Bethe and M-Bethe Permanent Inequalities

Bethe and M-Bethe Permanent Inequalities

Roxana Smarandache Martin Haenggi Departments of Mathematics and Electrical Engineering Department of Electrical Engineering University of Notre Dame University of Notre Dame Notre Dame, IN 46556, USA Notre Dame, IN 46556, USA [email protected] [email protected]

Abstract—In [1], it was conjectured that the permanent of proof of the inequality perm (θ) 6 perm(θ) that uses only P B a P-lifting θ↑ of a matrix θ of degree M is less than the combinatorial definition of the Bethe-permanent.3 or equal to the Mth power of the permanent perm(θ), i.e., P In this paper, we prove this conjecture and explore some perm(θ↑ ) 6 perm(θ)M and, consequently, that the degree-M θ θ properties of the permanent of block matrices that are lifts Bethe permanent permM,B( ) of a matrix is less than or equal θ θ θ θ of a matrix; these matrices are the matrices of interest when to the permanent perm( ) of , i.e., permM,B( ) 6 perm( ). In this paper, we prove these related conjectures and show some studying the degree-M Bethe permanent. Additional examples properties of the permanent of block matrices that are lifts of and explanations of the techniques used can be found in [8]. a matrix. As a corollary, we obtain an alternative proof of the inequality permB(θ) 6 perm(θ) on the Bethe permanent of the B. Related work base matrix θ, which, in contrast to the one given in [2], uses only the combinatorial definition of the Bethe-permanent. The literature on permanents and on adjacent areas (of The results have implications in coding theory. Since a P-lifting counting perfect matchings, counting 0-1 matrices with spec- corresponds to an M-graph cover and thus to a protograph- ified row and column sums, etc.) is vast. Apart from the based LDPC code, the results may help explain the performance previously mentioned papers, the most relevant papers to our of these codes. work are the one by Chertkov & Yedidia [4] that studies I.INTRODUCTION the so-called fractional free energy functionals and resulting lower and upper bounds on the permanent of a non-negative A. Background matrix, the papers [9] (on counting perfect matchings in The concept of the Bethe permanent was introduced in [3], random graph covers), [10] (on counting matchings in graphs [4] to denote the approximation of a permanent of a non- with the help of the sum-product algorithm)4, and [3], [11], negative matrix1 by solving a certain minimization problem [12] (on max-product/min-sum algorithms based approaches of the Bethe free energy with the sum-product algorithm. to the maximum weight perfect matching problem). Relevant In his paper [1], Vontobel uses the term Bethe permanent is also the work on approximating the permanent of a non- to denote this approximation and provides reasons for which negative matrix using Markov-chain-Monte-Carlo-based meth- the approximation works well by showing that the Bethe free ods [13], or fully polynomial-time randomized approximation energy is a convex function and that the sum-product algorithm schemes [14] or Bethe-approximation based methods or sum- finds its minimum efficiently.2 product-algorithm (SPA) based method [3], [15].5 In the recent paper [2], Gurvits shows that the permanent of a matrix is lower bounded by its Bethe permanent, i.e., C. Notation and definitions permB(θ) 6 perm(θ), and discusses conjectures on the A non-negative matrix is here a matrix with non-negative constant C in the inequality perm(θ) 6 C ·permB(θ). Related real entries. Rows and columns of matrices and entries of to the results of Gurvits, Vontobel [1] formulates a conjecture vectors will be indexed starting at 1. For a positive integer M, that the permanent of an M-lift θ↑P of a matrix θ is less than we will use the common notation [M] , {1,...,M}. We will or equal to the Mth power of the permanent perm(θ), i.e., also use the common notation hij or Hij to denote the (i, j)th perm(θ↑P) 6 perm(θ)M and, consequently, that the degree entry of a matrix H when there is no ambiguity in the indices H M-Bethe permanent permM,B(θ) of a matrix θ is less than and hi,j or i,j , respectively, when one of the two indices H or equal to the permanent perm(θ) of θ, i.e., permM,B(θ) 6 is not a simple digit, e.g., hi,m−1, i,m−1, respectively. |α| perm(θ). A proof of his conjecture would imply an alternative denotes the cardinality (number of elements) of the set α. For positive integers m,M, the set of all permutations on the set 1 A non-negative matrix contains only non-negative real entries. [m] is denoted by Sm, while the set of all M ×M permutation 2 Although its definition looks simpler than that of the determinant, the matrices is denoted by . In addition, will be the permanent does not have the properties of the determinant that enable efficient PM Mm(PM ) computation [5]. In terms of complexity classes, the computation of the 3 permanent is in the complexity class ♯P [6], where ♯P is the set of the counting The formal definition of the Bethe and M-Bethe permanents is given in problems associated with the decision problems in the class NP. Even the Definition 1. computation of the permanent of 0-1 matrices restricted to have only three 4Computing the permanent is related to counting perfect matchings. ones per row is ♯P-complete [7]. 5See [1] for a more detailed account of these and other related papers. set of all m × m block matrices with entries in PM , i.e., the A. Rewriting the permanent products of lifts of matrices entries are permutation matrices of size M × M: In this subsection, we present an algorithm that lets us P Mm(PM ) , {P = (Pij ) | Pij ∈PM , ∀i, j ∈ [m]}. rewrite the permanent-products of a -lifting of θ into a form useful for proving the conjecture. Finally, the permanent of an m × m-matrix with real entries Let θ = (θ ) be a non-negative matrix of size m × m and is defined to be ij let P = (Pij ) ∈ Mm(PM ). Let τ ∈ SmM be a permutation perm(θ) , θiσ(i) . on the set [mM] and let σ∈Sm i∈[m] X Y A , (θ↑P) Note that in contrast, the determinant of θ is τ iτ(i) i∈Y[mM] det(θ) , sgn(σ) θ , iσ(i) be a non-zero permanent-product of θ↑P, which is a non-zero σ∈Sm i∈[m] X Y term of where sgn(σ) is the signature operator. perm(θ↑P)= (θ↑P) . Definition 1. Let m,M be two positive integers and θ be a iτ(i) τ∈SmM i∈[mM] non-negative m × m matrix. X Y ↑P We first observe that, since Aτ is assumed to be non- • For a matrix P ∈ Mm(PM ), the P-lifting θ of θ of degree M is defined as zero, for each i ∈ [mM], there exists j, l ∈ [m] such that ↑P (θ ) = θjl. Indeed, let i ∈ [mM], then i ∈ J and θ P ... θ P iτ(i) 11 11 1m 1m τ(i) ∈ L, for some j, l ∈ [m], where I , {(j − 1)M + ↑P , . . θ  . .  , 1,...,jM} and L , {(l − 1)M + 1,...,lM}. Therefore ↑P θm1Pm1 ... θmmPmm (θ )iτ(i) is a non-zero entry in the matrix-entry θjlPjl of   θ↑P. Since all its nonzero entries of θ P are equal to i.e., as an m×m block matrix with its (i, j)-th entry equal jl jl θ , we obtain that (θ↑P) = θ . Therefore, the product to the matrix θ P , where P is an M ×M permutation jl iτ(i) jl ij ij ij A , θ↑P can be rewritten as a product of entries matrix in . (It results in an mM mM matrix.) τ ( )iτ(i) PM × i∈[mM] • The degree-M Bethe permanent of θ is defined as θjl of theQ matrix θ, j, l ∈ [m]. Let 1/M perm (θ) , perm(θ↑P) , ατ , {i ∈ J | τ(i) ∈ L} , (1) B,M P∈Mm(PM ) jl τ , τ where the angular brackets represent the arithmetic av- rjl |αjl|. (2) ↑P erage of perm(θ ) over all P ∈Mm(PM ). ↑P Then, (θ )iτ(i) = θjl, for all i ∈ αjl and for all j, l ∈ [m], • The Bethe permanent of θ is then defined as therefore perm (θ) , lim sup perm (θ). m B B,M r 1 r 2 r r M→∞ ↑P j j jm jl (θ )iτ(i) = θj1 θj2 ··· θjm = θjl , ∀j ∈ [m]. Since the permanent operator is invariant to the elementary i∈J l=1 Y Y operations of interchanging rows or columns, when taking the Since each row and each column of θ↑P must contribute to permanent, we can assume, without loss of generality, that τ the product exactly once, the matrix ατ , (αjl)j,l with the matrices P ∈Mm(PM ) have P1j = Pi1 = IM , for all i, j ∈ τ set αjl as its entry (j, l) satisfies [m], where IM is the identity matrix of size M × M. We call such matrices reduced. m τ τ ′ ′ τ αjl αjl′ = ∅, ∀j,l,l ∈ [m],l 6= l , αjl = J , (3) Definition 2. A matrix P = (Pij ) ∈Mm(PM ) is reduced if \ l[=1 P1j = Pi1 = IM , for all i, j ∈ [m]. τ from which it follows that 0 6 rjl 6 M, ∀j, l ∈ [m] and P Remark 1. Note that a -lifting of a matrix θ corresponds to m an M-graph cover of the protograph (base graph) described by rτ = M, ∀j ∈ [m], ↑P jl θ. Therefore we can consider θ to represent a protograph- l=1 based LDPC code and θ to be its protomatrix (also called its Xm τ base matrix or its mother matrix) [16].  rjl = M, ∀l ∈ [m]. (4) j=1 II. THE PERMANENT OF A MATRIX LIFT X τ In [1], it was conjectured that for any non-negative square Therefore, the matrix Rτ , (rij )i,j∈[m] corresponding to matrix θ and for any P ∈Mm(PM ), we have the inequality Aτ has positive entries and all row and column sums equal M. It will henceforth be referred to as the exponent matrix. perm(θ↑P) 6 perm(θ)M . For each σ ∈ Sm, let Pσ ∈ Pm be the m × m In this section we prove this conjecture and several related permutation matrix corresponding to σ and let tτσ , ↑P ↑P τ τ τ > results on the structure of the perm(θ ) of the lift θ of min{r1σ(1), r2σ(2),...,rmσ(m)} 0. Then Rτ − tτσPσ is the matrix θ, for any non-negative matrix θ. a positive matrix with the sums of all entries on each row and

2 3 2 2 3 4 4 2 2 2 2 each column equal to M − tτσ and with all its entries equal So a b c e f g h i = (aei)(bfg) (ceg) (afh) . It can be to the ones on the same positions of Rτ except for the entries easily seen that this factorization is unique (which is not corresponding to the permutation σ, which decreased by the always the case though). same amount tτσ. We can index the set {σ ∈ Sm} , {σk ∈ Sm, k ∈ [m!]} and compute sequentially B. Grouping entries in the permanent product

Rτ,1 , Rτ The rewriting algorithm presented in Section II-A provides m m k rjl a way to rewrite the product (θjl) as a product j=1 l=1 Rτ,k+1 , Rτ,k − tτσk Pσk = Rτ − tτσs Pσs , k > 2, tτσ s=1 θ1σ(1)θ2σ(2) ··· θmσ(m) QbutQ does not tell us exactly X σ∈Sm where the sums of all entries on each row and each column ↑P k howQ to combine the entries (θ
)iτ(i) to obtain this rewriting. of Rτ,k+1 are all equal to M − tτσs . The algorithm runs Is there a way to algorithmically combine the indices of the t s=1 sets ατ to form the products θ θ ··· θ τσ for until all non-zero entries get changedP into zero entries, see jl 1σ(1) 2σ(2) mσ(m) Example 1 for an illustration of this process. Consequently, all σ ∈ Sm? The answer is yes, as we explain in the next example of a concrete P-lifting of θ from Example 1
with P the matrix R− tτσPσ =0. This yields R = tτσPσ, σ∈Sm σ∈Sm reduced. P m m P , leading to A θ↑P θ rjl Before presenting it, let us introduce a new matrix ατ τ = i∈[mM]( )iτ(i) = ( jl) τ j=1 l=1 (αjl) obtained from ατ by substituting each index (j−1)M+k Q Q Q in an entry set by k, k ∈ [M]. Then, the properties (3) of the θ θ θ tτσ = 1σ(1) 2σ(2) ··· mσ(m) matrix ατ carry over to the following properties of the matrix σ∈S Ym
ατ : and tτσ = M. m σ∈Sm τ τ ′ ′ τ α α ′ = ∅, ∀j,l,l ∈ [m],l 6= l , α = [M]. (5) ThisP algorithm always terminates, which follows from the jl jl jl Birkhoff-von Neumann theorem on the decomposition of \ l[=1 doubly stochastic matrices6 into a convex combination of The following example uses the matrix α and provides a permutation matrices7. Hence the doubly stochastic matrix unique method of combining the indices ατ to obtain the 1 jl M Rτ can be written as a convex sum of permutation matrices. desired rewriting of the product Aτ . This method follows the We will refer to this algorithm of rewriting any permanent- steps of the algorithm illustrated in Example 1 for modifying ↑P product in perm(θ ) as a product of powers of permanent- the matrix Rτ . products in θ as the decomposition algorithm, and the decom- P position is called the standard decomposition. Example 2. Let θ be the 3 × 3 matrix in Example 1, = 3 ↑P 2 2 2 (Pij ) ∈P , θ and Aτ = a bdf h i as follows: a b c 3

Example 1. Let M = 7 and θ , d e f . Suppose that I3 I3 I3 0 0 1 0 1 0   2 2 g h i P , I3 QQ ,Q , 1 0 0 ,Q , 0 0 1 , 3 2 2 3 4 4 2 ↑P  2     Aτ , a b c e f g h i is a product in perm(θ ). Then, this I3 I3 Q 0 1 0 1 0 0 product corresponds to the following exponent matrix Rτ and       rτ Rτ ij the corresponding θ , (θij ): a1 0 0 b1 0 0 c1 0 0 3 2 2 a3 b2 c2  0 a2 0 0 b2 0 0 c2 0  R , , θRτ d0 e3 f 4 τ 0 3 4 = 0 0 a3 0 0 b3 0 0 c3 4 2 1 g4 h2 i1     d1 0 0 0 0 e1 0 f1 0        Following the algorithm we obtain ↑P   θ , 0 d2 0 e2 0 0 0 0 f2    3   2 2 2 2 2  0 0 d3 0 e3 0 f3 0 0  R 0 3 4 aei 4 bfg 2   τ = → ( ) → 0 2 → ( )  g1 0 0 h1 0 0 0 i1 0    4    4 2 1 2 0    0 g2 0 0 h2 0 0 0 i2   2   2    2 0 0 0  0 0 g3 0 0 h3 i3 0 0  → 0 2 2 → (ceg)2 → 0 0 2 → (afh)2.        (6) 2 2 0 0 2 0     where I3 denotes the identity matrix of size 3 and the entries 6A matrix is doubly stochastic if it has positive entries and both its rows boxed in θ↑P correspond to the permutation τ that gives the and columns sum to 1. 2 2 2 ↑P product Aτ = a bdf h i. Here we wrote the matrix θ with 7See http://staff.science.uva.nl/∼walton/Notes/Hall Birkhoff.pdf for a short presentation of the Birkhoff-von Neumann theorem and the decomposition its entries indexed by their row, e.g., a1 = a2 = a3 = a and algorithm. ai is on the ith row of the first block P11.

3 The matrices ατ , ατ and Rτ are a partition). Such a partition is surely given by the three-sets of the boxed entries in the first M columns, because each {1,3} {2} ∅ of these sets must be disjoint and they are exactly M, the 2 1 0 number of boxed entries from the first M columns, so the α =  {2} ∅ {1,3}  , R = 1 0 2 . τ τ   union of all entries in these products is equal to all entries in   0 2 1   the product Aτ . In fact, any three-sets associated to the boxed  ∅ {1,3} {2}      entries in a set (j −1)M +1,...,jM of columns corresponds   to a partition of the entries in A . For simplicity, however, Note that α corresponds to the indices of the boxed entries τ τ M in θ↑P. In the matrix α , we use circles and boxes to we choose the partition corresponding to the first columns, τ or, equivalently, to the matrix α . We call this decomposition show how to group the boxed entries of θ↑P: we combine τ same-index decomposition entries in θ↑P in rows indexed by the circled entries in . same-index decomposition of a permanent- α , and we combine entries in θ↑P in rows indexed by the Therefore, the τ product in θ↑P is the writing of the permanent-product as a boxed entries in ατ , thus obtaining a unique rewriting of the 2 product of M sub-products of m entries in θ each indexed by product Aτ as Aτ = (afh) (bdi), which is in correspondence the same row index, e.g., (a1f1h1)(b2d2i2)(a3f3h3). to the rewriting steps of the matrix Rτ . In terms of the ↑P indexed entries of θ , the above grouping corresponds to C. Decompositions that contain illegal sub-products A = (a f h )(a f h )(b d i ) which is exemplified through τ 1 1 1 3 3 3 2 2 2 So far in our example, the same-index decomposition of a circles, boxes and shaded boxes in the version of θ↑P with permanent-product is equal to its standard decomposition. In indexed entries in (6). the following section, we see that this is not always the case. Is a decomposition like the one drawn in ατ of Example 2 For example, this following decomposition in ακ could also always possible? The answer is yes due to the following simple occur: ↑P 2 3 fact. Each row and column of θ participates with exactly  1  one element to a permanent-product. In the matrix θ↑P of (6), 1 2 3 ,     once we choose d on the second column, or, equivalently, d2,  1 2 3  none of the entries a or g on that column can be part of the 2 2 ↑P permanent-product anymore and, therefore, the second row of yielding the following permanent-products of θ : matrix P (where a is positioned) and the second row of † † 11 2 a1a2a3 e1e2f3 h1i2i3 = (a1e1h1) (a2e2i2)(a3f3i3) the matrix P31 (where g2 is positioned) must contribute each = (a1e1i3)(a2e2i2)(a3f3h1). with exactly one entry other than the entries a2 and g2 that are not allowed. These are the boxed entries b2 and i2. We group In this case, not all of the products of 3 entries of the same these entries with d2 uniquely and continue the same way to index correspond to permanent-products in the matrix θ; we † group each of the a entries with the entries f and h that are marked with † the ones that do not, for example, (a1e1h1) on the two rows associated with the other two entries on the corresponds to aeh in θ which is not a permanent-product. columns of the entries a to obtain (a1f1h1) and (a3f3h3). We call such a product illegal. This illegal three-product In terms of the entries of the matrix ατ , this corresponds to needs to be grouped with another illegal three-product in P † the grouping we showed in Example 2 because the matrix is the same grouping, in this case (a3f3i3) , and rearranged reduced, so the first matrices Pl1 in each row and P1l in each as (a1e1i3)(a3f3h1) to obtain a standard decomposition, i.e., column are equal to the the identity matrix, for all l ∈ [m]. a product of permanent-products of θ. We call these sub- Therefore, for each of the first M columns, the nonzero entries products that correspond to a permanent-product in θ legal. on the jth column are all positioned on the jth row of the D. Mapping illegal products into legal products matrices Pl1, for all l ∈ [m]. Of course, this is not valid for a column that is not among the first M. Indeed, the boxed i Next we show that we can always assume that all ↑P ↑P of θ in (6) is on row 2 of matrix P33 and has the nonzero permanent-products in θ are products of θ-permanent- ↑P entries on rows 3 of matrix P13 and 2 of matrix P23. However, products by showing that any permanent-product of θ it still holds that the rows corresponding to these non-zero containing some illegal sub-products can be mapped uniquely entries must contribute to the product with exactly one entry into some product of M same-index permanent-products of that cannot be on the column of i. In this case, d2 on position θ. In addition, this product has the same exponent matrix (2, 2) in P21 and a3 on position (3, 3) of P11 are these entries. as the original permanent-product but is not a permanent- We can group these together as well. In fact any such grouping product of θ↑P. This way, we establish a one-to-one corre- of three where two of them are on the rows corresponding to spondence between permanent-products of θ↑P and products the non-chosen entries of the column of the third of the group of M permanent-products in θ. is a good association; the permanent-product Aτ is then a This correspondence illustrated in the previous example can product of some of these three-products with the property that be generalized to all permanent-products of θ↑P with same- the entries in the products are taken only once and they cover index decompositions that contain some illegal sub-products all the entries in the permanent-product Aτ (i.e., they form in the following way.

4 ↑P • Let θ be an m × m non-negative matrix and θ be a uct of same-index θ-permanent-products, thus guaranteeing the reduced matrix of degree M. existence of the map. • Let τ be a permutation on [mM] and Aτ be a permanent- The fact that no two permanent-productscan be mapped into ↑P ′ product in θ that is not trivially zero. Let Rτ be its the same Aτ,i is also ensured by the conditions of the mapping; exponent matrix. if two different permanent products Aτ and Aν map into the ′ • Write Aτ as the same-index decomposition; Aτ can or not same Aτ,i, then they must have a first entry in which they contain illegal same-index sub-products, i.e., products of differ; this entry must be necessarily after the first s entries. ′ m entries in θ of the same index that are not permanent- This means, however, that there must exist an Aτ,j that shares ′ products in θ. with Aν that entry but not with Aτ,i. Therefore, Aν cannot get ′ • List all distinct products of M same-index permanent- mapped into the same Aτ,i as Aτ , proving that the function products in θ corresponding to all standard decomposi- is one-to-one. In addition, if Aτ contains illegal same-index ′ ′ tions of Rτ that start with the entries in Aτ that are in sub-products, then Aτ,i such that Aτ 7→ Aτ,i cannot be a ↑P ′ ′ ↑P ↑P the first M columns of θ . Call them Aτ,1,...,Aτ,l permanent-product in θ . To see this, erase from θ all rows and reorder, if needed, the entries in the sub-products of and columns corresponding to the entries that the two share. ′ ′ Aτ and Aτ,1,...,Aτ,l such that the entries from the first Suppose that there are k entries in which the two products are ′ ′ ′ ′ M columns are always first in the subproduct, followed different, say, x1, x2,...,xk in Aτ and x1, x2,...,xk in Aτ,i. ′ by the entries ordered by the row index in θ increasingly Because the two products Aτ , Aτ,i have the same exponent ′ ′ ′ from 1 to m and such that the indices of the θ-permanent- matrix, so do the two products x1x2 ...xk and x1x2 ...xk. products are ordered increasingly from 1 to M. Therefore, in each block in which there exists some xi, i ∈ [k], ′ This procedure, henceforth called standard mapping, is for- there must exist also a j ∈ [k] such that xj is also in that block. ′ ′ ′ ′ malized in the following lemma. We can reorder x1, x2,...,xk so that each xl is in the same ′ block as xl. Note that there can be more entries in one block, Lemma 1 (Standard mapping). Initially, set L := but to each entry x corresponds a unique entry x′ in the same ′ ′ l l {Aτ,1,...,Aτ,l}. block. Since there is only one column in the k × k submatrix Start Let 6 s 6 M and 6 t

5 m m rjl ′ ′ m × m submatrices of H over the binary field or over the with Aτ = (θjl) , let Aτ,1,...,Aτ,l be the possible j=1 l=1 integers. products of MQ θQ-permanent-products associated with R. For each j ∈ [l], denote by Nτ,j the number of products of M θ- IV. ACKNOWLEDGMENTS ′ permanent-products that are equivalent to Aτ,j , i.e., they can The partial support of the U.S. National Science Founda- ′ be obtained from Aτ,j by applying an M-permutation on the tion through grants DMS-1313221, CIF-1252788, and CCF m m rjl ↑P 1216407 is gratefully acknowledged. indices. Then, the coefficient of (θjl) in perm(θ ) j=1 l=1 l Q Q REFERENCES is upper bounded by j=1 Nτ,j. [1] P. O. Vontobel, “The Bethe permanent of a non-negative matrix,” IEEE The following lemmaP determines Nτ,j for all j ∈ [l]. Trans. Inf. Theory, vol. 59, pp. 1866–1901, Mar. 2013. [2] L. Gurvits, “Unleashing the power of Schrijver’s permanental inequality Lemma 2. For each j ∈ [l] and σ ∈ Sm, let 0 6 with the help of the Bethe Approximation,” ArXiv e-prints, June 2011. ′ [3] B. Huang and T. Jebara, “Approximating the Permanent with Belief tj,σ 6 M such that tj,σ = M and A = τ,j Propagation,” ArXiv e-prints, Aug. 2009. σ∈Sm t [4] M. Chertkov and A. B. Yedidia, “Approximating the Permanent with θ θ ··· θ P j,σ . Then N = M where 1σ(1) 2σ(2) mσ(m) τ,j tj Fractional Belief Propagation,” Journal of Machine Learning Research, σ∈Sm vol. 14, pp. 2029–2066, July 2013. M Q is the multinomial coefficient
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matchings in random lifts,” Comb., Prob., and Comp., vol. 19, pp. 791– Theorem 2. Let θ = (θij ) be a non-negative matrix of size 817, Nov. 2010. m × m and let P = (P ) ∈M (P ). Then [10] M. Bayati and C. Nair, “A rigorous proof of the cavity method for ij m M counting matchings,” in Proc. 44th Allerton Conf. on Comm., Control, perm(θ↑P) 6 perm(θ)M . and Computing, (Monticello, IL, USA), Sep. 2006. [11] M. Bayati, D. Shah, and M. Sharma, “Max-product for maximum weight Proof: The upper bound follows immediately from matching: convergence, correctness, and LP duality,” IEEE Trans. Inf. Theory, vol. 54, pp. 1241–1251, Mar. 2008. M Lemma 2 and the expansion of perm(θ) as [12] S. Sanghavi, D. Malioutov, and A. Willsky, “Belief propagation and LP relaxation for weighted matching in general graphs,” IEEE Trans. Inf. M Theory, vol. 57, pp. 2203–2212, Apr. 2011. M perm(θ) = θ1σ(1)θ2σ(2) ··· θmσ(m) [13] P. Dagum and M. Luby, “Approximating the permanent of graphs with large factors,” Theoretical Computer Science, vol. 102, no. 2, pp. 283– σ∈S ! Xm 305, 1992. M tj,σ [14] M. Jerrum, A. Sinclair, and E. Vigoda, “A polynomial-time approxima- = θ1σ(1)θ2σ(2) ··· θmσ(m) . tion algorithm for the permanent of a matrix with nonnegative entries,” tj |tj|=M   σ∈Sm J. ACM, vol. 51, pp. 671–697, July 2004. X Y
[15] J. S. Yedidia, W. T. Freeman, and Y. Weiss, “Constructing free-energy approximations and generalized belief propagation algorithms,” IEEE III. CONCLUSIONS Trans. Inf. Theory, vol. 51, pp. 2282–2312, July 2005. The consequences of the results in this paper are more [16] J. Thorpe, “Low-density parity-check (LDPC) codes constructed from than just purely theoretical. They provide new insight into protographs,” IPN Progress Report, vol. 42-154, pp. 1–7, 2003. [17] D. MacKay and M. Davey, “Evaluation of Gallager codes for short block the structure of the permanent of a P-lifting of a matrix, length and high rate applications,” in Codes, Systems, and Graphical which can be exploited algorithmically to decrease the com- Models, vol. 123 of The IMA Vol. in Math. and Applications, pp. 113– putational complexity of the permanent of the P-liftings. Such 130, Springer New York, 2001. [18] R. Smarandache and P. O. Vontobel, “Quasi-cyclic LDPC codes: In- an algorithm can search for products of groups of entries fluence of proto- and Tanner-graph structure on minimum Hamming formed according to the groupings we presented in this paper distance upper bounds,” IEEE Trans. Inform. Theory, vol. 58, pp. 585– to check if they form valid permanent-products. In addition, 607, Feb. 2012. [19] R. Smarandache and P. O. Vontobel, “Absdet-pseudo-codewords and the structure of the permanent-products of P-liftings of a perm-pseudo-codewords: definitions and properties,” in IEEE Intern. matrix may have some implications on the constant C in the Symp. on Inform. Theory, (Seoul, Korea), June 2009. inequality θ 6 C θ . [20] B. K. Butler and P. H. Siegel, “Bounds on the Minimum Distance of perm( ) · permB( ) Punctured Quasi-Cyclic LDPC Codes,” IEEE Trans. Inform. Theory, Lastly, since a P-lifting of a matrix θ corresponds to an vol. 59, pp. 4584–4597, July 2013. M-graph cover of the protograph (base graph) described by [21] Y. Wang, S. C. Draper, and J. S. Yedidia, “Hierarchical and High-Girth θ, which, in turn, correspond to LDPC codes, these results QC LDPC Codes,” IEEE Trans. Inform. Theory, vol. 59, pp. 4553–4583, July 2013. may help explain the performance of these codes through the techniques presented in [17]–[21], which are based on explicit constructions of codewords and pseudo-codewords with components equal to determinants or permanents, of some

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