When Linear and Weak Discrepancy are Equal

Transcription

1 When Linear and Weak Discrepancy are Equal David M. Howard b,2, Sephen J. Young b,1, a School of Mahemaics, Georgia Insiue of Technology, Alana, GA Absrac The linear discrepancy of a pose P is he leas k such ha here is a linear exension L of P such ha if x and y are incomparable, hen h L (x) h L (y) k. Whereas he weak discrepancy is he leas k such ha here is a weak exension W of P such ha if x and y are incomparable, hen h W (x) h W (y) k. This paper resolves a quesion of Tanenbaum, Trenk, and Fishburn on characerizing when he weak and linear discrepancy of a pose are equal. Alhough i is shown ha deermining wheher a pose has equal weak and linear discrepancy is NP-complee, his paper provides a complee characerizaion of he minimal poses wih equal weak and linear discrepancy. Furher, hese minimal poses can be compleely described as a family of inerval orders. Key words: linear discrepancy, weak discrepancy, pose 2000 MSC: 06A07 Corresponding Auhor addresses: dmh@mah.gaech.edu (David M. Howard), s7young@mah.ucsd.edu (Sephen J. Young) 1 Presen Address: Deparmen of Mahemaics, Universiy of California, San Diego, La Jolla, CA Suppored by Naional Science Foundaion VIGRE gran DMS Preprin submied o Elsevier Ocober 31, 2010

2 When Linear and Weak Discrepancy are Equal David M. Howard b,2, Sephen J. Young b,1, b School of Mahemaics, Georgia Insiue of Technology, Alana, GA Inroducion In [11], Tanenbaum, Trenk, and Fishburn inroduce he noion of he linear discrepancy of a pose as a measure of he disance of a pose from a linear order. In essence, he linear discrepancy of a pose measures how far apar incomparable elemens are forced in a linear exension of he pose. One can analogously define weak discrepancy as how far apar incomparable elemens of a pose are forced in a weak exension [5]. Inuiively, i is clear ha he weak discrepancy should be a mos he linear discrepancy, and in fac his bound is igh. In his paper we answer a quesion of Fishburn, Tanenbaum, and Trenk [11] and characerize he igh examples. More precisely, we expand upon he idea of irreducibiliy wih respec o linear discrepancy, inroduced in [1] and expanded upon in [7, 8], o define and characerize he class of irreducible poses wih equal linear and weak discrepancy Preliminaries More formally, for a pose P, le O (P ) be he collecion of order preserving maps from P o N, le I (P ) be he collecion of injecive order preserving maps from P o N, and le F(P ) be he collecion of fracional order preserving maps from P o Q. More specifically, F(P ) is he collecion of maps f from P o Q such ha if x < y hen f(x) f(y) + 1. The linear discrepancy of P, denoed ld(p ), is min f(x) f(y), max f I(P ) x y where x y means ha x is incomparable o y in P. Similarly, he weak discrepancy of P, denoed wd(p ), is min max f(x) f(y). f O(P ) x y Corresponding Auhor addresses: dmh@mah.gaech.edu (David M. Howard), s7young@mah.ucsd.edu (Sephen J. Young) 1 Presen Address: Deparmen of Mahemaics, Universiy of California, San Diego, La Jolla, CA Suppored by Naional Science Foundaion VIGRE gran DMS Preprin submied o Elsevier Ocober 31, 2010

3 Finally, he fracional weak discrepancy of P, denoed wd f (P ), is min max f(x) f(y). f F(P ) x y Since F (P ) I (P ) O (P ), i is clear ha wd f (P ) wd(p ) ld(p ). Tanenbaum, e al. provide explici formulas for he linear and weak discrepancy of he disjoin union of chains in [11]. From hese formulas i is easy o see ha he disjoin union of a chain wih 2d elemens and a chain wih 1 elemen has linear and weak discrepancy equal o d, and hus he las inequaliy is igh. A his poin i is worh noing ha calculaing he linear discrepancy of a pose is NP-complee, via a reducion o he bandwidh of is co-comparabiliy graph [4, 11], while he fracional weak discrepancy and weak discrepancy can be calculaed in polynomial ime [5, 10]. Thus i is naural o hope ha he answer o he quesion of Tanenbaum, e al. [11] is in he form of a polynomial ime algorihm, however, he following reducion indicaes ha his is unlikely o be he case. Tha is, here is no a polynomial ime algorihm unless P = NP. A key componen of he reducion is he following lemma from [11]. Lemma 1. If P can be pariioned ino wo ses U and V such ha u < v for all u U and v V, hen ld(p ) = max {ld(u), ld(v )} and wd(p ) = max {wd(u), wd(v )}. Theorem 2. Deermining wheher ld(p ) = wd(p ) is NP-complee. Proof. Since deermining wheher he linear discrepancy is a mos k is in NP and calculaing he weak discrepancy is polynomial, deermining wheher hey are equal is clearly in NP. Thus i suffices o show ha here is an NP-complee problem ha can be reduced in polynomial ime o deermining wheher he linear and weak discrepancy are equal. The naural candidae o reduce o deermining wheher linear and weak discrepancy are equal is he problem of deermining he linear discrepancy of a pose P. If ld(p ) = wd(p ) he linear discrepancy may be deermined by finding he weak discrepancy of P, herefore we may assume ha wd(p ) < ld(p ). Now for all j, le P j be he pose consising of a chain wih 2j elemens and a single isolaed poin and observe ha ld(p j ) = wd(p j ) = j. Le X be he ground se of P and le Y j be he ground se of P j. For each j from 1 o X define he pose P j on he ground se X Y j by leing P j be equal o P on X, equal o P j on Y j and leing y < x for every y Y j and x X. Now by Lemma 1, ld ( P j) = max {ld(p ), ld(pj )} and wd ( P j) = max {wd(p ), wd(pj )}. Thus for 1 j < ld(p ) we have wd ( ( ) ( ) P j) ld P j and for j ld(p ) we have wd P j = ld ( ( ) ( ) P j) and hus ld(p ) is he firs j such ha ld P j = wd P j. Hence if calculaing wheher linear and weak discrepancy are equal were polynomial, hen deermining he linear discrepancy of P would be as well, and hus deermining wheher linear and weak discrepancy are equal is NP-complee. In ligh of Theorem 2, raher han aemping o explicily characerize all poses for which linear and weak discrepancy are he same, we follow he 3

4 work in [1, 7, 8] and deermine essenial characerisics of poses wih equal linear and weak discrepancy. To ha end, we recall ha a pose P is d-lineardiscrepancy-irreducible if ld(p ) = d and for any x P we have ld(p x) < d. We define d-weak-discrepancy-irreducible analogously. Addiionally, we say a pose P is (s, )-discrepancy irreducible (or simply (s, )-irreducible) if ld(p ) = s and wd(p ) = and for any poin x P eiher ld(p x) < s or wd(p x) <. If s = hen we may replace, wihou loss of generaliy, he second condiion wih for any x P, wd(p x) <. Tha is, if a pose is (d, d)-irreducible hen i is also d-weak-discrepancy-irreducible. Furher, we noe ha if a pose P is such ha ld(p ) = s and wd(p ) =, hen here are induced subposes of P, denoed P s, P and P (s,), such ha P s is s-linear-discrepancy-irreducible, P is -weak-discrepancy-irreducible, and P (s,) is (s, )-irreducible. Wih hese definiions in hand we review some preliminary work on weak discrepancy Weak Discrepancy Preliminaries In a pose P a forcing cycle is a sequence of elemens c 1, c 2,..., c k such ha for all i eiher c i < c i+1 or c i c i+1 aking all indices modulo k. Given a forcing cycle C, define up(c) as {i : c i < c i+1, 1 i k} and side(c) as 1+ {i : c i c i+1, 1 i k}. Tha is, up(c) is he number of up seps along he cycle and side(c) is he number of incomparable seps when viewing C cyclically. Using hese noions Gimbel and Trenk were able o provide a combinaorial characerizaion for he opimal weak discrepancy in erms of forcing cycles [5]. In [10], Schucha, Shull and Trenk were able o exend hese ideas and find he weak discrepancy via a linear programming relaxaion. In oaliy, hese resuls yield he following heorem. Theorem 3 ([5, 10]). If P is a pose ha is no a chain and C is he se of forcing cycles on P, hen wd(p ) = max up(c) C C side(c). Furhermore, if C is a forcing cycle, wih elemens c 1,..., c k, which is maximal wih respec o up(c) side(c) and f is a fracional labelling of C defined recursively by { f(ci ) + 1 c i < c i+1 f(c i+1 ) = f(c i ) up(c), side(c) c i c i+1 hen f can be exended o an order preserving map f on P and wd(p ) = max f(x) f(y). x y x,y C In fac, Schucha e al. [10] proved he sronger resul ha he f provided is in fac opimal over all fracional order preserving maps, yielding a fracional up(c) weak discrepancy of max C C side(c). In addiion o Theorem 3, which provides combinaorial cerificaion for wd(p ) k, he following heorem, which is implici in Choi and Wes s consrucion of he subposes forbidden by fracional weak discrepancy a mos k [2], will be key in characerizing he (d, d)-irreducible poses. 4

5 Theorem 4. A pose P on n poins is d-weak-discrepancy irreducible if and only if every forcing cycle C ha is maximal wih respec o up(c) side(c) has size side seps and (d 1) + 1 up seps and n = + (d 1) (d, d)-irreducible Poses Le W d be he collecion of d-weak-discrepancy-irreducible poses where here exiss a maximal forcing cycle wih all he up seps consecuive. Tha is, here exiss a forcing cycle a 1, a 2,..., a (d 1)+2, b 1, b 2,..., b 1 using all he elemens where a i < a i+1 for 1 i(d 1) + 2, a (d 1)+2 b 1, b j b j+1 for 1 j 2, and b 1 a 1. We claim ha W d is he se of all (d, d)-irreducible poses. Firs we show ha all elemens of W d are (d, d)-irreducible. Since he elemens of W d are d-weakdiscrepancy-irreducible by consrucion, i suffices o show ha hey all have linear discrepancy d. Lemma 5. If W W d, hen ld(w ) = d. Proof. Le W W d. By Theorem 4, W has d+1 poins for some ineger > 0. Le a 0 < c 1 1 < c 2 1 < < c d 1 1 < c 1 2 < < c2 d 1 < < c 1 < < c d 1 < a a 1 a 2 a 1 be a forcing cycle winessing he weak discrepancy, as provided by Theorem 3. Since W is d-weak-discrepancy irreducible, here is a funcion f winessing he opimal fracional weak discrepancy of (d 1) + 1 consruced as in Theorem 3. In paricular, f(a i ) = ( ) d i and f(c j i ) = (i 1)(d 1) + j. Define he funcion g : W {0,..., d} by g(a i ) = di and g(c j i ) = (i 1)d + j. We claim g is an injecive order preserving map of W winessing linear discrepancy a mos d. Firs we observe ha by consrucion if g(x) = g(y), hen x = y. Now if f(a i ) < f(c j î ), hen f(ai ) f(c j î ) d 1 d 1 (d )i (î 1)(d 1) + j d 1 d 1 i i + (î 1) + j (d 1) d 1 i + 1 î. 5

6 Thus i < î, so g(a i ) < g(c j î ). Similarly, if f(cj î ) < f(a i), hen f(c j î ) d 1 < f(a i) d 1 î 1 + j d 1 < i + i (d 1) î 1 + j i (d 1) < i. Since j i, we have î 1 < i and hence g(c j î ) < g(a i). In paricular, if f(x) < f(y) hen g(x) < g(y). Thus, since f is an order preserving map, g is also an order preserving map. Furhermore, since g is one-o-one, his implies ha g is an injecive order preserving map of W. Now suppose x y and g(x) g(y) > d. If x, y {a 0, a 1,..., a }, hen g(x) g(y) > d implies ha he indices of x and y differ by a leas 2 and hence f(x) f(y) 2 ( ) d and so x and y are comparable since f is winesses fracional weak discrepancy a mos d Thus precisely one of {x, y} is a poin of he form c j i and he oher is a poin of he form a k wih 1 k 1. We will show ha if g(c j i ) g(a k) > d, hen c j i and a k are comparable. In paricular, we wish o show ha if g(c j i ) g(a k) > d, hen c j i > a k, and if g(a k ) g(c j i ) > d, hen a k > c j i. Since he cj i form a chain, i suffices o consider he minimal c j i such ha g(cj i ) g(a k) > d and he maximal c j i such ha g(a k) g(c j i ) > d. We noe ha g(a k ) g(c j i ) = dk (i 1)d j = d(k i + 1) j d k i j d k i (d 1), and hus, if g(c j i ) g(a k) > d, hen 1 < k i + 1 and hence, i k + 1. However, for i = k we have g(a k ) g(c j i ) = d j < d, and hus we need only consider i k 1 or i k + 2. Since g(c 1 k+2 ) g(a k) = d + 1 = g(a k ) g(c d 1 k 1 ) i suffices o only consider c1 k+2 and cd 1 k 1. Now observe ha c 1 k+2 exiss only if k 2, and we have [ f(c 1 k+2) f(a k ) = (k + 1)(d 1) + 1 (d 1)k + k ] = (d 1) + k > (d 1)

7 Thus a k < c 1 k+2 since f winesses fracional weak discrepancy a mos d Similarly, c d 1 k 1 exiss only if k 2, and hen f(a k ) f(c d 1 k 1 ) = (d 1)k + k (k 2)(d 1) (d 1) = (d 1) + k > (d 1) + 1. Thus c d 1 k 1 < a k, and hence g is an injecive order preserving map of W ha winesses linear discrepancy a mos d. Since d = wd(w ) ld(w ) d, he linear discrepancy of W is now exacly d. The following heorem shows ha no only are all elemens of W d (d, d)- irreducible, every (d, d)-irreducible pose is a member of W d. Theorem 6. Le P be a pose wih ld(p ) = d. Then wd(p ) = d if and only if here exiss a subpose W of P such ha W W d. Proof. Firs suppose here is some subpose W of P such ha W W d. Since d = ld(p ) wd(p ) wd(w ) = d, we have wd(p ) = d. If ld(p ) = wd(p ) = d, hen i is clear ha here is some subpose W of P such ha W is (d, d)-irreducible. Since he removal of any poin from W decreases eiher he weak discrepancy or he linear discrepancy and wd(p ) ld(p ) for all P, we know ha W is d-weak-discrepancy irreducible. Thus i suffices o show ha he maximal forcing cycle has all he up seps consecuive. Since W is d-weak-discrepancy irreducible, W = d+1 for some and here is a maximal forcing cycle C using d + 1 poins. This forcing cycle naurally pariions he elemens of W ino chains C 1, C 2,..., C by using he side seps as break poins in he chain. For all chains C i, le a i be he minimal elemen and le b i be he maximal elemen (noe ha i is no necessarily he case ha a i b i ). We say ha a side move (b, a) {(b i, a i+1 ) 1 i 1} {(b, a 1 )}, encompasses a poin x wih respec o a linear exension L if b < L x < L a or a < L x < L b. Fix an arbirary linear exension L of W. Suppose x C i, a i x < b i (and hence x is no in a rivial chain), and x is no encompassed by any side move. Since x < b i and x is no enclosed by he side move (b i, a i+1 ), we have x L a i+1. In paricular, repeaedly using ha x is no enclosed in a side move, a i x < b i L a i+1 b i+1 L a i 1 b i 1 L a i. Hence x = a i, and for any y W, we have x L y. Similarly, if a i < x b i, hen x is he maximum elemen of L. Thus he only elemens of W ha are no encompassed by a side sep wih respec o L are he minimum and maximum elemens of L and he elemens belonging o a rivial chain. Now le T be he se of rivial chains. Since here are side seps, here exiss some side move (b L, a L ) encompassing a leas d+1 (2+ T ) = d 1+ T elemens in he linear exension L. Thus if T < 1, hen (b L, a L ) encompasses a leas d elemens wih respec o L, 7

8 and hence h L (b L ) h L (a L ) d+1. Since L was an arbirary linear exension, his implies ha ld(w ) d + 1, a conradicion. Thus T = 1, and so all bu one of he chains is rivial. Hence all he up seps are consecuive in he forcing cycle. 3. Characerizaion of W d In examining he naure of W d, i is clear ha, conrary o mos resuls on poses, W d is specified hrough explici local resricions on he se of comparabiliies and incomparabiliies raher han global resricion on he srucure of he pose. Tha is, W d is defined as he se of soluions o a collecion of ransiively oriened sandwich problems [6] where he order among some pairs of elemens is defined and oher pairs of poins are defined o be incomparable. However, we can exploi he srucure of elemens of W d o provide a more naural descripion of he class as inerval orders. This characerizaion of W d as a collecion of inerval orders joins wih resuls such as he forbidden subpose characerizaion of poses wih linear discrepancy a mos 2 [7, 8], he NP-compleeness of linear discrepancy [4], and he behavior of online algorihms for linear discrepancy [9] in emphasizing he cenraliy of inerval orders in he sudy of linear and weak discrepancy. Le W W d and le a 0 < c 1 1 < c 2 1 < < c1 d 1 < c 1 2 < < c d 1 2 < < < a a 1 a 2 a 1 be an opimal forcing cycle of W. We and cd 1 j 1 < a j. Since j i + 2, his implies ha every elemen of he chain a 0 < c 1 1 < < c d 1 < a is comparable o eiher a i or a j. Thus W does no conain a and hence is an inerval order [3]. Now in order o represen he elemens of W d as inerval orders, i suffices o provide a collecion of inervals or rules for generaing he inervals ha will realize every elemen of W d. We noe ha since a i < a j implies ha every elemen of he chain a 0, c 1 1, c 2 1,..., c d 1 1, c 1 2,..., c2 d 1,..., c 1,..., c d 1, a is c 1 < < c d 1 firs noe ha if a i < a j, hen a i < c 1 i+2 comparable o eiher a i or a j, we may assume ha he inervals associaed wih he long chain are degenerae. In paricular, we assume ha he inerval for c j i is {(i 1)d + j} and ha he inervals for a 0 and a are {0} and {d}, respecively. For 1 i 1, le he endpoins of he inerval associaed wih a i be l i and r i. Using ha c j i is assigned o he degenerae inerval {(i 1)d + j}, i is clear ha we may assume for 1 i 1 ha [l i, r i ] (d(i 1) 1, d(i + 1) + 1). The consrains a i a i+1 and a i < a i+2 require ha l i+1 < r i < l i+2. In fac, any inerlaced sequence 1 < l 2 < r 1 < l 3 < l < r 1 < d + 1 such ha r i < d(i + 1) + 1 for 1 i < 1 and d(j 1) 1 < l j for 1 < j will yield an inerval represenaion of an elemen of W d. For example, see Figure 1. Acknowledgemens The auhors would like o hank Michel T. Keller, he anonymous referees, and he edior for heir helpful commens on an earlier version of his manuscrip, which helped improve he clariy and exposiion of he curren manuscrip. 8

9 a 4 c 2 4 a 3 c 1 4 c 2 3 a 0 c 1 1 c 2 1 c 1 2 c 2 2 c 1 3 c 2 3 c 1 4 c 2 4 a 4 c 1 3 a 2 c 2 2 a 1 a 3 c 1 2 a 2 c 2 1 a 1 c 1 1 a 0 (a) Hasse Diagram (b) Inerval Represenaion Figure 1: A elemen of W 3 on 13 poins. References [1] G.-B. Chae, M. Cheong, S.-M. Kim, Irreducible poses of linear discrepancy 1 and 2, Far Eas J. Mah. Sci. (FJMS) 22 (2) (2006) [2] J.-O. Choi, D. B. Wes, Forbidden subposes for fracional weak discrepancy a mos k, European Journal of Combinaorics (2010) o appear. [3] P. C. Fishburn, Inransiive indifference wih unequal indifference inervals, J. Mahemaical Psychology 7 (1970) [4] P. C. Fishburn, P. J. Tanenbaum, A. N. Trenk, Linear discrepancy and bandwidh, Order 18 (3) (2001) [5] J. G. Gimbel, A. N. Trenk, On he weakness of an ordered se, SIAM J. Discree Mah. 11 (4) (1998) (elecronic). [6] M. Habib, D. Kelly, E. Lebhar, C. Paul, Can ransiive orienaion make sandwich problems easier?, Discree Mah. 307 (16) (2007) [7] D. M. Howard, G.-B. Chae, M. Cheong, S.-M. Kim, Irreducible widh 2 poses of linear discrepancy 3, Order 25 (2) (2008) [8] D. M. Howard, M. T. Keller, S. J. Young, A characerizaion of parially ordered ses wih linear discrepancy equal o 2, Order 24 (3) (2007)

10 [9] M. T. Keller, N. Sreib, W. T. Troer, Online linear discrepancy of parially ordered ses, in: An Irregular Mind, vol. 21 of Bolyai Soc. Mah. Sud., Springer, [10] A. Shucha, R. Shull, A. N. Trenk, The fracional weak discrepancy of a parially ordered se, Discree Appl. Mah. 155 (17) (2007) [11] P. J. Tanenbaum, A. N. Trenk, P. C. Fishburn, Linear discrepancy and weak discrepancy of parially ordered ses, Order 18 (3) (2001)