Equitable Resolvable Coverings

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Tilburg University

Equitable Resolvable Coverings

van Dam, E.R.; Haemers, W.H.; Peek, M.B.M.

Publication date:

2001

Document Version

Publisher's PDF, also known as Version of record Link to publication in Tilburg University Research Portal

Citation for published version (APA):

van Dam, E. R., Haemers, W. H., & Peek, M. B. M. (2001). Equitable Resolvable Coverings. (CentER Discussion Paper; Vol. 2001-103). Operations research.

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No. 2001-103

EQUITABLE RESOLVABLE COVERINGS

By Edwin R. van Dam, Willem H. Haemers and

Maurice B.M. Peek

December 2001

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Surri1 Wdnh dq dqh sodqh ri rughu t1 Il{ d sdudooho fodvv F/ dqg n ri lwv olqhv1 Odeho wkhvh

olqhv e| o> l @ 4> ===> n1 Il{ dqrwkhu sdudooho fodvv F/ dqg odeho lwv olqhv e| p> m @ 4> ===> t1 Dv

srlqw vhw ri wkh fryhulqj zh wdnh doo srlqwv rq wkh olqhv o> l @ 4> ===> n1 Zh odeho wkhvh srlqwv e|

wkh sdluv +l> m,> l @ 4> ===> n> m @ 4> ===> t/ zkhuh +l> m, lv wkh srlqw ri lqwhuvhfwlrq ri o dqg p1

Wkh uvw t sdudooho fodvvhv ri wkh fryhulqj duh lqkhulwhg iurp wkh dqh sodqh> wkh| duh wkh sdudooho fodvvhv glhuhqw iurp F +lqgxfhg rq wkh qhz srlqw vhw,1 Wkxv wkhvh sdudooho fodvvhv frqvlvw ri t eorfnv ri vl}h n> dqg doo sdluv ri srlqwv i+l> m,> +l_{> m}_{,j zlwk l 9@ l} _{duh fryhuhg h{dfwo|}

rqfh e| wkh eorfnv lq wkhvh sdudooho fodvvhv1

Wr fryhu wkh uhpdlqlqj sdluv zh frqvwuxfw wkuhh dgglwlrqdo sdudooho fodvvhv1 Wkh uvw sdudooho fodvv frqvlvwv ri wkh eorfnv

i+l> m, = m @ 4> ===> nj> l @ 4> ===> n dqg i+l> m, = l @ 4> ===> nj> m @ n . 4> ===> t/ zkloh wkh vhfrqg frqvlvwv ri wkh eorfnv

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i+l> m, = m @ 4> ===> w> n . 4> ===> tj ^ i+l . k > w . k, = k @ 4> ===> n 5wj> l @ 4> ===> n

+zkhuh l . k vkrxog eh frqvlghuhg prgxor n, dqg i+l> m, = l @ 4> ===> nj> m @ n w . 4> ===> n1 Lw lv hdv| wr fkhfn qrz wkdw wklv frqvwuxfwlrq jlyhv dq htxlwdeoh 50+tn> n> 4, fryhulqj +zlwk t . 6 sdudooho fodvvhv,1

Li pruhryhu t w +prg 5w,/ wkhq zh fdq gr ehwwhu1 Lq wkdw fdvh n @ t w 3+prg 5w,/ vd| n @ 5wv1 Qrz wkh qdo sdudooho fodvv lq wkh deryh frqvwuxfwlrq lv uhsodfhg e| wkh sdudooho fodvv frqvlvwlqj ri wkh eorfnv

i+l> m, = l @ kv . 4> ===> kv . v> m @ 4> ===> w> n . 4> ===> tj> k @ 3> ===> 5w 4 dqg i+l> m, = l @ 4> ===> nj> m @ w . 4> ===> n1

Exw qrz doo olqhv p +lqgxfhg rq wkh qhz srlqw vhw, rffxu wzlfh/ vr zh fdq +dqg pxvw, uhpryh

wkh sdudooho fodvv F_{/ wr rewdlq dq htxlwdeoh uhvroydeoh fryhulqj zlwk t . 5 sdudooho fodvvhv1}

Zh uhpdun wkdw wkh dgglwlrqdo uhvwulfwlrq t w +prg 5w, lv iru h{dpsoh vdwlvhg zkhq t lv rgg dqg n @ t 41

54@** M*LU!t

Iru vpdoo eorfn vl}h n/ l1h1 iru n 7/ dqg iru doprvw doo t/ uhvroydeoh 50+tn> n> 4, fryhulqjv zlwk plqlpdo u kdyh ehhq frqvwuxfwhg1 Pdq| ri wkrvh duh htxlwdeoh/ dv zh vkdoo vhh1

Frqvlghu d uhvroydeoh 50+tn> n> 4, fryhulqj zlwk u sdudooho fodvvhv/ dqg { d srlqw s1 Ohw | eh

wkh qxpehu ri sdluv rq s wkdw duh fryhuhg l wlphv1 Wkhq S| @ y 4 dqg Sl| @ u+n 4,/

khqfhS+l 4,|@ u+n 4, y . 41 Qrz vxssrvh wkdw y @ u+n 4,/ wkhq lw iroorzv wkdw | @ 3

iru l A 5/ dqg |2 @ 41 Wklv lpsolhv wkdw wkh fryhulqj lv htxlwdeoh/ dqg wkdw hyhu| srlqw lv lq

h{dfwo| rqh sdlu zklfk lv fryhuhg wzlfh1

Iru n @ 6 wkhuh h{lvw 50+6t> 6> 4, fryhulqjv zlwk u @

2t iru doo hyhq t 9@ 5> 7 +fi1 ^9`,1 Wkhvh

fryhulqjv duh htxlwdeoh e| wkh deryh dujxphqwv1 Iru t @ 5 wkhuh fdq eh qr htxlwdeoh fryhulqj e| wkh uhvxowv lq Vhfwlrq 6141 Iru t @ 7/ d fryhulqj zlwk 9 sdudooho fodvvhv grhv qrw h{lvw +fi1 ^5`,> krzhyhu/ wkh frqvwuxfwlrq ri Sursrvlwlrq 6 jlyhv dq htxlwdeoh uhvroydeoh fryhulqj zlwk : sdudooho fodvvhv1 Zh uhpdun wkdw dovr iru doo rgg t wkhuh h{lvw 50+6t> 6> 4, fryhulqjv zlwk u @ ^3₂ 1 Wkhvh fryhulqjv duh ghvljqv +Nlunpdq wulsoh v|vwhpv,/ dqg khqfh wkh| duh htxlwdeoh1

Iru n @ 7 wkhuh h{lvw 50+7t> 7> 4, fryhulqjv zlwk u @ ge^3 h iru doo exw qlwho| pdq| t +fi1 ^:`,1 Iru t 4 +prg 6, wkhvh fryhulqjv duh ghvljqv/ dqg khqfh wkh| duh htxlwdeoh1 Iru t 3 +prg 6, wkh fryhulqjv duh htxlwdeoh e| wkh deryh dujxphqwv1 Iru t 5 +prg 6, wkh fryhulqjv pd| ru pd| qrw eh htxlwdeoh1 Iurp wkh deryh dujxphqwv lw iroorzv wkdw lq wkhvh fryhulqjv hdfk srlqw lv lq wzr sdluv wkdw duh fryhuhg wzlfh/ ru lq rqh sdlu wkdw lv fryhuhg wkuhh wlphv1 W|slfdoo| wkhvh fryhulqjv dv frqvwuxfwhg e| Odpnhq/ Ploov/ dqg Uhhv ^:` duh qrw htxlwdeoh +wkh fryhulqjv zlwk t 5 i8q> 44q> 4:q> 56q = q 4 +prg 6,j ^ i7:> 8<> :4> ;6> 43:> 464> 4:<j duh htxlwdeoh,1 Iru t @ 5 qr htxlwdeoh uhvroydeoh fryhulqj h{lvwv +exw d uhvroydeoh fryhulqj zlwk u @ 6 h{lvwv,1 Iru t @ 8 wkh fryhulqj iurp ^:` +dqg Sursrvlwlrq 6, lv htxlwdeoh1 Wkh vpdoohvw rshq fdvh lv t @ ;= wkh fryhulqj +zlwk u @ 44, iurp ^:` lv qrw htxlwdeoh/ mxvw olnh wkh fryhulqj rewdlqhg iurp wkh fryhu ri S J+7> 5, zlwk +surmhfwlyh, olqhv +vhh Vhfwlrq 515,1

Lq jhqhudo/ Ehxwhovsdfkhu*v ^6` fryhu ri S J+q 4> 5,/ q rgg/ e| olqhv jlyhv d uhvroydeoh 50 +5?_{> 7> 4, fryhulqj zlwk} 2?_n

sdudooho fodvvhv wkdw lv qrw htxlwdeoh1 Lw lv krzhyhu hdv| wr dgmxvw

wklv fryhu ri S J+q 4> 5, vxfk wkdw hyhu| srlqw lv fryhuhg rqfh ru wzlfh1 Wr gr wklv/ wdnh rqh ri wkh olqhv wkdw fryhuv wkh xqltxh srlqw wkdw lv fryhuhg wkuhh wlphv/ dqg uhpryh wklv olqh o1 Qrz dgg wzr olqhv wkdw gr qrw lqwhuvhfw/ hdfk olqh fryhulqj rqh ri wkh wzr srlqwv wkdw zhuh xqfryhuhg e| wkh uhprydo ri wkh olqh o1 E| Hlvihog*v ^7` fkdudfwhul}dwlrq/ wklv lv d plqlpxp %htxlwdeoh%

(9)

fryhu ri S J+q 4> 5, zlwk olqhv1 Wkh fruuhvsrqglqj htxlwdeoh uhvroydeoh 50+5?_{> 7> 4, fryhulqj}

zlwk 2?_n

. 4 sdudooho fodvvhv pd| krzhyhu qrw eh plqlpdo +lq wkh vhqvh wkdw zh gr qrw nqrz

li wkhuh h{lvwv dq htxlwdeoh uhvroydeoh 50+5?_{> 7> 4, fryhulqj zlwk} 2?_n

sdudooho fodvvhv,1

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Vlpxodwhg dqqhdolqj lv d orfdo vhdufk phwkrg +vhh ^4`,/ zklfk kdv suryhg wr eh iuxlwixo iru wkh frqvwuxfwlrq ri fryhulqjv/ fi1 ^;`1 Khuh zh dsso| wkh whfkqltxh lq wkh iroorzlqj zd|1 Vxssrvh zh zdqw wr qg d uhvroydeoh 50+tn> n> 4, fryhulqj zlwk u sdudooho fodvvhv iru {hg n/ t dqg u1 Zh vhdufk lq wkh vhw ri uhvroydeoh 40+tn> n> u, ghvljqv +l1h1 u sdudooho fodvvhv frqvlvwlqj ri t eorfnv ri vl}h n,1 Zh ghqh wkh frvwv i+G, ri vxfk d ghvljq G wr eh wkh qxpehu ri sdluv ri srlqwv iurp G qrw fryhuhg e| wkh eorfnv ri G1 Fohduo|/ li i+G, @ 3 wkhq G lv d uhvroydeoh fryhulqj1 D qhljkerxu ri G lv rewdlqhg e| lqwhufkdqjlqj wzr srlqwv iurp glhuhqw eorfnv zlwklq d sdudooho fodvv1 D qhljerxu G _{ri G lv dffhswhg dv d qh{w lwhp lq wkh vhdufk zlwk suredelolw|}

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Wkh sdudphwhu f lv wkh frqwuro sdudphwhu zklfk ghshqgv rq wkh qxpehu ri vwhsv l lq wkh

vhdufk1 Lw frqyhujhv wr 3li l $ 41 Zh kdyh vxffhvixoo| dssolhg wkh vhdufk zlwk t @ 7/ u @ 9/ n 5 i8> 9> <> 46> 4:j/ dqg frqwuro sdudphwhu f @ ₂+=<<,f3D01 Wkh uhvxowv duh h{klelwhg lq

Wdeohv 4 wr 8 ri wkh dsshqgl{1 Iru pruh ghwdlov/ vhh ^<`1

E| wkh vdph phwkrg zh vhdufkhg iru htxlwdeoh uhvroydeoh fryhulqjv1 Wkh rqo| glhuhqfh zlwk wkh deryh lv wkh frvw ixqfwlrq1 Iru d uhvroydeoh 40ghvljq G/ ohw { ghqrwh wkh qxpehu ri

sdluv wkdw duh fryhuhg rqfh/ dqg ohw {2 eh wkh qxpehu ri sdluv wkdw duh fryhuhg wzlfh1 Zh vdz

wkdw iru dq htxlwdeoh uhvroydeoh fryhulqj/ wkh qxpehuv { dqg {2 duh ghwhuplqhg e| n/ t dqg

u1 Ghqrwh wkhvh qxpehuv e| a{ dqg a{21 Wkh frvwv ri d uhvroydeoh 40ghvljq duh qrz jlyhq e|

m{ a{m . m{2 a{2m1 Zh dssolhg wkh phwkrg iru n @ 8/ t @ 7 dqg u @ : zlwk wkh vdph frqwuro

sdudphwhu dv deryh1 Diwhu pdq| krxuv ri vhdufklqj wkh frpsxwhu irxqg wkh vroxwlrq jlyhq lq Wdeoh 91

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4ih@?|4 bS Ebbc b

dbo ii!c ?44 +itL*@M*i Lih?}tc @t|ih<t |itt E? #|Uc A*Mh} N?iht|)c 2fff dfo #+ 5|?tL?c Lih?}tc @T|ih WVH ? Ai + O@?_MLL! Lu L4M?@|Lh@* #it}?t Ea L*MLh? @?_

(10)

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H 2f 2 . S S D 2 . b e H f D e b S H e D 2f f e . 2 b b D S . 2 H f S H 2 b . H e 2f . D 2 e D b S D e 2 . S S b 2f H f . 2 D b H e . H b 2 D e . e S H 2f b 2 S D f . . S 2f e f b 2 2 D H e D H S b

Wdeoh 4= d plqlpxp uhvroydeoh 50+53> 8> 4, fryhulqj

e H 2e S . D S f 2 . D 2 22 2 e H 2 2f b b . . 2 2f S 2 S e f H D b 2 b 22 e 2e 2 H D b f e S 2e 2f 2 2 D D H . H . S 22 2 b e 2 2 H H e 22 2f e 2 b D f D S . S b 2 . 2e 2 2 b f H 2 S S b . 2 2 22 2f D 2e H D . e e e 2 . D S b e 22 S f 2f H . b 2e 2 H D 2 2

2 S H 2. 2 2H 2 D b 2 2D f 2e 2S S f e 2 2f 2b . D D . e 22 e S b H f 2 . . b 2e 2. e e 2 H 2f 2 2D 2 S D b S S 2 2b H f D 2S 22 e D 2H b S . 2 D 2D 2 f 22 2 e . f b 2 D e D 2. 2 S 2S 2f S H 2H H e 2e 2b 2 2 S 2D 2. 2b e f D 22 2 D S 2e f S e D 2S . H H . b e 2 2f b 2H 2 . S D 2b 2 b 2e 2S H 2 D f e S . 2D 2H e b 2 2 D 2. H e S 2 f 2f 22 2S D b D 2 2 . 2b 2. 2 22 2 S H 2D D f H e f . b 2H S e S e 2f 2 2e

Wdeoh 6= d plqlpxp uhvroydeoh 50+69> <> 4, fryhulqj

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2 S H 2 2 f H Df D2 D e eb 2 D b . H 2D b e eS ef e2 S f e 2b 2 2S 2f . S eD D D . e 2H 22 2. 2e e eH b e. ee f 2 . 2e 2b eS e2 H Df D ee e e 2 f 2f H 2. e S . e eH S D b S 2 2 2D 22 b D2 b eD e. H D 2S 2H . 2 eb D ef e H e S 2. 2 2 eD eH D ef e2 S 2 . 2e H 2 2S Df D2 e ee 2 D e D e. 2f 22 b . eS eb S H f b f 2D 2b 2H b . D e e 2 2 S eb H 2H 2 e e eD Df eS D 2S 22 f e2 b e e. f S e D 2b 2 2. . H eH D D2 ef H . b e 2e 2 2D 2f . S D ee b . S D 2 e . eb ee 2e f ef eD H 2 D f D 2b 22 2 b H D e e. e b 2 2. b 2S H eH Df eS 2D e S 2H 2 2f e . D2 e2 S H . S 2 2S e f eS 2 D ee D2 S 2 eD e H H 2H 2 2e e D e 2 D . 2f b e. 22 Df S . ef f e b e2 2b 2D D eb 2. b eH

2 S H 2H 22 2. 2e . b eS De SS SH D Df SD 2 D b . H 2D e e. e DD D. S2 DS DH S f e 2b 2 2S 2f eD eH e2 S D D2 S eb S. D . e 2 2 f b ee H e ef Db Se D S Sf f 2 . 2S 2H . 2 ef eD S2 e. DH De SS S. Sf e e 2 f 2f H 2. eS S e e Df D2 D D. Se S D b S 2 2 2D 22 eH . e H D SH DD S S H D 2e 2b b e2 ee b eb SD D DS Db H e S 2e f 2 . e b eH e. S Se D DS DH S 2 . 22 H 2b 2 ef e . e2 SS SH eb D. Sf 2 D e D 2H 2 2f S S H D D S2 SD D2 De f b 2. 2S b 2D eS e eD ee DD S. Db Df 2 2 S b H 2H 2 SD e ee eH D. Db S SS S2 D 2b 2 2. . e2 H eS e. DH D eb Df S f S e D 2S 22 f eD . e De Se S. SH DS H . b e 2e 2 2D 2f ef b e S D D2 D Sf DD . S D 2. 2 2 eH Df SD Sf ef eS DS S H 2 D f 2e H 2 2S D eD H b D e S. D. S e b 2 f 2D 2b 2H e DD e2 De Se SS eb S2 e e S . 2f 22 b ee . S e. Db D SH DH D2 H S . 2e 2 2. 22 . e2 Df eS S2 b D Sf SH S 2 2 2b H b eb S Db e De ee eH ef D. D D e 2 2f 2H 2 S D S H SS D2 D DS e b f 2S f 2D . DH e. eD e S. SD e DD Se

Wdeoh 8= d plqlpxp uhvroydeoh 50+9;> 4:> 4, fryhulqj

H e S S H e b b . . D f D 2 2f 2 D 2f H . S S f H 2 e . e b b 2 D f . D H 2 S . 2f 2 e b H D S e b H S b 2 D . D 2 S e H 2f . e b f b 2 S H D H . 2 f b e . 2f D e S . 2 S e D 2f 2 . H D e S b f b H S e . f H 2 H 2 2f . S b e D D b

Equitable Resolvable Coverings

Tilburg University

Equitable Resolvable Coverings

van Dam, E.R.; Haemers, W.H.; Peek, M.B.M.

No. 2001-103

EQUITABLE RESOLVABLE COVERINGS

By Edwin R. van Dam, Willem H. Haemers and

Maurice B.M. Peek

December 2001

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