WHEN LINEAR AND WEAK DISCREPANCY ARE EQUAL

Transcription

1 WHEN LINEAR AND WEAK DISCREPANCY ARE EQUAL DAVID M. HOWARD AND STEPHEN J. YOUNG 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. 1. Inroducion In [10] Fishburn, Tanenbaum, and Trenk 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 measure 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 [4]. 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 [10] 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 [6, 7], o define and characerize he class of irreducible poses wih equal linear and weak discrepancy Preliminaries. More formally, if P is a pose le O (P ) be he collecion of order preserving maps from P o N and le I (P ) be he collecion of injecive order preserving maps from P o N. Then he linear discrepancy of P, denoed ld(p ), is min max f(x) f(y), 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 Now since I (P ) O (P ) i is clear ha wd(p ) ld(p ). Tanenbaum, e al. provide an explici formula for he linear and weak discrepancy of he disjoin union of chains in [10]. From hese formula i is easy o see ha he disjoin union of a chain of lengh 2d 1 and a chain of lengh 1 has linear and weak discrepancy equal o d, and hus he inequaliy is igh. 1

2 2 DAVID M. HOWARD AND STEPHEN J. YOUNG 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 [3, 10] while he weak discrepancy can be calculaed in polynomial ime [4, 9]. Thus i is naural o hope ha he answer o he quesion of Tanenbaum, e al. 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 [10]. Lemma 1. If P can be pariioned ino wo ses U and V such ha for all u U and v V, u < 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 he linear discrepancy is in NP and deermining 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 for his is 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 of lengh 2j 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. Thus 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. Thus, raher han aemping o explicily characerize all poses for which linear and weak discrepancy are he same, we follow he work in [1, 6, 7] and deermine essenial characerisics of poses wih equal linear and weak discrepancy. To ha end, we recall ha a pose P is d-linear-discrepancy-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-lineardiscrepancy-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 = c 1, c 2,..., c k such ha for all i eiher c i < c i+1 or c i c i+1

3 WHEN LINEAR AND WEAK DISCREPANCY ARE EQUAL 3 and (wihou loss of generaliy) c 1 c k. Given a forcing cycle C, define up(c) as {i c i < c i+1, 1 i k 1} and side(c) as 1 + {i c i c i+1, 1 i k 1}. 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 since c 1 c k. Using his noaion, Gimbel and Trenk, prove he following heorem [4]. Theorem 3. Le P be a pose and C be he se of forcing cycles on P, hen wd(p ) = max up(c) C C side(c). Furhermore, if C = c 1, c 2,..., c k is a maximal forcing cycle and f is a fracional labelling of P where f(c 1 ) = 0 and f(c i+1 ) = f(c i ) + 1 if c i < c i+1 and f(c i+1 ) = f(c i ) up(c) side(c) if c i c i+1. Then f is an opimal weak discrepancy labelling. In fac, Gimbel and Trenk prove he sronger resul ha he f provided is in fac opimal over all fracional weak 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 he work of Choi and Wes [2], will be key in characerizing he (d, d)-irreducible poses. 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, in paricular, here exiss a forcing cycle C = a 1, a 2,..., a (d 1)+2, b 1, b 2,..., b 1 using all he elemens where a i < a j if i < j, b j b j+1 for 1 j 2, a (d 1)+2 b 1, a 1 b 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-weak-discrepancyirreducible 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 have d + 1 poins and le C = a 0 < c 1 1 < c 2 1 < < c1 d 1 < c 1 2 < < c d 1 2 < < c 1 < < c d 1 < a a 1 a 2 a 1 be he opimal forcing cycle. Now since W is d-weak-discrepancy irreducible le f be he funcion winessing he opimal fracional weak discrepancy of (d 1) + 1 as provided 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 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 ),

4 4 DAVID M. HOWARD AND STEPHEN J. YOUNG hen f(ai ) f(c j î ) d 1 d 1 (d )i (î 1)(d 1) + j d 1 d 1 i j i + î 1 (d 1) d 1 i + 1 î. 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. Bu hen, since j i, we have î 1 < i and hence g(c j î ) < g(a i). Thus since f is a weak exension and for any x, y W if f(x) < f(y), hen g(x) < g(y), hen g is a weak order preserving map of W. Bu, since g is one-o-one, his implies ha g is an 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 wo 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 cj i such ha g(c j 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 i k + 2 and if g(a k ) g(c j i ) > d hen i k. However, for i = k we have g(a k ) g(c j i ) = d j < d, and hus we need only consider i < k. Since g(c 1 k+2 ) g(a k) = d+1 = g(a k ) g(c d 1 k 1 i suffice o only consider c 1 k+2 and cd 1 k 1. Now observe ha c1 k+2 exiss only if

5 WHEN LINEAR AND WEAK DISCREPANCY ARE EQUAL 5 k 2, we have f(c 1 k+2) f(a k ) = (k + 1)(d 1) + 1 (d 1)k + k = (d 1) + k > (d 1) + 1. 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 order preserving map of W ha winesses linear discrepancy a mos d. Bu hen since d = wd(w ) ld(w ) d, he linear discrepancy of W is 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. Then since d = ld(p ) wd(p ) wd(w ) = d, we have wd(p ) = d. Suppose hen ha ld(p ) = wd(p ) = d. Then i is clear ha here is some subpose W of P such ha W is (d, d)-irreducible. Now 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 and a i x < b i (and hence x is no in a rivial chain) and x is no encompassed by any side move. Then, since x < b i, by raversing he cycle we can conclude ha x L a j for any 1 j. Bu hen x L y for any y W and hence is he minimum elemen of L. Similarly if a i < x b i, hen x is he maximum elemen of L. Thus he only elemens of P 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. Then, as here are side seps, here exiss some side move (b L, a L ) encompassing a leas d+1 (2+ T ) = d elemens 1+ T

6 6 DAVID M. HOWARD AND STEPHEN J. YOUNG in he linear exension L. Thus if T < 1, hen (b L, a L ) encompasses a leas d elemens wih respec o L and hence h L (b L ) h L (a L ) d + 1. Bu 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 and 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 [5] where he order among some pairs of elemens are 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 wo [6, 7], he NP-compleeness of linear discrepancy [3], and he behavior of online algorihms for linear discrepancy [8] in emphasizing he cenraliy of inerval orders in he sudy of linear and weak discrepancy. Le W W d and le C = a 0 < c 1 1 < c 2 1 < < c1 d 1 < c 1 2 < < c2 d 1 < < c 1 < < c d 1 < a a 1 a 2 a 1 be an opimal forcing cycle of W. We firs noe ha if a i < a j, hen a i < c 1 i+2 and cd 1 j 1 < a j. Bu hen, 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. Now in order o characerize he elemens of W d 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 o < c 1 1 < c 2 1 < < c d 1 1 < c 1 2 < < c2 d 1 < < c 1 < < c d 1 < a is 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. Now 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, [l i, r i ] (d(i 1) 1, d(i + 1) + 1). The consrain ha a i a i+1 and ha a i < a i+2 imposes 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 for his helpful commens on an earlier version of his manuscrip which helped improve he clariy and exposiion of he curren manuscrip.

7 WHEN LINEAR AND WEAK DISCREPANCY ARE EQUAL 7 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 2 a 4 c 1 2 a 3 c 1 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] Gab-Byung Chae, Minseok Cheong, and Sang-Mok Kim, Irreducible poses of linear discrepancy 1 and 2, Far Eas J. Mah. Sci. (FJMS), 22 (2006), pp [2] Jeong-Ok Choi and Douglas B. Wes, Forbidden subposes for fracional weak discrepancy a mos k. preprin. [3] Peer C. Fishburn, Paul J. Tanenbaum, and Ann N. Trenk, Linear discrepancy and bandwidh, Order, 18 (2001), pp [4] John G. Gimbel and Ann N. Trenk, On he weakness of an ordered se, SIAM J. Discree Mah., 11 (1998), pp (elecronic). [5] Michel Habib, David Kelly, Emmanuelle Lebhar, and Chrisophe Paul, Can ransiive orienaion make sandwich problems easier?, Discree Mah., 307 (2007), pp [6] David M. Howard, Gab-Byung Chae, Minseok Cheong, and Sang-Mok Kim, Irreducible widh 2 poses of linear discrepancy 3, Order, 25 (2008), pp [7] David M. Howard, Michel T. Keller, and Sephen J. Young, A characerizaion of parially ordered ses wih linear discrepancy equal o 2, Order, 24 (2007), pp [8] Michel T. Keller, Noah Sreib, and William T. Troer, Online linear discrepancy. in preperaion, [9] Alan Shucha, Randy Shull, and Ann N. Trenk, The fracional weak discrepancy of a parially ordered se, Discree Appl. Mah., 155 (2007), pp [10] Paul J. Tanenbaum, Ann N. Trenk, and Peer C. Fishburn, Linear discrepancy and weak discrepancy of parially ordered ses, Order, 18 (2001), pp