w de j + olde j wd0 j + old0 j + gd0 j f B C. ocw de + ocg d0 ocw d0 = ocw d0 j ocol de ol de j + ocol d0 ocol de = old0 j f D. (6) = ocol d0 f A. + ocg d0 f B C. = ocol d0 f A. (7) 8. Updang Domnance Durng Search As he search progresses, s ofen possble o force he assgnmen of ceran games. The mos mporan domnance s o noce ha only wn varables whn he e brea and elmnaon se mus be se va search. Once hose varables have been se, all ha remans s o ensure eams mee or exceed p and w and o mae sure eams ryng o bea earn as many of her necessary overme losses whn he e brea se and wns as many of hem as possble ou of he e brea se. These domnances lead o a correc soluon and maes sure eams n B earn as many pons whn he e brea as possble. Anoher opporuny s when eams have sasfed (4). Specfcally, once a eam n A has me he condons of (4), hey may gve pons o oher eams n A whou any consequences. Anoher domnance s ha once a eam n he e brea se has acheved boh p and w hey mus lose any remanng games n regulaon me. 8.3 Prunng Values from Consraned Teams va Flow Manpulaon As menoned n Sec. 7, he feasbly of he e brea se depends on wheher here exss a max flow equal o he needs of he eams n he flow graph represened by Fg. 3. An mporan observaon ha can be made s ha any feasble flow s a vald assgnmen of he wn varables of he eams n he elmnaon and e brea ses. We can prune he varables whn he solver by aempng o updae an already exsng flow o conan a specfc es value usng a mehod adaped from Maher e al.[0]. If here s a flow ha conans he value hen here s a suppor for ha value and ha value s ep. If no, hen we prune he value from he doman of he varable. The dea s smlar o he dea used n he Ford-Fulerson algorhm. However, n our case, we are ryng o fnd a pah no from s o bu from j o. We mus repea he updae a mos d mes where d s he sze of he doman of w j and w j.
0/3 0/ 0/3 v, 0/4 0/ 0/3 0/ 0/ 0/3 w v, 3 4 3 w 0/ 0/ s (a) 4 (b) s Fg. 4. (a) An example flow graph for hree eams where Team mus earn beween and 3 games, Team mus earn exacly games and Team 3 s unbounded. (b) The resdual graph conanng a maxmum flow. 4 + 4 4 4 s s s (a) (b) (c) (d) s Fg. 5. Reduced Prunng Graph. (a) shows he reduced resdual graph of Fgure 4. In (b), we reduce he ln beween nodes and and ncrease he ln beween nodes and, whch ensures he consran ha he flow beween hem equals some muual capacy. (c) shows he pah ha s found from node o node correcng he mbalance. Once a pah s found, he flow s redreced and he oppose edges are updaed by he change. (d) shows he new sable soluon showng suppor for he assgnmens of w = and w =. To reduce he praccal complexy of he algorhm, we reduce he resdual graph o only hose componens ha wll be updaed. In a graph conanng a feasble flow, he edges ou of v and no w are compleely sauraed. Snce any modfcaon mus also be a feasble flow, hese edges mus reman sauraed and any modfcaon should no aler hese edges. The oher reducon ha we can mae o he graph depends on he symmery beween nodes represenng eams and he lns o her mached games. Ths allows us o remove he nodes represenng he mached games and ln he nodes drecly ogeher. Example. Examne Fg. 4b and noe ha n he resdual graph lns no v and ou of w are sauraed and can be removed. Also observe ha he edge from node o node (, ) s he same as he edge from node (, ) o node. Therefore, we can remove node (, ) and drecly ln (, ). Fgure 5 shows he reduced prunng graph along wh a sngle varable updae.
Solver Sage & Resul Number of Insances (/5430) Solved va Enumeraon Solved va Wn Checs (Posvely) 49 Solved va Wn Checs (Negavely) 54 Solved va Bacracng Search (Posvely) 338 Solved va Bacracng Search (Negavely) 07 Table. The couns of problems solved va he varous sages of he solver. Posvely solved nsances means a soluon was found and he bound mus be ncreased. Negavely solved nsances means ha bound was vald for ha nsance. Any problem whou a defnve soluon was passed o he nex phase of he solver. Torono Psburgh 60 40 Bound Pons Max Pons 60 40 Bound Pons Max Pons 0 0 Pons 00 80 Pons 00 80 60 60 40 40 0 0 0 40 60 80 00 0 40 60 80 (a) Days 0 40 60 80 00 0 40 60 80 (b) Days Fg. 6. The mnmum number of pons needed by Torono and Psburgh o guaranee a playoff spo n he 006-07 NHL season. 9 Resuls We mplemened he solver n C++ usng he Boos Graph Lbrary [] for he feasble flow calculaons and ILOG Solver[] o solve he fnal consran model. In order o es our solver, we used he 006-07 season resuls o calculae he mnmum pons needed o clnch a playoff spo. Table shows he resuls of hose calculaons. In oal, deermnng he bound for all 30 eams on all 8 game days of he 006-07 NHL season (5430 problems) oo a lle over 46 hours. Each ndvdual nsance, represenng a eam a a gven dae, oo less han en mnues o calculae he bound and hose problems near he end of he season, where he resuls maer he mos, were calculaed n seconds. We noe ha our enumeraon echnques solves of he 5430 of he problems and when we add frs level e breang wh wns we solve a furher 3773 problems, whch maes
Feaure Value Team(s) Earles Day where a Team could 64 days S. Lous no Guaranee Mos Days where a Team could no 8 days S. Lous Guaranee Mos Tmes a Team go Lucy 4 Torono, Boson and NY Rangers Number of Teams ha go Lucy NY Islanders and NY Rangers and Earned a Spo Number of Teams ha go Lucy bu Faled o Earn a Spo 7 Torono, Boson, Washngon, Carolna, Edmonon, Phoenx and Columbus Table 3. Shows some of he feaures ha can be hghlghed by calculang he mnmum number of pons needed o guaranee a playoff spo. up abou 9% of he problems. However, he remanng 8% problems requre a bacracng consran solver o calculae he fnal number. Also, noe ha n 47% of he oal nsances, whch amouns o 6% of he nsances no solved drecly by enumeraon, he answer dffers from he nal lower bound gven by enumeraon. We plo he resul agans boh he curren pons of he eam and maxmum possble pons of he eam. If he resul s greaer han he maxmum possble pons, hen he eam s no longer able o guaranee a playoff spo. If he resul s equal o he number of pons needed by he eam hen ha eam has clnched a playoff spo. Fgure 6 shows he resul calculaed for Torono and Psburgh. Noe ha Torono dd no mae he playoffs because hey never reached he bound value. Also noe ha Torono placed hemselves n a poson where hey could no guaranee a playoff spo and go lucy four mes. In oher words, hey los a mus wn game fve mes durng he 006-07 season whle Psburgh was never n ha suaon. Anoher neresng feaure s ha we can see, n boh graphs, he bound on pons, 45, needed a he sar of he season o guaranee a playoff spo. Table 3 shows an overvew of he resuls of he 006-07 NHL season n erms of he mnmum pons needed o guaranee a playoff spo. One neresng observaon ha can be made from hs able s ha of he nne eams ha go a second chance only wo of hose eams ended up earnng a playoff spo. As well, of hose seven eams, wo of hem had four chances o mae he playoffs afer losng a mus wn game. Anoher neresng noe s ha S. Lous could no guaranee a playoff spo afer only he sxy-fourh game day and never recovered durng he fnal one hundred and egheen game days. 0 Concluson As he season wnds down, he fans of he NHL are neresed n nowng how far her eam s from clnchng a playoff spo. We presen a mehod for calculang
he mnmum number of pons ha mus be earned n order o ensure ha he eam reaches a playoff spo. We preform hs calculaon effcenly by usng a mul-sage solver ha combnes enumeraon, flow newor calculaons and bacracng search. A sde effec of hs calculaon s he ably o deermne when he eam s n danger of losng conrol of her desny. These games, ofen descrbed by coaches as mus wn games, can be denfed by her loss reducng he maxmum possble pons o below he bound of he eam. We denfed nne dfferen eams n he 006-07 NHL season ha los conrol of her fae and hen ganed ha conrol bac hrough msaes by her opponens. We also noed ha only wo of hese eams oo full advanage of hs suaon and clnched a playoff spo. Our solver used a decomposon of he problem o allow us o effecvely apply several dfferen sraeges n several sages o ensure a quc soluon o he problem. The resuls of hs wor could be appled o oher spors. One area ha seems o be mssed enrely s baseball, especally NBA baseball, where ha league shares many smlares wh he NHL. The e breang condons vary slghly and he NBA uses a smpler scorng model wh only wns and losses. References. Russell, T., van Bee, P.: Mahemacally clnchng a playoff spo n he NHL and he effec of scorng sysems. In: Proceedngs of he s Canadan Conference on Arfcal Inellgence. (008). Rbero, C.C., Urrua, S.: An applcaon of neger programmng o playoff elmnaon n fooball champonshps. Inernaonal Transacons n Operaonal Research (005) 375 386 3. Adler, I., Erera, A.L., Hochbaum, D.S., Olnc, E.V.: Baseball, opmzaon and he world wde web. Inerfaces 3 (00) 4. Robnson, L.W.: Baseball playoff elmnaons: an applcaon of lnear programmng. Operaons Research Leers 0 (99) 67 74 5. Gusfeld, D., Marel, C.E.: The srucure and complexy of spors elmnaon numbers. Algorhmca 3 (00) 73 86 6. Wayne, K.D.: A new propery and a faser algorhm for baseball elmnaon. SIAM Journal on Dscree Mahemacs 4 (00) 3 9 7. Brown, J.R.: The sharng problem. Operaons Research 7 (979) 34 340 8. Ahuja, R.K., Magnan, T.L., Orln, J.B.: Newor Flows: Theory, Algorhms and Applcaons. Prence Hall (993) 9. Schwarz, B.: Possble wnners n parally compleed ournamens. SIAM Revew 8 (966) 30 308 0. Maher, M., Narodysa, N., Qumper, C.G., Walsh, T.: Flow-based propagaors for he sequence and relaed global consrans. In: Proceedngs of he 4h Inernaonal Conference on Prncples and Pracce of Consran Programmng. (008). Se, J., Lee, L.Q., Lumsdane, A.: Boos Graph Lbrary: User Gude and Reference Manual. Addson-Wesley (00). ILOG S.A.: ILOG Solver 4. user s manual (998)