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Linear Algebra and its Applications 434 (2011) 232 238 Contents lists available at ScienceDirect Linear Algebra and its Applications journal homepage: www.elsevier.com/locate/laa Minimum permanents on two faces of the polytope of doubly stochastic matrices Kyle Pula a, Seok-Zun Song b,,1, Ian M. Wanless c a b c Department of Mathematics, University of Denver, Denver, CO 80208, USA Department of Mathematics, Jeju National University, Jeju 690-756, Republic of Korea School of Mathematical Sciences, Monash University, Vic 3800, Australia A R T I C L E I N F O A B S T R A C T Article history: Received 6 July 2010 Accepted 18 August 2010 Available online 19 September 2010 Submitted by R.A. Brualdi AMS classification: 15A15 Keywords: Permanent Doubly stochastic Cohesive Barycentric We consider the minimum permanents and minimising matrices on the faces of the polytope of doubly stochastic matrices whose nonzero entries coincide with those of, respectively, [ ] [ ] In J U m,n = n,m In J and V J m,n 0 m,n = n,m. m J m,n J m,m Here J r,s denotes the r s matrix all of whose entries are 1, I n is the identity matrix of order n and 0 m is the m m zero matrix. We conjecture that V m,n is cohesive but not barycentric for 1 < n < m + m and that it is not cohesive for n m + m. We prove that it is cohesive for 1 < n < m + m and not cohesive for n 2m and confirm the conjecture computationally for n < 2m 200. We also show that U m,n is barycentric. 2010 Elsevier Inc. All rights reserved. This research was undertaken while the first and second authors were visiting Monash University. The first author s visit was supported by the Australian Academy of Science, the Australian-American Fulbright Commission, and the National Science Foundation. The second author s visit was supported by the National Research Foundation of Korea and the Australian Academy of Science. The third author s research is supported by ARC discovery Grant DP1093320. Corresponding author. E-mail addresses: jpula@math.du.edu (K. Pula), szsong@jejunu.ac.kr (S.-Z. Song), ian.wanless@monash.edu (I.M. Wanless). 1 His research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 2010-0003011). 0024-3795/$ - see front matter 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.laa.2010.08.015
K. Pula et al. / Linear Algebra and its Applications 434 (2011) 232 238 233 1. Introduction and preliminaries Let Ω n be the polytope of n n doubly stochastic matrices, that is, the n n nonnegative matrices whose row and column sums are all equal to 1. The permanent of an n n matrix A =[a ij ] is defined by per A = σ a 1σ(1) a nσ(n) where σ runs over all permutations of {1, 2,...,n}. Let D =[d ij ] be an n n nonnegative matrix with per D > 0, and let Ω(D) ={[x ij ] Ω n x ij = 0 whenever d ij = 0}. Then Ω(D) is a face of Ω n, and since it is nonempty and compact, Ω(D) contains at least one minimising matrix Y such that per Y per X for all X Ω(D). Let J r,s denote the r s matrix all of whose entries are 1, I n the identity matrix of order n and 0 m the m m zero matrix. In this paper, we study minimising matrices on the faces Ω(U m,n ) and Ω(V m,n ), where [ ] [ ] In J U m,n = n,m In J and V m,n = n,m. J m,n 0 m J m,n J m,m If per D > 0 then the barycenter b(d) of Ω(D) is given by b(d) = 1 P, per D P D where the summation extends over the set of all permutation matrices P with P D. Brualdi [1] defined an n-square (0, 1) matrix D to be cohesive if there is a matrix Z in the interior of Ω(D) for which per Z = min{per X X Ω(D)}. He defined an n-square (0, 1) matrix D to be barycentric if per b(d) = min{per X X Ω(D)}. Since b(d) always falls in the interior of Ω(D), being barycentric is a stronger property than being cohesive. We will consider the question of which values of parameters m, n make U m,n and V m,n barycentric, cohesive, or neither. In Section 2 we prove that V m,n is cohesive for 1 < n < m + m and not cohesive for n 2m. In Section 3 we show that U m,n is barycentric. Our results contribute towards solution of two problems from Minc s well-known catalogue of unsolved problems on permanents (see [2] for the most recent update). Problems 14 and 15 in Minc s list ask for a characterisation of cohesive and barycentric matrices, respectively. These problems were originally posed by Brualdi [1], who determined the minimising matrix on Ω(V 1,n ). Minc had resolved the face Ω(V m,2 ) in [4]. Song [5,7] determined the minimum permanent on Ω(V m,3 ), while the faces Ω(V 2,n ) and Ω(V 3,n ) were resolved by Song [6] and Song et al. [8,9], respectively. Taken together, the prior literature determines the minimising matrices of Ω(V m,n ) for m < 4orn < 4. Recall that an n n nonnegative matrix is said to be fully indecomposable if it contains no k (n k) zero submatrix for 1 k < n. We will use the following well-known Lemma from [3]. Lemma 1.1. Let D =[d ij ] be an n n fully indecomposable (0, 1) matrix, and suppose Y =[y ij ] is a minimising matrix on Ω(D). Then Y is fully indecomposable and per Y(i j) = per Y if d ij = 1 and y ij > 0, per Y(i j) per Y if d ij = 1 and y ij = 0. As usual, for any matrix M and lists L 1 and L 2 of row and column indices, respectively, M(L 1 L 2 ) denotes the submatrix formed by omitting the rows L 1 and columns L 2 from M. Terms of the form 0 0 occurring in our calculations should always be interpreted as 1. 2. The minimising matrices of Ω(V m,n ) In this section, we consider the minimum permanents and minimising matrices on the faces Ω(V m,n ). Throughout this section, we assume that m, n 2.
234 K. Pula et al. / Linear Algebra and its Applications 434 (2011) 232 238 Let Y be a minimising matrix on Ω(V m,n ). Since the last m rows and last m columns of V m,n are the same, we can use the averaging method on those rows and columns of Y (by Theorem 1 in Minc [4]). Without loss of generality, we may therefore assume Y =[y ij ] is a minimising matrix of the form: x i if i = j n, 0 if i n and j n and i /= j, a y ij = i if i n and j > n, (1) a j if i > n and j n, x if i > n and j > n. Note that Y is doubly stochastic so x i = 1 ma i for 1 i n. Since V m,n is fully indecomposable, it follows from Lemma 1.1 that Y is also fully indecomposable. In particular a i > 0 for all i, although it is plausible that x i = 0 for some i or that x = 0. We next consider the possible choices of {a i } in (1), i.e. those that minimise per Y. Theorem 2.1. The minimising matrix Y has a 1 = a 2 = = a n. Proof. Without loss of generality we assume that a 1 a 2 a n and hence x 1 x 2 x n. Aiming for a contradiction, assume that a n > a 1.Let p 0 = per Y(1,n 1,n), p 1 = per Y(1,n,n+ 1 1,n,n+ 1), p 2 = per Y(1,n,n+ 1,n+ 2 1,n,n+ 1,n+ 2). If x n > 0 then by Lemma 1.1, we have that and hence x 1 p 0 + m 2 a 2 1 p 1 = per Y(n n) = per Y = per Y(1 1) = x n p 0 + m 2 a 2 n p 1 p 0 = m2 (a 2 n a2 1 ) x 1 x n p 1 = m(a 1 + a n )p 1. (2) Also, as a 1 > 0, per Y = per Y(1 n + 1), = ma 1 per Y(1,n+ 1 1,n+ 1), = ma 1 (x n p 1 + (m 1) 2 a 2 n p 2). ) and similarly per Y = ma n (x 1 p 1 + (m 1) 2 a 2 1 p 2, which leads to (m 1) 2 p 2 = a nx 1 a 1 x n a 1 a n (a n a 1 ) p 1 = p 1 a 1 a n. However, expanding per Y along the first and nth rows we find per Y = x 1 x n p 0 + m 2 a 2 n x 1p 1 + m 2 a 2 1 x np 1 + m 2 (m 1) 2 a 2 1 a2 n p 2, (3) = (1 ma 1 )(1 ma n )m(a 1 + a n )p 1 + m 2 a 2 n (1 ma 1)p 1 + m 2 a 2 1 (1 ma n)p 1 + m 2 a 1 a n p 1, = mp 1 (a 1 + a n ma 1 a n ). (4) Note that although (2) is only valid for x n > 0 we are free to substitute it in (3) in the case x n = 0as well, since in that case p 0 is being multiplied by 0. Examining (4) we see that by varying a 1,a n while preserving a 1 + a n we could decrease per Y unless a 1 = a n. By assumption, Y is a minimising matrix so a 1 = a n, from which the result follows.
K. Pula et al. / Linear Algebra and its Applications 434 (2011) 232 238 235 In light of Theorem 2.1, for any given values of m and n, we are left with a one variable optimisation to find the minimising matrix in Ω(V m,n ) since Y is now determined by the value of x in (1). Let A x =[a ij ] be the (n + m) (n + m) matrix defined by 1 n (n m + m2 x) if i = j n, 0 if i n and j n and i /= j, a ij = (5) x if i > n and j > n, 1 (1 mx) otherwise. n ] when m n,orx [ ] m n, 1 in the case m > n. m 2 m Note that A x is doubly stochastic provided x [ 0, 1 m We next consider the problem of finding x in the stated range that minimises per A x (and hence satisfies per A x = per Y). Theorem 2.2. For n 2m and x > 0 we have per A x > per A 0 and thus A 0 is the unique minimising matrix in Ω(V m,n ). In contrast, for n < m + m it is never the case that A 0 is a minimising matrix in Ω(V m,n ). Proof. If m > n then A 0 is not even doubly stochastic, and if m = n then A 0 is not fully indecomposable. So by Lemma 1.1 we may assume that n > m. From(5) wehave per A x = In particular, and hence = m m ( ) 2 m i!n!(m i)! i (n m + i)! xi ( ) m i ( )2m 2i ( ) 1 1 n m+i n (1 mx) n (n m + m2 x), n!m! (n m + i)! n n+m i xi (1 mx) 2m 2i (n m + m 2 x) n m+i. (6) n! m! (n m)n m per A 0 =, (7) (n m)! nn+m per A x per A 0 = m ( ( = ( (n m + i)! xi (1 mx) 2m 2i ) n m m ( ) m ni (n m)! i 1 + m2 x n m 1 + m2 x n m (1 + mx) m (1 mx + 2m 2 x 2 ) m, = (1 + m 2 x 2 + 2m 3 x 3 ) m, 1 + m2 x n m ) n m (n m + m 2 x) i, ( ) m (1 mx) ( 2m 2i nx mx + m 2 x 2) i, i ) n m m ((1 m mx) 2 + nx mx + m 2 x 2) m, whenever n 2m. The first statement of the theorem follows. Next, consider x 0in(6), where per A x = n!m!(1 mx)2m (n m+m 2 x) n m (n m)!n n+m [ n!m!(n m)n m = (n m)!n n+m 1 m 2 x + + m 2 n!(m 1)!(n m) n m+1 (n m + 1)!n n+m 1 x + O(x 2 ) mn(n m) (n m + 1) x ] + O(x 2 ).
236 K. Pula et al. / Linear Algebra and its Applications 434 (2011) 232 238 It follows that when n(n m) <(n m + 1)m (or in other words, n < m + m), per A x < per A 0 for small positive x. This proves the second statement in the Theorem. For m n we know x [0, 1/m]. Having examined the situation at the lower end of that interval, we now turn our attention to the upper end. Theorem 2.3. A 1/m is not a minimising matrix of Ω(V m,n ) for m n. Proof. From (6) there is a polynomial q(x) such that per A x = m! n n x m (n m + m 2 x) n + (1 mx) 2 q(x). Thus the derivative of per A x at x = 1/m is 2m! m 2 m. Not only is per A x increasing at x = 1/m, but in fact the rate of increase depends only on m. Next we consider similar questions for the case m n. Theorem 2.4. V m,n is cohesive for m n. Proof. From (5), we have per A x = m!2 n n 2n for x [ m n, 1 m 2 m per A (m n)/m 2 Therefore, per A 1/m per A (m n)/m 2 ( n i ) n i ]. In particular, we have (m n + i)! (n m + m2 x) i (1 mx) (2n 2i) x m n+i (8) = m!2 (m n) m n (m n)!m 2m and per A 1/m = m! m m. = mm (m n)! m!(m n) m n > 1 for 0 < n m, and A 1/m cannot be a minimising matrix. Now consider that per A x = m! ( (1 mx) 2n x m n + n2 (1 mx) 2n 2 x m n+1 ) n 2n (n m + m 2 x) (m n)! (m n + 1)! +(n m + m 2 x) 2 r(x), for some polynomial r(x). Therefore the derivative of per A x at x = (m n)/m 2 is m! (m n + 1)! (m n)m n m 2 2m. In particular, it is negative so A (m n)/m 2 is not a minimising matrix either. Combining Theorems 2.2 2.4, wehave: Corollary 2.5. V m,n is cohesive for n < m + m but not for n 2m. For the cases not covered by this corollary, i.e. m + [ m n ] < 2m, we have demonstrated that per A x is increasing at both end points of the interval m n, 1 but it remains to be determined m 2 m whether the minimum actually occurs at x = (m n)/m 2. As reported below, we have investigated this question computationally for m 100.
K. Pula et al. / Linear Algebra and its Applications 434 (2011) 232 238 237 For n m, the barycenter of V m,n is located at A β where ( )( ) m n m 2 1 i=1 (i n m + i i 1)!(m i)! 2 1 β = ( )( ) 2 = m n m (i n m + 1)!(m i)! i i 2 β = m n m 2 + m i ((n m + i)!(m i)!i!) 1 m 2. m ((n m + i)!(m i)!i!) 1 The value of per b(v m,n ) can then be calculated from (6). For n < m, the barycenter is located at A β where n i ((m n + i)!(n i)!i!) 1 m 2 n ((m n + i)!(n i)!i!) 1 and the value of per b(v m,n ) can then be calculated from (8). In order to investigate small cases not covered by the preceding theory, two of the authors independently wrote programs for the computer algebra systems Maple and Mathematica. The results of their computations agreed and are as follows. Let P(x) = per A x and let P (x) denote its derivative. For 2 n < m 100, we found that P (x) has no rational roots in the interval [(m n)/m 2, ) and that P(x) is increasing at the barycenter. Either of these facts shows that V m,n is not barycentric in these cases although we know from Corollary 2.5 that it is cohesive. For 2 m n < 2m 200, we found that P(x) is increasing at the barycenter and that P (x) has no nonnegative rational roots when n /= m + m.ifn = m + m the only nonnegative rational root is x = 0. Again, either fact shows that V m,n is not barycentric. For 2 < m + m n < 2m 200,P(x) is monotone increasing throughout the interval [0, 1/m]. Once again, the only case in which P (x) has a root in this interval is for n = m + m and this root occurs at x = 0. Taken together with Corollary 2.5, this data suggests the following conjecture. Conjecture 2.1. V m,n is cohesive but not barycentric for 1 < n < m + m, while for n m + m, V m,n is not cohesive and A 0 is a minimising matrix. 3. The face Ω(U m,n ) We finish by determining the minimum permanent and minimising matrix on the face Ω(U m,n ). Note that for Ω(U m,n ) to be nonempty we require n m. Relying on Brualdi [1] for the case m = 1 and using a proof identical to that of Theorem 2.1 for m 2weget: Theorem 3.1. For any n m the unique minimising matrix in Ω(U m,n ) is A 0. By symmetry it is obvious that b(u m,n ) = A 0 and thus we also have: Corollary 3.2. U m,n is barycentric for any n m. The minimum permanent is given by (7). Corollary 3.3. For any n m the minimum permanent in U m,n is n! m! (n m)n m per A 0 = per b(u m,n ) = (n m)!. n n+m For example, the minimum permanent on Ω(U 4,n ) is 1)(n 2)(n 3)(n 4)n 4 per b(u 4,n ) = 4! (n n n+3, which is also the minimum permanent on Ω(V 4,n ) for n 6.
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