Ranking the vertices of a complete multipartite paired comparison digraph

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1 Rankng the vertces of a complete multpartte pared comparson dgraph Gregory Gutn Anders Yeo Department of Mathematcs and Computer Scence Odense Unversty, Denmark September 1, 2003 Abstract A pared comparson dgraph (abbrevated to PCD) D = (V, A) s a weghted dgraph n whch the sum of the weghts of arcs, f any, onng two dstnct vertces equals one. A one-to-one mappng α from V onto {1, 2,..., V } s called a rankng of D. For every rankng α, an arc vu A s sad to be forward f α(v) < α(u), and backward, otherwse. The length of an arc vu s l(vu) = ɛ(vw) α(v) α(u), where ɛ(vw) s the weght of vu. The forward ( backward) length f D (α) (b D (α)) of α s the sum of the lengths of all forward (backward) arcs of D. A rankng α s forward (backward) optmal f f(α) s maxmum ( b(α) s mnmum). M. Kano (Dsc. Appl. Math., 17 (1987) ) characterzed all backward optmal rankngs of a complete multpartte PCD D and rased the problem to characterze all forward optmal rankngs of a complete multpartte PCD L. We show how to transform the last problem nto the sngle machne ob sequencng problem of mnmzng total weghted completon tme subect to precedence parallel chans constrants. Ths provdes an algorthm for generatng all forward optmal rankngs of L as well as a polynomal algorthm for fndng Correspondng author. Ths work was supported by the Dansh Research Councl under grant no The support s gratefully acknowledged. 1

2 the average rank of every vertex n L over all forward optmal rankngs of L. 1 Introducton Kano and Sakamoto [7, 8, 9] ntroduced a few new methods (forward, backward and mutual) of rankng the vertces of a pared comparson dgraph (abbrevated to PCD). Advantages and applcatons of these methods were descrbed n [4, 7, 8, 9]. We only note that, for tournaments, all these methods, unlke some others, concde wth the most popular approach consstng of computng the scores, the number of games won by each player, and comparng them (see [7]). It s not dffcult to fnd all optmal mutual rankngs of any PCD usng Theorem 1 n [7]. At the same tme the problems of fndng a forward or backward optmal rankng are NP -hard (see Theorem 6.5 n [9] and Theorem 3.1 here). Moreover, Brghtwell and Wnkler [1] showed that the problem of countng the number of backward optmal rankngs of an acyclc dgraph s #P -complete. In contrast, Kano characterzed all optmal backward rankngs of a complete multpartte PCD (Theorem 1 n [6]). Kano and Sakamoto derved a characterzaton of all forward optmal rankngs of a complete multpartte PCD contanng not more than two vertces n each colour class (Theorem 4 n [7]) and Kano [6] rased the problem to characterze all forward optmal rankngs of any complete multpartte PCD L. We show how to transform the last problem nto the sngle machne ob sequencng problem of mnmzng total weghted completon tme subect to precedence parallel chans constrants. The sngle machne ob sequencng problem of mnmzng total weghted completon tme subect to varous precedence constrants has been consdered n a number of papers (see [3]). The transformaton above provdes an algorthm for generatng all forward optmal rankngs of L as well as a polynomal algorthm for computng the average rank (called the proper forward rank) of every vertex n L over all forward optmal rankngs of L. A polynomal algorthm for fndng the proper backward rank of every vertex n L was descrbed n [4]. In contrast, the problem of calculatng the proper backward rank of a vertex of an acyclc dgraph s proved [1] to be polynomally equvalent to a #P -complete 2

3 problem. Kano [6] proved hs man theorem usng an approach based on Lemma 2.3 [9],.e. he consdered dfferences f D (α xy ) f D (α) (see Secton 2.) Our approach s based on consderng the explct form of the functon f D (α). Usng our approach the man result of [6] can be proved n a somewhat easer manner. 2 Termnology and notaton Let D = (V, A) be a weghted dgraph n whch every arc xy has a postve real weght ɛ(xy). A dgraph D s called a pared comparson dgraph f D satsfes the followng condtons: 1. ɛ(xy) + ɛ(yx) = 1 f both xy and yx are arcs; 2. ɛ(xy) = 1 f xy A but yx / A. A PCD D s sad to be multpartte f the vertex set V of D can be parttoned nto colour classes V 1,..., V r such that there s no arc between any par of vertces from the same colour class. A multpartte PCD D s complete f every two vertces from dfferent colour classes are oned by at least one arc. A complete multpartte PCD D s a complete PCD f every colour class of D conssts of a sngle vertex. The postve (negatve, resp.) score of a vertex x V s σ + (x) = xy A ɛ(xy), (σ (x) = yx A ɛ(yx), resp.) A one-to-one mappng α from V onto {1, 2,..., V } s called a rankng of D. For α(x) =, α 1 () = x. Sometmes, we shall determne a rankng α by the strng (α 1 (1),..., α 1 ( V )). For a subset X of V, a rankng α of D nduces the followng permutaton α X : for a vertex x X, α X (x) = {y X : α(y) α(x)}. Let x and y be two vertces n D and let α be a rankng of D. Then α xy denotes a rankng of D as follows: α xy (z) = α(z) for every z / {x, y}, and α xy (x) = α(y), α xy (y) = α(x). For a rankng α, an arc vu A s called forward f α(v) < α(u). The length of an arc vu s ɛ(vu) α(v) α(u). The forward length f D (α) of α s the sum of the lengths of all forward arcs. A rankng α s forward optmal f 3

4 f D (α) s maxmum. The set of all forward optmal rankngs of D s denoted by F OR(D). The man obectve s to calculate the proper forward rank of every vertex x of D,.e. π(x) = 1 F OR(D) α F OR(D) α(x). Let D = (V, A) be a multpartte PCD and let α be a rankng of D. Then, for a vertex x V, we defne ψ (α, x) = σ (x) + {y U : α(y) > α(x)} and ψ + (α, x) = σ + (x) + {y U : α(y) < α(x)}, where U s the colour class of D contanng x. 3 NP -hardness Theorem 3.1 The problem of fndng a forward optmal rankng of a PCD s NP -hard. Proof: Ths proof s smlar to that of Theorem 4 n [9] but we shall use the followng problem nstead of the optmal lnear arrangement problem (OLAP), see [2], p.200. Maxmum lnear arrangement problem (MLAP). Instance: Graph G = (V, E(G)) and postve nteger k. Queston: Is there a rankng α so that {x,y} E(G) α(x) α(y) k. (If we replace by we get the OLAP.) The MLAP s NP -complete snce the OLAP s NP -complete and snce {x,y} E(G) α(x) α(y) + {x,y} E(G) α(x) α(y) s a constant dependng only on the number of vertces n G (G s the complement of G.) 4

5 Let G = (V, E) be a graph and let D = (V, A) be the symmetrc dgraph correspondng to G,.e. A = {xy, yx : {x, y} E}. Let also ɛ(xy) = 0.5 for every xy A. Then {x,y} E α(x) α(y) = 2f D (α). Hence, the MLAP s polynomal reducble to the problem of fndng an optmal forward rankng of a PCD.. 4 Lemmas Lemma 4.1 [7] Let K = (V, A) be a complete PCD wth n vertces, and let α be a rankng of K. Then f K (α) = σ (x)α(x) 1 6 n(n2 1) = 1 3 n(n2 1) σ + (x)α(x). (1) In the sequel, let D = (V, A) be a complete multpartte PCD wth n vertces contaned n r colour classes V 1,..., V r. The result of Lemma 4.2 (nvolvng ψ (α, x) only) was proved n [6]. Note that our proof s shorter. Lemma 4.2 Let α be a rankng of D. Then f D (α) = ψ (α, x)α(x) 1 6 n(n2 1) = 1 3 n(n2 1) ψ + (α, x)α(x). (2) Proof: For every colour class U of D, add the set of arcs {vw : v, w U, α(w) < α(v)} (all of weght one) to A. The new PCD H s complete. Note that the negatve (postve, resp.) score of a vertex x n H equals ψ (α, x) (ψ + (α, x), resp.). Now (2) follows from (1) and an obvous fact that f D (α) = f H (α).. 5

6 Lemma 4.3 Let α be a forward optmal rankng and β be a rankng of D, and let x and y be vertces n the same colour class of D. Then () f D (β xy ) f D (β) = m(σ + (y) σ + (x)), where m = α(y) α(x); () f σ + (x) = σ + (y), then α xy F OR(D) as well, and n partcular, π(x) = π(y); () α(x) < α(y) mples σ + (x) σ + (y). Proof: () can be proved by (2), and () and () are easy consequences of ().. 5 Forward optmal rankngs Lemma 4.3 () provdes a unque optmal rankng of the vertces n any colour class of D up to the vertces havng the same postve score. Now we wsh to fx completely the order of vertces havng the same postve score and contaned n the same colour class of D. To do that we consder a collecton Λ = {λ 1,..., λ r } of rankngs of the colour classes of D, where λ s a rankng of V, such that σ + (λ 1 (1)) σ + (λ 1 (2))... σ + (λ 1 ( V )), (3) for every = 1,..., r. We wsh to fnd all forward optmal rankngs α so that α V = λ for each = 1,..., r. We call such rankngs Λ-optmal. By Lemma 4.3, Λ-optmal rankngs exst for every Λ satsfyng (3). Fx a collecton Λ satsfyng (3). Then, for every Λ-optmal rankng α of D, ψ + (α, x) does not depend on α. Hence, we can defne ω(x) = ψ + (α, x), for a Λ-optmal rankng α and a vertex x of D. By (2), the problem to fnd all Λ-optmal rankngs of D s equvalent to the followng problem. Fnd all rankngs α of D whch provde mnmum to the functon subect to the constrants F (D, α) = ω(x)α(x) α V = λ, for every = 1,..., r. (4) Consder the followng sngle machne ob sequencng problem (for basc termnology on the theory of schedulng see, e.g., [3].) Let us be gven n obs 6

7 whch, for smplcty, are the vertces of D and a collecton Λ satsfyng (3). Let ω(x) be the weght and p(x) be the processng tme of a ob (vertex) x. Then, the total weghted completon tme C(D, α), when the obs are sequenced accordng to a permutaton α, s C(D, α) = ω(x) {p(y) : α(y) α(x)}. We wsh to mnmze C(D, α) subect to the constrants (4). But these constrants are ust the parallel chans constrants [10]. Horn [5] was the frst to propose an algorthm for fndng an optmal permutaton,.e. a permutaton provdng the mnmum total completon tme subect to the parallel chans constrants (he has really consdered more general forestlke constrants.) Obvously, the ob sequencng problem above, when all p(x) = 1, s equvalent to the problem of fndng Λ-optmal rankngs of D. We shall deal wth a modfcaton of Horn s algorthm due to Sdney (see Algorthm 2 n [10].) Sdney proved (Theorem 9 and Lemma 14 n [10]) that a permutaton s optmal f and only f t can be generated by hs algorthm. Below we descrbe Sdney s algorthm adopted to our problem. Let λ = (v () 1, v () 2,..., v n () ) (n = V ) for every = 1,..., r. For an ndex {1,..., r} and a par, k so that 1 k n, defne the set S () k = {v (), v +1, ()..., v () k } and ts average of ω: ρ(s() k ) = (k + 1) 1 k m= ω(v m () ). For {1,..., n }, a set S () k s called ρ -maxmal f, for every l {, + 1,..., n }, ρ(s () k ) ρ(s() l ) and, for every t {, + 1,..., k 1}, ρ(s() k ) > ρ(s () t ). We shall denote such a ρ -maxmal set by S () and ts average of ω by ρ (). Algorthm Choose a collecton Λ = {λ 1,..., λ r } satsfyng (3). Call λ = (v () 1, v () 2,..., v () n ) the th chan. 2. The ntal current permutaton α s the empty permutaton. For every = 1,..., r, the current frst vertex on the th chan has ndex m = 1 and set S () n +1 = and ρ () n +1 =. 3. For every = 1,..., r and = 1,..., n, compute S () and ρ (). 7

8 4. Choose an ndex so that ρ () m = max 1 t r ρ (t) m t. Let S m () = S () m,k. Append the elements of S m (), n ther natural order, to α. Set m = k If all S () are empty, stop; α s optmal. Otherwse, return to Step 4. By the dscusson above we obtan the followng characterzaton of forward optmal rankngs. Theorem 5.1 A rankng α s forward optmal f and only f t can be generated by Algorthm 1. To llustrate the last theorem consder the complete multpartte PCD H treated n [7], Fg.4. The PCD H has colour classes V 1 = {a, b}, V 2 = {c} and V 3 = {d, e}. The postve scores are σ + (a) = 2.6, σ + (b) = 1.3, σ + (c) = 1.7, σ + (d) = 1, σ + (e) = 1.4. The only collecton Λ satsfyng (3) s Λ = {(a, b), (c), (e, d)}. Hence, we obtan ω(a) = 2.6, ω(b) = 2.3, ω(c) = 1.7, ω(e) = 1.4, ω(d) = 2. The ρ -maxmal sets and ther averages of ω are S 1 = {a}, ρ 1 = 2.6, S 2 = {b}, ρ 2 = 2.3, S 1 = {c}, ρ 1 = 1.7, S 1 = {e, d}, ρ 1 = 1.7, S 2 = {d}, ρ 2 = 2. Therefore, by Algorthm 1, F OR(H) = {(a, b, e, d, c), (a, b, c, e, d)}. Ths set concdes wth that constructed n [7] usng Theorem 4 of [7]. A fast mplementaton of Algorthm 1 s based on the followng procedure (Procedure 2) for fndng S () and ρ () (n Step 3) due to Horn [5]. For = n, n 1,..., 1 (n that order) do the followng 3 steps: 1. F () = + 1 and ρ () 2. Whle F () n and ρ () 3. S () = ω(v () ) < ρ () F () do the followng: ρ () = ρ() (F () ) + ρ () F ()(F (F ()) F ()) F (F ()) F () = F (F ()) = {v (), v +1, ()..., v () F () 1 } 8

9 Lemma 5.2 Procedure 2 computes all S () 1, 2,..., n ) n tme O(n). and ρ () ( = 1, 2,..., r = Proof: Defne a potental functon Φ() as follows. Φ() s the mnmum nteger so that F Φ()+1 () = n + 1 (e.g. Φ(n ) = 0 snce F (n ) = n + 1). Clearly all Φ() 0, when = 1, 2,..., n. Let T () be the total tme used n the procedure, up to and ncludng the pont when we have computed, where a performance of Step 1 takes a total of one unt of tme, and each teraton of the whle-loop (.e. computng ρ () and F ()) takes one S () unt of tme. Step 1 sets T (n ) = 1, T () = T ( + 1) + 1, Φ(n ) = 0 and Φ() = Φ( + 1) + 1 when = 1, 2,..., n 1. Each tme one goes down nto the whle-loop t decreases Φ() by one, but ncreases T () by one. Ths means that T () + Φ() = 2(n ) + 1. Snce Φ(1) 0, we get that T (1) < 2n, whch provdes the desred complexty.. By Lemma 5.2 Steps 1, 2 and 3 can be completed n O(n) tme. Now by usng an approprate prorty queue, such as a heap (see [11]), we can perform Steps 4 and 5 n O(log r) tme per teraton. Snce there are at most n teratons of Steps 4 and 5, we obtan the followng: Theorem 5.3 Algorthm 1 runs n tme O(n log n). Algorthm 1 can be easly modfed to an algorthm for generatng all optmal forward rankngs. It s less trval, but stll not dffcult to obtan an algorthm for computng proper forward ranks of the vertces of D. By Lemma 4.3 (), we may fx a collecton Λ satsfyng (3) n the begnnng of the algorthm and, then, n the end recalculate the average rank of any vertex as the mean of the ranks of all vertces from the same colour class havng the same postve score. The only non-trval problem whch arses here s how to fnd the proper forward ranks of the vertces n ρ -maxmal sets wth the same average of ω. It s easy to see that, n order to solve the problem above, t s suffcent to solve the followng auxlary problem. Let P = {P 1, P 2,..., P r } be a collecton of sequences P = (R () 1, R () 2,..., R q () ), where R () k are dsont sets of V and = 1, 2,..., r; q 1, q 2,..., q r 1. A rankng, β, of all the sets R () k s feasble f, for every = 1, 2,..., r, 1 s < q mples that ) < β(r () ). Let F P denote the set of all feasble rankngs of the sets β(r () s 9

10 n P. We wsh to fnd E P (k, ), the average number of all vertces n front of taken over all feasble rankngs of P,.e. R () k E P (k, ) = 1 F P β F P β(x)<β(r () k ) X For each = 1, 2,..., r and k = 1, 2,..., q, E P (k, ) can be computed as follows. E P (k, ) = k 1 k =1 R() k + r =1 where Q P (k, k, q, q ) = q k =0 Q P (k, k, q, q ) k ( )( ) k+k 1 q +q k k k ( q) k q +q q k =1 R ( ) k, To show that the formula above s correct we ust notce that Q P (k, k, q, q ) s the probablty that a randomly chosen β F P wll have β(r ( ) k ) < β(r () k ) < ) β(r( k +1 ) where ) β(r( 0 ) = and β(r ( ) q +1) =. Indeed, there are ( ) k+k 1 k ways of permutng the frst k 1 sets of P together wth the frst k sets of P, there are also ( ) q +q k k q k ways of permutng the last q k sets of P together wth the last q k sets of P. The total number of permutatons of the sets n P together wth the sets n P s ( ) q +q q. Obvously, the formula above leads to a polynomal algorthm for fndng all E P (k, ). In order to keep the paper n approprate length we omt a detaled consderaton of an algorthm for fndng the proper forward ranks of the vertces n D. 6 Acknowledgments The authors would lke to thank N. Alon, G. Brghtwell and M. Kano for some helpful correspondence, and both referees for some useful suggestons. References [1] G. Brghtwell and P. Wnkler, Countng lnear extensons. Order 8 (1991)

11 [2] M.R. Garey and D.S. Johnson, Computers and Intractablty: a Gude to the Theory of N P -Completeness, Freeman, San Franssco, [3] R.L. Graham, E.L. Lawler, J.K. Lenstra and A.H.G. Rnnooy Kan, Optmzaton and approxmaton n determnstc sequencng and schedulng: a survey. Annals Dsc. Math. 5 (1979) [4] G. Gutn, Determnng the ranks of vertces n a complete multpartte graph of pared comparsons. Automaton and Remote Control no. 10 (1989) [5] W.A. Horn, Sngle machne ob sequencng wth treelke precedence orderng and lnear delay penaltes. SIAM J. Appl. Math. 23 (1972) [6] M. Kano, Rankng the vertces of an r-partte pared comparson dgraph. Dsc. Appl. Math. 17 (1987) [7] M. Kano and A. Sakamoto, Rankng the vertces of a weghted dgraph usng the length of forward arcs. Networks 13 (1983) [8] M. Kano and A. Sakamoto, Rankng the vertces of a pared comparson dgraph wth normal completeness theorems. Bull. Faculty Engneerng, Tokushma Unv. 20 (1983) [9] M. Kano and A. Sakamoto, Rankng the vertces of a pared comparson dgraph. SIAM J. Alg. Dsc. Meth. 6 (1985) [10] J.B. Sdney, Decomposton algorthms for sngle-machne sequencng wth precedence relatons and deferral costs. Operatons Res. 23 (1975) [11] J.W.J. Wllams, Algorthm 232: Heapsort. Comm. ACM 7 (1964)