On Leonid Gurvits s proof for permanents

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1 On Leond Gurvts s proof for permanents Monque Laurent and Alexander Schrver Abstract We gve a concse exposton of the elegant proof gven recently by Leond Gurvts for several lower bounds on permanents. 1. Permanents The permanent of a square matrx A = (a, ) n,=1 s defned by (1) pera = π S n =1 a,π(), where S n denotes the set of all permutatons of {1,...,n}. (The name permanent has ts root n Cauchy s fonctons symétrques permanentes [2], as a counterpart to fonctons symétrques alternées the determnants.) Despte ts appearance as the smpler twn-brother of the determnant, the permanent has turned out to be much less tractable. Whereas the determnant can be calculated quckly (n polynomal tme, wth Gaussan elmnaton), determnng the permanent s dffcult ( number-p-complete ). As yet, the algebrac behavour of the permanent functon has appeared to a large extent unmanageable, and ts algebrac relevance moderate. Most frutful research on permanents concerns lower and upper bounds for the permanent (see the book of Mnc [12]). In ths paper we wll consder only lower bounds. Indeed, most nterest n the permanent functon came from the famous Van der Waerden conecture [16] (n fact formulated as queston), statng that the permanent of any n n doubly stochastc matrx s at least n!/n n, the mnmum beng attaned only by the matrx wth all entres equal to 1/n. (A matrx s doubly stochastc f t s nonnegatve and each row and column sum s equal to 1.) Ths conecture was unsolved for over ffty years, whch, when contrasted wth ts smple form, also contrbuted to the reputaton of ntractablty of permanents. Fnally, Falkman [6] and Egorychev [4] were able to prove ths conecture, usng a classcal nequalty of Alexandroff and Fenchel. The proof wth egenvalue technques also revealed some unexpected nce algebrac behavour of the permanent functon (see, also for background, Knuth [9] and van Lnt [10,11]). Before the proof of the Van der Waerden conecture was found, a weaker conecture was formulated by Erdős and Rény [5]. It clams the exstence of a real number α 3 > 1 such that, for each nonnegatve nteger-valued n n matrx A wth all row and column sums equal to 3, the permanent of A s at least α3 n. Ths would follow from the Van der Waerden conecture, snce 1 3A s doubly stochastc, hence 1

2 (2) pera = 3 n per( 1 n! ( 3 ) n. 3A) 3n n n e Erdős and Rény also asked for the largest value of α 3 one can take n ths bound. More generally, for any natural number k, they asked for the largest real number α k such that each nonnegatve nteger-valued n n matrx A = (a, ) wth all row and column sums equal to k has permanent at least αk n. Note that ths permanent s equal to the number of perfect matchngs n the k-regular bpartte graph wth vertces u 1,...,u n, v 1,...,v n, where u and v are connected by a, edges. In 1979, before the Van der Waerden conecture was settled, the frst conecture of Erdős and Rény was proved, by Bang [1], Fredland [7], and Voorhoeve [15]. Bang and Fredland n fact showed that the permanent of any n n doubly stochastc matrx s at least e n. Note that lm n (n!/n n ) 1/n = e 1, so ths may be seen as an asymptotc proof of the Van der Waerden conecture. It also mples that the number α k of Erdős and Rény s at least k/e; n partcular, α 3 3/e > 1. The proof of Voorhoeve gves a better bound: α 3 4/3. In fact, ths bound s best possble. Indeed, t follows from a theorem of Wlf [18] that α 3 4/3, and more generally (3) α k (k 1)k 1 k k 2, and Schrver and Valant [14] conectured that equalty holds for each k. For k = 1, 2, ths s trval, and for k = 3 ths follows from Voorhoeve s theorem. The proof of Voorhoeve that α 3 4/3 s very short and elegant, and t seduces one to search for smlar arguments for general k. However, t was only at the cost of frghtenng techncaltes that Schrver [13] found a proof that equalty ndeed holds n (3) for each k. Ths amounts to a lower bound for permanents of doubly stochastc matrces n whch all entres are nteger multples of 1/k. Under ths restrcton, ths bound s larger than the Van der Waerden bound. In fact, both the bound of Falkman-Egorychev and that of Schrver are best possble, n dfferent asymptotc drectons. Let µ(k, n) denotes the mnmum permanent of n n doubly stochastc matrces wth all entres beng nteger multples of 1/k. Then the two bounds state (4) µ(k, n) n! ( k 1 ) (k 1)n. n n and µ(k, n) k They are best possble n the followng sense: (5) nf k µ(k, n)1/n = n!1/n n and nf n µ(k, n)1/n = ( k 1 k ) k 1. The proof of Falkman and Egorychev requres some nontrval theorems, and the proof of Schrver s combnatorally complex. It was a bg surprse when Leond Gurvts [8] gave an amazngly short proof of the two bounds. En route, he extended Schrver s theorem to: each doubly stochastc n n matrx wth at most k nonzeros n each column has permanent at least ((k 1)/k) (k 1)n. In fact, Gurvts proved that each doubly stochastc n n matrx 2

3 A satsfes (6) pera g(mn{, λ A ()}) =1 (Gurvts s nequalty), where λ A () s the number of nonzeros n the th column of A, and where ( k 1 ) k 1 (7) g(0) := 1 and g(k) := for k = 1, 2,... k (settng 0 0 = 1). Gurvts bound mples both the bound of Falkman and Egorychev and the bound of Schrver see Secton 4. We gve here a proof based on Gurvts s proof. The buldng blocks of the proof are from Gurvts [8], but we take a few shortcuts. 2. Descrpton of Gurvts s approach As usual, let R + := {x R x 0}. Recall the geometrc-arthmetc mean nequalty, sayng that f λ 1,...,λ n, x 1,...,x n R + wth n =1 λ = 1, then (8) n λ x =1 =1 x λ. It amounts to the concavty of the log functon. For any n n matrx A, defne the followng multvarate polynomal p A n the varables x 1,...,x n : (9) p A (x 1,...,x n ) := a x = =1 =1 =1 n a, x, where a denotes the th row of A (n a x we take a as a row vector and x = (x 1,...,x n ) T as a column vector). So p A s homogeneous of degree n. Then the coeffcent of the monomal x 1 x n n p A s equal to pera. Ths can also be stated n terms of partal dervatves as (10) pera = n p A x 1 x n. Note that the latter expresson s a constant functon. The crux of the method s to consder more generally the followng dervatves of p A, for any = 0,...,n: (11) q (x 1,...,x ) := n p A x +1 x n. x +1 = =x n=0 3

4 So q R[x 1,...,x ]. Then q n = p A and q 0 = pera. The polynomals q wll be related through the followng concept of capacty of a polynomal. The capacty cap(p) of a polynomal p R[x 1,...,x n ] s defned as (12) cap(p) := nf p(x), where the nfmum ranges over all x R n + wth n =1 x = 1. So cap(q 0 ) = pera. Moreover, we have: Proposton 1. If A s doubly stochastc, then cap(p A ) = 1. Proof. For any x R n + wth n =1 x = 1 we have, usng the geometrc-arthmetc mean nequalty (8): (13) p A (x) = a x x a, = x a, = Hence cap(p A ) 1. As p A (1,...,1) = 1, ths gves cap(p A ) = 1. P x a, = Then Gurvts s nequalty (6) follows nductvely from the nequalty (14) cap(q 1 ) cap(q )g(mn{, λ A ()}) x = 1. for = 1,...,n, assumng A to be nonnegatve. Ths s the basc nequalty n Gurvts s proof, whch s establshed usng the concept of H-stable polynomal, as follows. Defne C + := {z C Rez 0} and C ++ := {z C Rez > 0}. A polynomal p C[x 1,...,x n ] s called H-stable f p has no zeros n C n ++. (Here H stands for halfplane. ) Note that for any doubly stoachastc matrx A, the polynomal p A ndeed s H- stable. For any polynomal p R[x 1,...,x n ], let the polynomal p R[x 1,...,x n 1 ] be defned by (15) p (x 1,...,x n 1 ) := p. x n x n=0 Then the polynomals ntroduced n (11) satsfy q 1 = (q ) for = 1,...,n. As noted above, we need to show nequalty (14). Ths s what the followng key result of Gurvts does, relatng cap(p) and cap(p ). Theorem 1. Let p R + [x 1,...,x n ] be H-stable and homogeneous of degree n. Then p 0 or p s H-stable. Moreover, (16) cap(p ) cap(p)g(k), where k = deg xn (p) denotes the degree of x n n p. 4

5 Note that the degree of the varable x n q s at most mn{, λ A ()}. Hence, as g s monotone nonncreasng, (14) ndeed follows. 3. Proof of Theorem 1 We frst prove a lemma. For any x C n, let Rex := (Rex 1,...,Rex n ). Lemma 1. Let p C[x 1,...,x n ] be H-stable and homogeneous. Then for each x C n +: (17) p(x) p(re x). Proof. By contnuty, we can assume x C n ++. Then, as p s H-stable, p(rex) 0. Fxng x, consder p(x + srex) as a functon of s C. As p s homogeneous, we can wrte (18) p(x + srex) = p(rex) m (s b ) =1 for b 1,...,b m C, where m s the total degree of p. For each, as p(x + b Rex) = 0 and as p s H-stable, we know x + b Rex C n ++, and so Re(1 + b ) 0, that s, Reb 1, whch mples b 1. Therefore, (19) p(x) = p(x + 0Rex) = p(rex) m b p(rex). =1 Now we can prove Theorem 1. n 1 =1 Rey = 1: It suffces to prove that for each y C n 1 ++ wth (20) () f p (y) = 0 then p 0, and () f y s real, then p (y) cap(p)g(k). Before provng (20), note that for any real t > 0, p(rey, t) (21) cap(p). t Indeed, let λ := t 1/n and x := λ (Rey, t). We chose λ so that ( =1 x n 1 ) = λ n =1 Rey t = 1. Hence, as p s homogeneous of degree n, (22) cap(p) p(x) = λ n p(rey, t) = p(rey, t), t and we have (21). We now prove (20). Frst assume that p(y, 0) = 0. Then by (17) we have p(rey, 0) = 0 (snce 0 = p(y, 0) p(rey, 0) 0). Moreover, 5

6 (23) p p(y, t) p(y, 0) p(y, t) (y) = lm = lm, t 0 t t 0 t and a smlar expresson holds for p (Rey). By (17), as all coeffcents of p are nonnegatve, p(rey, t) p(y, t) for all t 0. So, usng (21), p(rey, t) (24) cap(p) lm = p p(y, t) (Rey) lm = p (y). t 0 t t 0 t Ths mples (20): we have () snce f p (y) = 0 then p (Rey) = 0, so p 0 (as all coeffcents of p are nonnegatve); () follows as g(k) 1 for each k. Second assume that p(y, t) has degree at most 1, as a polynomal n t. Then also p(re y, t) has degree at most 1, snce p(re y, t) p(y, t). Moreover, (25) p p(y, t) (y) = lm, t t and a smlar expresson holds for p (Rey). Now agan usng (21), p(rey, t) (26) cap(p) lm = p p(y, t) (Rey) lm = p (y), t t t t agan mplyng (20). So we can assume that p(y, 0) 0 and that p(y, t) has degree at least 2, as a polynomal n t. Ths mples k 2. Snce p(y, 0) 0, we can wrte (27) p(y, t) = p(y, 0) k (1 + a t) =1 for some a 1,...,a k C. Hence p (y) = p(y, 0) k =1 a. As p(y, t) has degree at least 2, not all a are 0. Moreover, for = 1,...,k: (28) f a 0, then a 1 s a nonnegatve real lnear combnaton of y 1,...,y n 1. Otherwse there s a lne n the complex plane C through 0 that separates a 1 from y 1,...,y n 1. So there exsts a λ C such that Re(λa 1 ) < 0 and Re(λy ) > 0 for = 1,...,n 1. Hence (λy, λa 1 ) belongs to C n ++. However, p(λy, λa 1 ) = λ n p(y, a 1 ) = 0 (ths follows by substtutng t = a 1 n (27)). Ths contradcts the H-stablty of p, and thus proves (28). As the y belong to C ++, (28) n partcular mples that Rea > 0 f a 0. Hence, as not all a are 0, k =1 a 0. Therefore p (y) = p(y, 0) k =1 a 0. Ths gves (20)(). To prove (20)() we now assume that y s real. Then by (28), all a are real nonnegatve. By scalng p we can assume that p(y, 0) = 1. Set (29) t := k (k 1)p (y). 6

7 Then, usng the geometrc-arthmetc mean nequalty (8) and the fact that p (y) = k =1 a, (30) p(y, t) = k ( 1 (1 + a t) k =1 ( k ) k. k 1 k k ( 1 ) k ( (1 + a t)) = k (k + p (y)t) = ) k = k 1 =1 Therefore, by (21), (31) cap(p) p(y, t) t 1 t ( k ) k ( k ) k 1. = p (y) k 1 k 1 The value (29) for t was determned by Gurvts to yeld the best nequalty relatng cap(p ) and cap(p). 4. Applcatons to permanents Corollary 1a. For any nonegatve n n matrx A: (32) pera cap(p A ) g(mn{, λ A ()}). =1 Proof. We may assume that A has no zero row. Then, as p A (x) = 0 mples that a x = 0 for some, p A s H-stable. Defne q R + [x 1,...,x ] as n (11). Then by Theorem 1, for = 1,...,n, (33) cap(q 1 ) cap(q )g(deg x (q )) cap(q )g(mn{, λ A ()}), snce deg x (q ) mn{, λ A ()} and g s monotone nonncreasng. As cap(q 0 ) = pera, (32) follows by nducton. If A s doubly stochastc, cap(p A ) = 1 (Proposton 1), and hence Corollary 1a gves Gurvts s nequalty (6). Ths mples the theorem of Falkman [6] and Egorychev [4] (Van der Waerden s conecture [16]): Corollary 1b. If A s a doubly stochastc n n matrx, then (34) pera n! n n. Proof. By (32), 7

8 (35) pera ( 1 ) 1 n ( 1) 1 = = n! n n. =1 =1 Another consequence of Corollary 1a s the bound of Voorhoeve [15] (for k = 3) and Schrver [13]: Corollary 1c. If A s a nonnegatve nteger n n matrx wth all row and column sums equal to k, then (36) pera ( (k 1) k 1 k k 2 ) n. Proof. Let B := 1 ka. Then B s doubly stochastc and each column has at most k nonzeros. Hence by Corollary 1a, (37) pera = k n perb k n( k 1 ) (k 1)n ( (k 1) k 1) n. = k k k 2 For each fxed k, the base (k 1) k 1 /k k 2 n (36) s best possble (Wlf [18]). It s even best possble when A s restrcted to 0, 1 matrces (Wanless [17]). As was observed by Henryk Mnc, Corollary 1c for k = 6 mples the (currently best known) lower bound of for the 3-dmensonal dmer constant (see Cucu [3]). One can also derve unqueness of the doubly stochastc matrx havng mnmum permanent (a result of Egorychev [4]). Corollary 1d. Let A = (a, ) be a doubly stochastc n n matrx wth pera = n!/n n. Then a, = 1/n for all,. Proof. By symmetry t suffces to show that all entres n the last column of A are equal. Let the polynomals q be as n (11). Then, snce equalty holds n (33), (38) nf y q n 1(y) = cap(q n 1 ) = ( n 1 ) n 1cap(qn ( n 1 ) = n n ) n 1, where y ranges over y R+ n 1 wth n 1 =1 y = 1. Now for any such y we have the followng, where and k range over 1,...,n and ranges over 1,...,n 1, and where a denotes the th row of A wth the last entry chopped off: (39) q n 1 (y) = a k,n a y k k k k(a y) ak,n = k (a y) ak,n = (a y) 1 a,n = ( ) 1 a,n ( a, y = (1 a,n ) a ), 1 a,n y 1 a,n ( ) ( (1 a,n ) 1 a,n y a )( ), = (1 a,n ) 1 a,n y a, ( )( ) = (1 a,n ) 1 a,n y = ( (1 a,n ) 1 a,n n 1 ) n 1. n 8

9 Here we used, n the frst two nequaltes, the geometrc-arthmetc mean nequalty (8) and, n the last nequalty, the log-convexty of the functon x x (note that (n 1)/n s the average of the 1 a,n ). By (38) we must have equalty n the last nequalty of (39). Hence 1 a,n s the same for all, and so the last column of A s constant. Acknowledgements. We thank the referees for helpful comments mprovng the presentaton of the paper. References [1] T. Bang, On matrx-functons gvng a good approxmaton to the v.d.waerden permanent conecture, Preprnt Seres 1979 No. 30, Matematsk Insttut, Københavns Unverstet, Copenhagen, [2] A.-L. Cauchy, Mémore sur les fonctons qu ne peuvent obtenr que deux valeurs égales et de sgnes contrares par sute des transpostons opérées entre les varables qu elles renferment, Journal de l École Polytechnque 27, no. 10 (1812) [3] M. Cucu, An mproved upper bound for the 3-dmensonal dmer problem, Duke Math. J. 94 (1998) [4] G. P. Egorychev, Proof of the van der Waerden conecture for permanents [n Russan], Sbrsk. Mat. Zh. 22, no. 6 (1981) [Englsh translaton: Sberan Math. J. 22 (1981) ]. [5] P. Erdős, A. Rény, On random matrces II, Studa Sc. Math. Hungar. 3 (1968) [6] D. I. Falkman, Proof of the van der Waerden conecture regardng the permanent of a doubly stochastc matrx [n Russan], Mat. Zametk 29 (1981) [Englsh translaton: Math. Notes 29 (1981) ]. [7] S. Fredland, A lower bound for the permanent of a doubly stochastc matrx, Ann. of Math. (2) 110 (1979) [8] L. Gurvts, Van der Waerden/Schrver-Valant lke conectures and stable (aka hyperbolc) homogeneous polynomals: one theorem for all, Electron. J. Combn. 15 (2008) # R66. [9] D. E. Knuth, A permanent nequalty, Amer. Math. Monthly 88 (1981) [10] J. H. van Lnt, Notes on Egortsev s proof of the van der Waerden conecture, Lnear Algebra Appl. 39 (1981) 1 8. [11], The van der Waerden conecture: two proofs n one year, Math. Intellgencer 4, no. 2 (1982) [12] H. Mnc, Permanents, Addson-Wesley, Readng MA, [13] A. Schrver, Countng 1-factors n regular bpartte graphs, J. Combn. Theory Ser. B 72 (1998) [14] A. Schrver, W. G. Valant, On lower bounds for permanents, Indag. Math. (N.S.) 42 (1980) [15] M. Voorhoeve, A lower bound for the permanents of certan (0, 1)-matrces, Indag. Math. (N.S.) 41 (1979) [16] B. L. van der Waerden, [Aufgabe] 45, Jahresber. Deutsch. Math.-Veren. 35 (1926)

10 [17] I. M. Wanless, Addendum to Schrver s work on mnmum permanents, Combnatorca 26 (2006) [18] H. S. Wlf, On the permanent of a doubly stochastc matrx, Canad. J. Math. 18 (1966) Monque Laurent receved her Ph.D. n mathematcs at the Unversty Pars Dderot n After that she became researcher at CNRS n Pars, frst wth the Unversty Pars Dauphne and later wth Ecole Normale Supéreure. Snce 1997 she s researcher at CWI (Centrum Wskunde & Informatca) n Amsterdam and, snce 2009, she s also professor at Tlburg Unversty. CWI, Scence Park 123, 1098 XG Amsterdam, The Netherlands, and Department of Econometrcs and Operatons Research, Tlburg Unversty, Tlburg, The Netherlands [email protected] Alexander Schrver receved hs Ph.D. n mathematcs at the Free Unversty n Amsterdam n He was a professor of mathematcs at Tlburg Unversty, and snce 1989 researcher at CWI (Centrum Wskunde & Informatca) n Amsterdam and professor of mathematcs at the Unversty of Amsterdam. CWI, Scence Park 123, 1098 XG Amsterdam, The Netherlands, and Department of Mathematcs, Unversty of Amsterdam, Amsterdam, The Netherlands [email protected] 10