Symmetric Determinantal Representation of Formulas and Weakly Skew Circuits

Transcription

1 Contemporary Mathematic Symmetric Determinantal Repreentation of Formula and Weakly Skew Circuit Bruno Grenet, Erich L. Kaltofen, Pacal Koiran, and Natacha Portier Abtract. We deploy algebraic complexity theoretic technique to contruct ymmetric determinantal repreentation of formula and weakly kew circuit. Our repreentation produce matrice of much maller dimenion than thoe given in the convex geometry literature when applied to polynomial having a concie repreentation (a a um of monomial, or more generally a an arithmetic formula or a weakly kew circuit). Thee repreentation are valid in any field of characteritic different from 2. In characteritic 2 we are led to an almot complete olution to a quetion of Bürgier on the VNP-completene of the partial permanent. In particular, we how that the partial permanent cannot be VNP-complete in a finite field of characteritic 2 unle the polynomial hierarchy collape. 1. Introduction 1.1. Motivation. A linear matrix expreion (ymmetric linear matrix form, affine ymmetric matrix pencil) i a ymmetric matrix with the entrie being linear form in the variable x 1,..., x n and real number coefficient: (1) A(x 1,..., x n ) = A 0 + x 1 A x n A n, A i ymmetric in R t t. A linear matrix inequality (LMI) retrict to thoe value ξ i R of the x i uch that A(ξ 1,..., ξ n ) 0, i.e., i poitive emidefinite. The et of all uch value define a pectrahedron. A real zero polynomial i a polynomial p with real coefficient uch that for every x R n and every µ C, p(µx) = 0 implie µ R. The Lax conjecture and generalized Lax conjecture eek for repreentation of real zero polynomial f(x 1,..., x n ) a f = det(a) with A a in (1) and A 0 0. Thi i in fact an equivalent formulation of the original Lax conjecture which wa tated in term of hyperbolic polynomial (ee [18] for thi equivalence). Furthermore, the matrice are required to have dimenion d where d i the degree of the polynomial. For 2000 Mathematic Subject Claification. Primary 03D15; Secondary 68Q05, 15A15, 90C22, 90C25. Thi material i baed on work upported in part by the National Science Foundation under Grant CCF and CCF (E. L. Kaltofen). Partially funded by European Community under contract PIOF-GA of the 7th PCRD (N. Portier). 1 c 0000 (copyright holder)

2 2 B. GRENET, E. L. KALTOFEN, P. KOIRAN, AND N. PORTIER n = 2 uch repreentation alway exit while a counting argument how that thi i impoible for n > 2 [12] (actually, Lewi, Parrilo and Ramana [18] give the firt proof of the Lax conjecture in it original form baed on the reult of Helton and Vinnikov [12]). Two relaxation have been uggeted to evade thi counting argument: At firt it wa uggeted to remove the dimenion contraint and eek for bigger matrice, and thi wa further relaxed by eeking for repreentation of ome power of the input polynomial. Counterexample to both relaxation have recently been contructed by Brändén [3]. Another relaxation i to drop the condition A 0 0 and repreent any f a det(a) [11, 24]. However, the purely algebraic contruction of Quarez [24] lead to exponential matrix dimenion t. Here we continue the line of work initiated in [11, 24] but we proceed differently by ymmetrizing the complexity theoretic contruction by Valiant [27]. Our contruction yield maller dimenional matrice not only for polynomial repreented a um of monomial but alo for polynomial repreented by formula and weakly kew circuit [22, 14]. Even though in the mot general cae the bound we obtained are lightly wore than Quarez [24], in a lot of intereting cae uch a polynomial with a polynomial ize formula or weaklykew circuit, or in the cae of the permanent, our contruction yield much maller matrice (ee Section 4). Our contruction are valid for any field of characteritic different from 2. For field of characteritic 2, it can be hown that ome polynomial (uch a e.g. the polynomial xy+z) cannot be repreented a determinant of ymmetric matrice [9]. Note a a reult that the 2-dimenional permanent xw+yz cannot be ymmetrized over characteritic 2 with any dimenion. It would be intereting to exactly characterize which polynomial admit uch a repreentation in characteritic 2. For the polynomial x + y, we have 0 x 0 y x x x + y = det( y 0 0 1/2 ) = det( 0 y ), /2 0 where the firt matrix i derived from our contruction, but the econd i valid over any commutative ring. It i eaily hown that for every polynomial p, it quare p 2 admit a ymmetric determinantal repreentation in characteritic 2. Thi i related to a quetion of Bürgier [4]: I the partial permanent VNP-complete over field of characteritic 2? We give an almot complete negative anwer to thi quetion. Our reult give a a by-product an intereting reult which wa not known to the author knowledge: Let A be an (n n) matrix with indeterminate coefficient (ranging over a field of characteritic different from 2), then there exit a ymmetric matrix B of dimenion O(n 5 ) which entrie are the indeterminate from A and contant from the field uch that det A = det B. Thi relie on the exitence of a ize-o(n 5 ) weakly-kew circuit to compute the determinant of an (n n) matrix [2, 22], and thi weakly-kew circuit can be repreented by a determinant of a ymmetric matrix a proved in thi paper. The dimenion of B can be reduced to O(n 4 ) if we replace the weakly kew circuit from [2, 22] by the kew circuit of ize O(n 4 ) contructed by Mahajan and Vinay [20]. Thee author contruct

3 SYMMETRIC DETERMINANTAL REPRESENTATION 3 an arithmetic branching program for the determinant with O(n 4 ) edge, 1 and the arithmetic branching program can be evaluated by a kew circuit of ize O(n 4 ). After learning of our reult, Meena Mahajan and Prajakta Nimbhorkar have noticed that the arithmetic branching program for the determinant can be tranformed directly into a ymmetric determinant of dimenion O(n 3 ) with technique imilar to the one ued in thi paper. We give a detailed proof in Subection 3.2. We add that the aymptotically mallet known diviion-free algebraic circuit for the n n determinant polynomial have ize O(n 2.70 ) [13, 15]. The circuit actually can compute the characteritic polynomial and the adjoint and are baed on algebraic rather than combinatorial technique. Weakly kew circuit of uch ize appear not to be known. Organization. Section 1.2 i devoted to an introduction to the algebraic complexity theoretic ued in our contruction, a well a a reminder of the exiting related contruction in algebraic complexity. Section 2 deal with ymmetric repreentation of formula while Section 3 focue on weakly-kew circuit. Table 2 page 34 give an overview of all the different contruction ued in thi paper. Section 4 then proceed to the comparion between the reult obtained o far and Quarez [24]. The pecial cae of field of characteritic 2 i tudied in Section 5. A horter verion of thi paper [8] ha been publihed in Proceeding of STACS It contain material from Section 3 and Section 5. Acknowledgment. We learned of the ymmetric repreentation problem from Marku Schweighofer ISSAC 2009 Tutorial ~chweigh/preentation/dcblmi.pdf. We thank Meena Mahajan for pointing out [20], ketching the contruction of a ymmetric determinant of dimenion O(n 3 ) from a determinant of dimenion n and reading our proof of it Known reult and definition. In hi eminal paper Valiant [27] expreed the polynomial computed by an arithmetic formula a the determinant of a matrix whoe entrie are contant or variable. If we define the kinny ize e of the formula a it number of arithmetic operation then the dimenion of the matrix are at mot e+2. The proof ue a weighted digraph contruction where the formula i encoded into path from a ource vertex to a target, ometime known a an Algebraic or Arithmetic Branching Program [23, 1]. Thi theorem how that every polynomial with a ub-exponential ize formula can be expreed a a determinant with ub-exponential dimenion, enhancing the prominence of linear algebra. A light variation of the theorem i alo ued to prove the univerality of the permanent for formula which i one of the tep in the proof of it VNPcompletene. In a tutorial, von zur Gathen [31] give another way to expre a formula a a determinant: hi proof doe not ue digraph and hi bound i 2e + 2. Refining hi technique, Liu and Regan [19] gave a contruction leading to an upper bound of e + 1 in a lightly more powerful model: multiplication by contant are free and do not count into the ize of the formula. Our purpoe here i to expre a formula a a determinant of a ymmetric matrix. Multiplication by contant are alo given for free. Our contruction ue path in graph, imilar to the path in digraph in Valiant original proof. In fact, thi original contruction appear to have a little flaw in it. Interetingly enough, 1 Thi bound can be found on p.11 of their paper.

4 4 B. GRENET, E. L. KALTOFEN, P. KOIRAN, AND N. PORTIER thi flaw ha never been mentioned in the literature to the author knowledge. A light change in the proof i given in [5, Exercie 21.7 (p570)] that ettle a part of the problem. And the ame flaw appear in the proof of the univerality of the permanent in [4]. When adding two formula, the reulting digraph can have two arc between the ource and the target, which can lead to the um of two variable being an entry of the matrix, and thi i not allowed a we eek for ymmetric matrice where each entry i either a contant or a variable. The firt idea to correct the proof i to keep the ame parity for all -t-path a in Valiant original proof, adding two new vertice and replacing one of the arc by a length-three path. Thi method i very imple but it diadvantage i that it increae the dimenion of the final matrix to 2e + 3. In the ymmetric cae we will ue a coefficient to correct the parity difference between path intead of adding new vertice. Uing thi technique in the non-ymmetric cae allow u to prove Valiant theorem with (e + 1) intead of (e + 2). Our technique alo give for free multiplication by contant a in [19]. It ue digraph and i to our opinion more intuitive than direct work on matrice. In [26, 22], reult of the ame flavor were proved for a more general cla of circuit, namely the weakly-kew circuit. Malod and Portier [22] can deduce from thoe reult a fairly imple proof of the VQP-completene of the determinant (under qp-projection). Moreover, they define a new cla VP w of polynomial repreented by polynomial-ize weakly-kew circuit (with no explicit retriction on the degree of the polynomial) for which the determinant i complete under p-projection. A formula i a circuit in which every vertex ha out-degree 1 (but the output). Thi mean in particular that the underlying digraph i a tree. A weakly-kew circuit i a kind of generalization of a formula, with a le contrained tructure on the underlying digraph. For an arithmetic circuit, the only retriction on the digraph i the abence of directed cycle (that i the underlying digraph i a directed acyclic graph). A circuit i aid weakly-kew if every multiplication gate α ha the following property: the ub-circuit aociated with one of it argument β i connected to the ret of the circuit only by the arrow going from β to α. Thi mean that the underlying digraph i diconnected a oon a the multiplication gate α i removed. In a ene, one of the argument of the multiplication gate wa pecifically computed for thi gate. Toda [26] proved that the polynomial computed by a weakly-kew circuit of kinny ize e can be repreented by the determinant of a matrix of dimenion (2e + 2). Thi reult wa improved by Malod and Portier [22]: The contruction lead to a matrix of dimenion (m+1) where m i the fat ize of the circuit (i.e. it total number of gate, including the input gate). Note that for a circuit in general and for a weakly-kew circuit in particular m 2e + 1. The latter contruction ue negated variable in the matrix. It i actually poible to get rid of them [14]. Although the kinny ize i well uited for the formula, the fat ize appear more appropriate for weakly-kew circuit. In Section 3, we ymmetrize thi contruction o that a polynomial expreed by a weakly-kew circuit equal the determinant of a ymmetric matrix. Our contruction yield a ymmetric matrix of dimenion (2m + 1). In fact, thi can be refined a well a the non-ymmetric contruction. An even more appropriate ize for a weakly-kew circuit i (e + i) where e i the kinny ize and i the number of input labelled by a variable (clearly e + i m).

5 SYMMETRIC DETERMINANTAL REPRESENTATION 5 We can how that the bound are till valid if we replace m by (e + i) and even when multiplication by contant are free a in [19] (ee Section 3.3). Let u now give ome formal definition of the arithmetic circuit and related notion. Definition 1. An arithmetic circuit i a directed acyclic graph with vertice of in-degree 0 or 2 and exactly one vertex of out-degree 0. Vertice of in-degree 0 are called input and labelled by a contant or a variable. The other vertice, of in-degree 2, are labeled by or + and called computation gate. The vertex of out-degree 0 i called the output. The vertice of a circuit are commonly called gate and it arc arrow. An arithmetic circuit with contant input in a field k and variable in a et x naturally compute a polynomial f k[ x]. Definition 2. If α i a gate of a circuit C, the ub-circuit aociated to α i the ubgraph of C made of all the gate β uch that there exit a oriented path from β to α in C, including α. A gate α receiving arrow from β and γ i aid to be dijoint if the ub-circuit aociated to β and γ are dijoint from one another. The gate β and γ are called the argument of α. Definition 3. An arithmetic circuit i aid weakly-kew if for any multiplication gate α, the ub-circuit aociated to one of it argument β i only connected to the ret of the circuit by the arrow going from β to α: it i called the cloed ub-circuit of α. A gate which doe not belong to a cloed ub-circuit of C i aid to be reuable in C. A formula i an arithmetic circuit in which all the gate are dijoint. The reuability of a gate depend of coure on the conidered circuit C. For intance, in Fig. 1(b), the weakly-kew circuit ha two cloed ub-circuit. The input z i in the right cloed ub-circuit and i therefore not reuable. But inide thi cloed ub-circuit, it i reuable, and actually ued a argument to the ummation gate twice. Figure 1(a) and (c) are repectively an equivalent arithmetic circuit and an equivalent formula, that i the two circuit and the formula compute the polynomial (x + y) 2 + 2yz. Let u remark a fact that will be ueful later: all the multiplication gate of a weakly-kew circuit are dijoint (but thi i not a ufficient condition). In our contruction, we hall ue graph and digraph. In particular, the improved contruction baed on Valiant repreent formula by path in a digraph. On the other hand, to obtain ymmetric determinantal repreentation the digraph have to be ymmetric. Thee correpond to graph. In order to avoid any confuion between directed and undirected graph, we hall excluively ue the term graph for undirected one, and otherwie ue the term digraph. It i well-known that cycle cover in digraph are in one-to-one correpondence with permutation of the vertice and therefore that the permanent of the adjacency matrix of a digraph can be defined in term of cycle cover of the digraph. Let u now give ome definition for thoe fact, and ee how it can be extended to graph. Definition 4. A cycle cover of a digraph G = (V, A) i a et of cycle uch that each vertex appear in exactly one cycle. The weight of a cycle cover i defined to be the product of the weight of the arc ued in the cover. Let the ign of a vertex cover be the ign of the correponding permutation of the vertice, that i () N

6 6 B. GRENET, E. L. KALTOFEN, P. KOIRAN, AND N. PORTIER x y z x y x y z (a) An arithmetic circuit (b) A weakly-kew circuit: The cloed ub-circuit are marked with gray boxe. x y x y y z z (c) A formula Figure 1. An arithmetic circuit, a weakly-kew circuit and a formula computing the ame polynomial (x + y) 2 + 2yz. where N i the number of even cycle. Finally, let the igned weight of a cycle cover be the product of it weight and ign. For a graph G = (V, E), let G d = (V, A) be the correponding ymmetric digraph. Then a cycle cover of G i a cycle cover of G d, and the definition of weight and ign are extended to thi cae. In particular, if there i a cycle cover of G with a cycle C = (u 1,..., u k ), then a new cycle cover i defined if C i replaced by the cycle (u k,..., u 1 ). Thoe two cycle cover are conidered a different cycle cover of G. Definition 5. Let G be a digraph. It adjacency matrix i the (n n) matrix A uch that A i,j i equal to the weight of the arc from i to j (A i,j = 0 i there i no uch arc). The definition i extended to the cae of graph, een a ymmetric digraph. In particular, the adjacency matrix of a graph i ymmetric. Lemma 1. Let G be a (di)graph, and A it adjacency matrix. Then the permanent of A equal the um of the weight of all the cycle cover of G, and the determinant of A i equal to the um of the igned weight of all the cycle cover of G. Proof. The cycle cover are obviouly in one-to-one correpondence with the permutation of the et of vertice, and the ign of a cycle cover i defined to match the ign of the correponding permutation. Suppoe that the vertice of V are {1,..., n} and let A i,j be the weight of the arc (i, j) in G. Let C a cycle cover and σ the correponding permutation. Then it i clear that the weight of C i A 1,σ(1) A n,σ(n), hence the reult.

7 SYMMETRIC DETERMINANTAL REPRESENTATION 7 Figure 2. A graph with a cycle cover (the arc in the cover are in black). The validity of thi proof for graph follow from the definition of the cycle cover of a graph in term of the cycle cover of the correponding ymmetric digraph. In the following, the notion of perfect matching i ued. A perfect matching in a graph G i a et M of edge of G uch that every vertex i incident to exactly one edge of M. The weight of a perfect matching i defined in thi a the weight of the correponding cycle cover (with length-2 cycle). Thi mean that thi i the product of the weight of the arc it ue, or equivalently it i the quare of the product of the weight of the edge it ue. Note that thi i the quare of the uual definition. A path P in a digraph i a ubet of vertice {u 1,..., u k } uch that for 1 i k 1, there exit an arc from u i to u i+1 with nonzero weight. The ize P of uch a path i k. 2. Formula 2.1. Non-ymmetric cae. In thi ection, a in Section 2.2 and 3, a field k of characteritic different from 2 i fixed and the contant input of the formula and the weakly-kew circuit are taken from k. The variable are uppoed to belong to a countable et x = {x 1, x 2,... }. Following [19], we define a formula ize that doe not take into account multiplication by contant. Definition 6. Conider formula with input being variable or contant from k. The green ize gize(ϕ) of a formula ϕ i defined inductively a follow: The green ize of a contant or a variable i 0; If c i a contant then the green ize of c ϕ i equal to the green ize of ϕ; If ϕ 1 and ϕ 2 are formula, then gize(ϕ 1 + ϕ 2 ) = gize(ϕ 1 ) + gize(ϕ 2 ) + 1. If ϕ 1 and ϕ 2 are non-contant formula, then gize(ϕ 1 ϕ 2 ) = gize(ϕ 1 )+ gize(ϕ 2 ) + 1 An even maller ize can be defined by deciding that every variable-free formula ha ize zero and Theorem 1 can eaily be extended to thi cae. A formal definition of thi ize i given i Section 3.3 in the context of weakly-kew circuit.

8 8 B. GRENET, E. L. KALTOFEN, P. KOIRAN, AND N. PORTIER ϕ 1 ϕ 2 G 1 G 2 t Figure 3. G 1, c 1 and G 2, c 2 are repectively aociated to ϕ 1 and ϕ 2 ; ϕ = ϕ 1 ϕ 2. Theorem 1 ([19]). For every formula ϕ of green ize e with at leat one addition there i a quare matrix A of dimenion e + 1 whoe entrie are input of the formula and element of {0, 1,, 1/2} uch that ϕ = det(a). We remark that if ϕ ha no addition it i of the form cx 1... x n and it ha ize (n 1). Then a uitable matrix i the (n + 1) (n + 1) diagonal matrix made of the n variable and the contant c. Thu the dimenion of the matrix are at mot n + 1 = e + 2, and are n = e + 1 if c = 1. Note that thi latter bound i minimal a the determinant of a (d d) matrix i a degree-d polynomial. The dimenion (n + 1) are not minimal when c 1 a hown by the (3 3) matrix 0 x y x 0 z y z 0 repreenting 2xyz. One can alo ee that the n bound cannot be general a there i no (2 2) matrix repreenting the polynomial 2xy. Lemma 2. Let ϕ be an arithmetic formula of green ize e. Then there exit a contant c 0 and an edge-weighted digraph G with at mot e + 2 vertice and two ditinct vertice and t uch that c 0 () P w(p ) = ϕ. -t-path P Proof of Lemma 2. We prove the lemma by induction on formula. If ϕ i equal to a variable x (rep. a contant c) then G ha two vertice and t and an edge (, t) labelled by x (rep. c) and the contant c 0 i equal to 1. If ϕ = c ϕ let G be the digraph and c 0 the contant atifying the lemma for the formula ϕ. Then obviouly G = G and c 0 = c 0c atify the lemma for ϕ. If ϕ = ϕ 1 ϕ 2, let G 1 and c 1 (rep. G 2 and c 2 ) atifying the lemma for ϕ 1 (rep. ϕ 2 ). Then let c = c 1 c 2 and G be the dijoint union of G 1 and G 2, except for t 1 and 2 which are merged (ee Fig 3). The ize of G i equal to G 1 + G 2 gize(ϕ 1 )+gize(ϕ 2 )+3 = gize(ϕ)+2. A -t-path P in G i a 1 - t 1 -path P 1 in G 1 followed by a 2 -t 2 -path P 2 in G 2 and we have P = P 1 + P 2 and w(p ) = w(p 1 ) w(p 2 ), hence the reult. If ϕ = ϕ 1 + ϕ 2, let G 1 and c 1 (rep. G 2 and c 2 ) atifying the lemma for ϕ 1 (rep. ϕ 2 ). If c 1 = 0 then ϕ and ϕ 2 compute the ame polynomial and we jut have to take G = G 2 and c = c 2. Suppoe now c 1 0. Then we define G a the dijoint union of G 1 and G 2, except for 1 and 2 which are merged, and with an

9 SYMMETRIC DETERMINANTAL REPRESENTATION 9 ϕ 1 ϕ 2 G 1 G 2 t 1 t 2 c 2 /c 1 Figure 4. G 1, c 1 and G 2, c 2 are repectively aociated to ϕ 1 and ϕ 2 ; ϕ = ϕ 1 + ϕ 2. edge (t 2, t 1 ) of weight c 2 /c 1 (ee Fig 4). The ize of G atifie the ame relation a in the multiplication cae. Let c 0 = c 1. A -t-path P in G i a 1 -t 1 -path in G 1 or a 2 -t 2 -path P 2 in G 2 followed by the edge (t 2, t 1 ), and in the econd cae we have w(p ) = w(p 2 )( c 2 /c 1 ) and P = P 2 + 1, hence the reult. Remark that t 2 ha only one outgoing edge and it weight i a contant, and that thi property will not be changed in the inductive contruction. Thi property will be ueful to prove the bound in the theorem. Proof of Theorem 1. Let ϕ be an arithmetic formula of green ize e and let G and c 0 be given by Lemma 2. Let Ḡ be the digraph obtained from G in the following way. We merge and t. A remarked in the proof of Lemma 2 there i a vertex v that ha only one outgoing edge and it weight i a contant c (a ϕ i uppoed to have at leat one addition). We change it weight to c 0 c and add a loop weighted by c 0 on v. We put a loop with weight 1 on every other vertex than v and. Let {1,..., e + 1} be the vertice of Ḡ and A it adjacency matrix. Let u have a cloer look at cycle cover of Ḡ. The cycle in Ḡ are cycle containing (which are in bijection with -t-path in G) and loop. In a cycle cover C the vertex belong to a cycle S. It weight w() i the weight of the correponding -t-path P in G and it cardinal i S = P 1. If the vertex v appear in S then w(s) = c 0 w(p ) and every other cycle in C i a loop of weight 1. Otherwie w(s) = w(p ) and C contain the loop v of weight c 0. In both cae w(c) = c 0 w(p ). Let u recall that gn(c) i the ignature of the underlying permutation: here it i if S i even and 1 otherwie, and o it i equal to () P. Uing Lemma 1 we get det(a) = gn(c)w(c) = c 0 () P w(p ) = ϕ. cycle cover C of Ḡ -t-path P G 2.2. Symmetric cae. The aim of thi ection i to write an arithmetic formula a a determinant of a ymmetric matrix, whoe entrie are contant or variable. Recall that in thi ection a in Section 3, a field k of characteritic different from 2 i fixed, and the input contant are taken from thi field. In the equel, every contructed graph i undirected. At firt, the reult i proved for the kinny ize of the formula. We recall that the kinny ize of ϕ i the number of arithmetic operator it contain.

10 10 B. GRENET, E. L. KALTOFEN, P. KOIRAN, AND N. PORTIER Theorem 2. Let ϕ be an arithmetic formula of kinny ize e. Then there exit a matrix A of dimenion at mot 2e + 3 whoe entrie are input of the formula and element of {0, 1,, 1/2} uch that ϕ = det A. Thi theorem i a corollary of the following lemma. Lemma 3. Let ϕ be an arithmetic formula of kinny ize e. Then there exit a graph G with at mot 2e + 2 vertice and two ditinct vertice and t uch that (1) The graph G ha an even number of vertice, every cycle in G i even and every -t-path ha an even number of vertice. (2) The ubgraph G \ {, t} i empty if e = 0 and for e 1 it ha only one cycle cover: It i a perfect matching of weight 1. For every -t-path P in G, the ubgraph G \ P i empty or ha only one cycle cover: a above it i a perfect matching of weight 1. (3) The following equality hold in G: () P /2+1 w(p ) = ϕ -t-path P The graph G i called the graph aociated to ϕ. The firt property of the lemma enure that becaue of a parity argument every cycle cover of the final contructed graph Ḡ ued in the proof of Theorem 2 (ee Fig. 5) include exactly one path between and t. The econd property enure that the weight of the cycle cover i the weight of the cycle involving and t, that i every other cycle ha weight 1, and that other cycle of the cover are of length 2. The third property give the relation between the graph and the formula. A in Valiant contruction for the not necearily ymmetric cae, the formula ϕ will be encoded in the weight of path between and t, but in a lightly different way. In Valiant contruction, a cycle cover of the digraph i made of a cycle including a -t-path, other cycle being loop. Moreover every -t-path ha the ame parity and o every cycle cover ha the ame parity of odd cycle and the underlying permutation ha the ame ignature. With thi property of the digraph the determinant of it adjacency matrix i equal to it permanent up to the ign. In our contruction a cycle cover of the graph i made of a cycle including a -tpath, other cycle being length-2 cycle. A length-2 cycle ha a negative ignature and every -t-path of the graph ha an even cardinality, o the ign of the cycle permutation i to the number of length 2 cycle. Thi how that the ign of the cycle permutation i a function of the length of the involved -t-path modulo 4. There i a way to enure that thi ign doe not depend on the choen -t-path: replace the graph G aociated to a ize-0 formula x in the proof of Lemma 3 by a 4-vertice path with weight x on it firt edge, and replace weight (Fig. 5, Fig. 7 and Fig. 8) by weight 1. Thi yield a matrix with entrie in k x whoe determinant and permanent are equal to ϕ, but it dimenion can be 4e + 5. To achieve the 2e + 3 bound, we contruct a matrix A whoe determinant can be very different from the permanent: For example, the permanent of the matrix aociated to ϕ = x+x i 0 when it determinant i 2x. Nonethele we can very eaily obtain a matrix B having the ame dimenion a A and uch that perm B = ϕ by replacing every entry in A by 1.

11 SYMMETRIC DETERMINANTAL REPRESENTATION 11 t G () G /2 c Figure 5. Contruction of Ḡ from G. Proof of Theorem 2. Let G be the graph aociated to ϕ and let Ḡ be the graph G augmented with a new vertex c and the edge tc of weight 1/2 and c of weight () G /2 (ee Fig. 5). Condition (1) and (2) imply that there i a bijection between path from to t or t to and cycle cover in Ḡ. More preciely, every cycle cover in Ḡ ha a unique odd cycle and it i of the form cp c where P i a -t-path or a t--path. Indeed, the graph Ḡ ha an odd number of vertice. Suppoe there i a cycle cover of Ḡ involving the length-2 cycle tct. Other cycle of thi cover are cycle of G and thu by (1) they are all even. Thi i not poible a an odd et can not be partitioned into even ubet. For the ame reaon, there i no cycle cover of Ḡ involving the cycle c. Thu every cycle cover of Ḡ ha a cycle including c and a path P between and t. Let u recall that the ign of a cycle cover i the ign of the underlying permutation, i.e. if it ha an odd number of even cycle and 1 otherwie, and let u define the igned weight of a cycle cover a the product of it weight and ign. Let C be a cycle cover of Ḡ involving the -t-path P. By property (2) there i only one way to complete the cover. Thu the weight of the cycle cover i the weight of P multiplied by (1/2 () G /2+1 ) and it ign i the ign of a perfect matching of cardinality G \ P, o it i () ( G\P )/2. By ymmetry, the invere cycle cover ha the ame igned weight. So the um of the igned weight of all cycle cover of Ḡ i equal to twice the um over all -t-path P of (1/2 () P /2+1 w(p )). According to Lemma 3 it i equal to ϕ. The reult follow from Lemma 1. Proof of Lemma 3. We proceed by tructural induction. In other word, we firt prove the lemma for the implet poible formula, namely x, and then how that the aertion of Lemma 3 i table under addition and multiplication. Let ϕ = x be an arithmetic formula of ize 0. Then the graph G aociated to ϕ by definition ha two vertice and t and an edge t of weight x. It verifie trivially propertie (1) and (2) and it only -t-path i t and we have: () 2/2+1 x = ϕ. Let ϕ = ϕ 1 + ϕ 2 and G 1 and G 2 be the graph aociated to ϕ 1 and ϕ 2. Firt let u uppoe 1 t 1 or 2 t 2 ha weight 0. Thi mean in particular that ϕ 1 or ϕ 2 i of ize at leat 1. Let = 1 = 2 and t = t 1 = t 2. Suppoe G 1 \ { 1, t 1 } and G 2 \ { 2, t 2 } have dijoint et of vertice and let G = G 1 G 2 (ee Fig. 6). Then G = G 1 + G ϕ ϕ = 2 ϕ. If 1 t 1 i an edge in G 1 and 2 t 2 i an edge in G 2 then the preceding contruction would lead to two edge between and t. They could be tranformed into a ingle edge if adding the two weight, but then the weight could be a um of two variable, and it i omething that i not allowed in thi context. So the graph G 1 i tranformed into a graph G 1 by adding two vertice u and v, removing the edge

12 12 B. GRENET, E. L. KALTOFEN, P. KOIRAN, AND N. PORTIER ϕ 1 ϕ 2 G 1 G 2 t Figure 6. Graph aociated to ϕ = ϕ 1 + ϕ x x u t 1 t 1 v Figure 7. Tranformation of G 1 into G 1. 1 t 1 with weight x and adding the edge 1 u with weight x, uv with weight 1 and vt 1 with weight (ee Fig. 7). We can verify eaily that G 1 atifie the three condition of Lemma 3. In particular for the third condition, the term x correponding to the path 1 t 1 in G 1 in the um i replaced by the term correponding to the path 1 uvt 1 in G 1: () 4/2+1 x = x. We then contruct the graph G aociated to ϕ a above but with G 1 replacing G 1. It ize i at mot 2 ϕ + 2. Now let u prove that the graph aociated to ϕ atifie the three propertie of the lemma. (1) G ha an even number of vertice and the cardinality of every -t-path i even. A cycle in G i a cycle in G 1, or a cycle in G 2, or a path from to t in G 1 or G 2 followed by path from t to in G 1 or G 2, and conequently every cycle in G i even. (2) If G 1 \ { 1, t 1 } and G 2 \ { 2, t 2 } are non-empty they are diconnected, and a cycle cover of the ubgraph G \ {, t} i contituted by a cycle cover of G 1 \ { 1, t 1 } and a cycle cover of G 2 \ { 2, t 2 }. So G \ {, t} ha only one cycle cover and it i a perfect matching of weight 1. If G 1 \ { 1, t 1 } i empty then G \ {, t} = G 2 \ { 2, t 2 } and ha only one cycle cover and it i a perfect matching of weight 1. Let P be a path between and t in G. We can uppoe wlog that the ubgraph G \ P i the union of the two graph G 1 \ P and G 2 \ { 2, t 2 }, which are diconnected from one another. The property to prove i then traightforward from the induction hypothei. (3) A path of G i a path of G 1 or a path of G 2, which prove the equality. Let ϕ = ϕ 1 ϕ 2 and G 1 and G 2 be the graph aociated to ϕ 1 and ϕ 2. Suppoe G 1 and G 2 have dijoint et of vertice and let G be G 1 G 2 with an additional edge t 1 2 of weight, and let = 1 and t = t 2 (ee Fig.8). Then G = G 1 + G 2 2 ϕ ϕ = 2 ϕ + 2. Let u prove that G atifie the three propertie of the lemma.

13 SYMMETRIC DETERMINANTAL REPRESENTATION 13 ϕ 1 ϕ 2 t 1 G 1 2 G 2 Figure 8. Graph aociated to ϕ = ϕ 1 ϕ 2. t (1) G ha an even number of vertice and every path from to t ha an even cardinality. A cycle in G i either a cycle in G 1, or a cycle in G 2 or the length-2 cycle t 1 2, and conequently every cycle in G i even. (2) Let u conider a cycle cover of G \ {, t}. The vertex t 1 can be in a cycle of G 1 or in the cycle t 1 2. If it i in a cycle of G 1 then we have a cycle cover of G 1 \ { 1 }, which i not poible becaue it i an odd et and all it cycle are even. Thu the cycle cover of G \ {, t} can be partitioned into t 1 2 of weight () 2, a cycle cover of G 1 \ { 1, t 1 } and a cycle cover of G 2 \ { 2, t 2 }. Thoe cycle cover are unique and o there i only one cycle cover of G \ {x, y} and it i a perfect matching of weight 1. Let P be a path between and t in G. It i a path P 1 from 1 to t 1 in G 1 followed by t 1 2 and a path P 2 from 2 to t 2 in G 2. So G \ P i the union of the two graph G 1 \P 1 and G 2 \P 2, which are diconnected (if non empty) from one another. The property to prove i then traightforward from the induction hypothei. (3) A -t-path P in G can be decompoed into three path: a 1 -t 1 -path P 1, t 1 2 which i of weight and a 2 -t 2 -path P 2. Thu () P 2 +1 w(p ) = () P 1 + P w(p 1 )()w(p 2 ) and o = () P w(p 1 ) () P w(p 2 ) () P 2 +1 w(p ) = P1 P = ϕ 1 ϕ 2 = ϕ. P 1 () 2 +1 w(p 1 ) () P w(p 2 ) P2 The upper bound (2e + 2) of Lemma 3 i tight a hown by Fig. 9. It can be hown eaily that thi contruction yield a graph of ize at leat ϕ + 2, and thi lower bound i tight a hown by Fig. 10. In fact, a in the non-ymmetric cae, the kinny ize can be replaced by the green ize of the formula defined in Definition 6.

14 14 B. GRENET, E. L. KALTOFEN, P. KOIRAN, AND N. PORTIER x 2 x3 x n+1 x 1... Figure 9. Graph aociated to ϕ = x x n+1 : ϕ = n and G = 2n + 2. x 1 x 2 x 3 x n y x 1... x x 3 2 x n Figure 10. Graph aociated to ϕ = x 1 x 1 + x 2 x 2 + x n x n + y: ϕ = 2n and G = 2n + 2. Theorem 3. For every formula ϕ of green ize e there i a quare matrix A of dimenion 2e + 3 whoe entrie are input of the formula and element of {0, 1,, 1/2} uch that ϕ = det A. Proof. It i ufficient to how how to have the contant for free in the contruction of Lemma 3. We alo proceed by tructural induction. In fact, the contruction remain almot the ame but with the lat property changed. For an arithmetic formula ϕ of green ize e, there exit a graph G that atifie the condition of Lemma 3 but the third one i replaced by the exitence of a contant c 0 uch that c 0 () P /2+1 w(p ) = ϕ. -t-path P Let ϕ = x be an arithmetic formula of ize 0. Then the graph G aociated to ϕ by definition ha two vertice and t and an edge t of weight x. The aociated contant i c 0 = 1. Let ϕ = cψ and G, c 0 be aociated to ψ. Then G, cc 0 i aociated to ϕ. Let ϕ = ϕ 1 ϕ 2 and G 1, c 1 (rep. G 2, c 2 ) be aociated to ϕ 1 (rep. ϕ 2 ). The graph G aociated to ϕ i exactly the ame a in the proof of Lemma 3 and the contant i c 1 c 2. Let ϕ = ϕ 1 +ϕ 2 and G 1, c 1 (rep. G 2, c 2 ) be the graph and contant aociated to ϕ 1 (rep. ϕ 2 ). We uppoe that G 1 and G 2 have ditinct et of vertice except for 1 = 2. The graph G i obtained by adding a new vertex u, an edge t 2 u with weight 1 and an edge ut 1 with weight c 2 /c 1, and the aociated contant i c 1 (ee Fig. 11). Thi define a ize-(2e + 2) graph G aociated to a green ize-e formula ϕ. It remain to turn thi graph into a matrix. Let Ḡ be the graph G augmented with a new vertex c and the edge tc of weight c 0 /2 and c of weight () G /2. The

15 SYMMETRIC DETERMINANTAL REPRESENTATION 15 ϕ 1 ϕ 2 G 2 G 1 u t 2 t 1 c 2 /c 1 Figure 11. ϕ = ϕ 1 + ϕ 2 ; G 1, c 1 and G 2, c 2 are repectively aociated to ϕ 1 and ϕ 2. adjacency matrix A of Ḡ atifie ϕ = det(a) and the proof i imilar to the one of Theorem 2. The bound obtained in Theorem 3 can be harpened when k = R or C. The idea i to build Ḡ by merging and t intead of adding a new vertex. Suppoe that ϕ ha at leat one addition gate. Let w = c 0 /2. In the contruction for thi addition gate (ee Fig. 11), multiply the weight of t 2 u and ut 1 by w. A cycle cover of the graph either goe through the path t 2 ut 1, or contain the edge ut 2 in it perfect matching part. In both cae, it weight i multiplied by w 2. Now if () G /2+1 c 0 /2 > 0, then the graph obtained ha the atifying propertie, and the new bound i 2e + 1. If it i negative, two olution can be applied. Either k i the field of complex number and it i ufficient to replace w by iw (where i 2 = ) to get the ame bound 2e + 1. Otherwie, if k i the field of real number, it i ufficient to add a new vertex with a loop of weight (thi correpond to adding a new line and a new column, filled with zeroe but the diagonal element with ) to get the bound (2e + 2). 3. Weakly kew circuit In thi ection, we extend the previou reult to the cae of weakly-kew circuit. Recall that thoe circuit are defined from arithmetic circuit by a retriction on the multiplication gate: the ub-circuit aociated to one of the argument of a multiplication gate α ha to be cloed, that i only connected to the ret of the circuit by the arrow going to α. A gate that i not in any uch cloed ub-circuit i aid to be reuable. The main difficulty to extend the reult i the exitence of everal reuable gate. In the cae of formula, there i a ingle output. Therefore, there i a ingle vertex t in the graph for which the um of the weight of the -t-path ha to equal a given expreion. Thi i no longer the cae for weakly-kew circuit. If the matrix we wih to contruct i not ymmetric, that i if the graph i oriented, thi difficulty i overcome by enuring that the graph i a directed acyclic graph. In that way, adding a new vertex cannot change the expreion computed at previouly added vertice. But in the ymmetric cae, adding a new vertex, for example in the cae of an addition gate, create ome new path in the graph. Thu it change the um of the weight of the -t α -path for ome vertex t α. A olution to thi problem i given in Lemma 4 by introducing the notion of acceptable path: A path P in a graph G i aid acceptable if G \ P admit a cycle cover.

16 16 B. GRENET, E. L. KALTOFEN, P. KOIRAN, AND N. PORTIER 3.1. Symmetric determinantal repreentation. For the weakly-kew circuit, the green ize i no longer appropriate. Hence, the reult of thi ection are expreed in term of the fat ize of the circuit: the fat ize of a circuit i it total number of gate, including the input gate. Thi meaure of circuit ize i refined in Section 3.3. Theorem 4. Let f be a polynomial computable by a weakly-kew circuit of fat ize m. Then there exit a ymmetric matrix A of dimenion at mot 2m + 1 whoe entrie are input of the circuit and element from {0, 1,, 1/2} uch that f = det A. The proof relie on the following lemma. It applie to o-called multiple-output weakly-kew circuit. Thi generalization jut conit of circuit for which there exit everal out-degree-0 gate. Lemma 4. Let C be a multiple-output weakly-kew circuit of fat ize m. There exit a graph G with at mot 2m + 1 vertice and a ditinguihed vertex uch that G i odd, every cycle in G i even, and for every reuable gate α C there exit a vertex t α G uch that (2) (1) Every -t α -path (whether acceptable or not) ha an odd number of vertice; (2) For every acceptable -t α -path P in G, the ubgraph G \ P i either empty or ha a unique cycle cover, which i a perfect matching of weight 1; (3) The following equality hold in G: acceptable -t α-path P P () 2 w(p ) = f α where f α i the polynomial computed by the gate α. Furthermore, the graph G\{} ha a unique cycle cover which i a perfect matching of weight 1. Proof. The graph G i built by induction on the (fat) ize of the circuit, the required propertie being verified at each tep of the induction. If α i a reuable gate of C, then t α i aid to be a reuable vertex of G. A ize-1 circuit i an input gate α with label x. The correponding graph G ha three vertice:, t α and an additional vertex v α. There i an edge between and v α of weight x, and an edge between v α and t α of weight. It i traightforward to check that G atify the condition of the lemma. Let m > 1 and uppoe that the lemma hold for any multiple-output weaklykew circuit of ize le than m. Let C be a multiple output weakly-kew circuit of ize m, and α be any of it output. If α i an input gate with label x, let C = C \ {α} and G the correponding graph with a ditinguihed vertex. The graph G i obtained from G by adding two new vertice v α and t α, an edge of weight x between and v α and an edge of weight between v α and t α (ee Fig. 12). The vertex i the ditinguihed vertex of G. The ize of G i G = G + 2 (2(m 1) + 1) + 2 = 2m + 1. Thu G i odd. A cycle in G i either a cycle in G or one of the two cycle v α or v α t α, o every cycle in G i even. The ize-3 path from to t α i acceptable (a G \ {} ha a unique cycle cover of weight 1) and atifie (2). Now, any other reuable gate β belong to C, o the condition are atified by induction hypothei (it i

17 SYMMETRIC DETERMINANTAL REPRESENTATION 17 x C v α x G Figure 12. Induction tep when α i an input gate. t α β C G G γ t β t γ β tβ 2 v α v α t α Figure 13. Induction tep when α i an addition gate. C t α ufficient to remark that when i removed, v α and t α are diconnected from the ret of the circuit, and a cycle cover ha to match thoe two vertice). If α i an addition gate, let C = C\{α} and uppoe that α receive arrow from gate β and γ. Note that β and γ are reuable. Let G be the graph correponding to C, and be it ditinguihed vertex. G contain two reuable vertice t β and t γ. The graph G i obtained by adding two vertice v α and t α, and the following edge: t β v α and t γ v α of weight 1, and v α t α of weight (ee Fig. 13). If β = γ, then G contain a vertex t β, and we merge the two edge adjacent to t β and t γ into an edge t β v α of weight 2. Then G = G + 2 2m + 1, and G remain odd. Every -t δ -path for ome reuable gate δ in C i even. A cycle in G i either a cycle in G, or the cycle v α t α, or i made of a t β -t γ -path P in G plu the vertex v α. Let P be a -t β -path and u the firt vertex of P belonging to P. Then, P =,..., u,..., t β and P =,..., u,..., t γ are both path with an odd number of vertice. In particular the ize of u,..., t β and u,..., t γ are of ame parity. Thu P i of odd ize and P {v α } i an even-ize cycle. Hence, every cycle in G i even. An acceptable path in G i either an acceptable path in G or a path from to t α. Indeed, the only way to cover t α in a cycle cover i to match it with v α. Therefore, no acceptable path goe through t β, v α and t γ. So, the reuable gate in C atify the condition of the lemma by induction. Any acceptable path P from to t α i an acceptable path P from to t β or t γ followed by a path from t β or t γ to t α. Thu P = P + 2 i odd and G \ P = G \ P ha a unique cycle cover which i a

18 18 B. GRENET, E. L. KALTOFEN, P. KOIRAN, AND N. PORTIER C 1 β C 2 G 2 γ t γ G 1 t α = t β Figure 14. Induction tep when α i a multiplication gate. perfect matching of weight 1. Finally, P () 2 w(p ) = acceptable -t α -path P () acceptable -t β -path P β P β +2 2 ( w(p β )) + = () P β 2 w(p β ) + () P γ 2 w(p γ ) P β Pγ =f β + f γ = f α. () acceptable -t γ -path P γ Pγ +2 2 ( w(p γ )) If α i a multiplication gate, α receive arrow from two ditinct gate β and γ. Exactly one of thoe gate, ay β, i not reuable and removing the gate α yield two dijoint circuit C 1 and C 2 (ay β belong to C 1 and γ to C 2 ). Let G 1 and G 2 be the repective graph obtained by induction from C 1 and C 2, with ditinguihed vertice 1 and 2 repectively. The graph G i obtained a in Fig. 14 a the union of G 1 and G 2 where t γ and 1 are merged, the ditinguihed vertex of G being the ditinguihed vertex 2 of G 2, and t α being equal to t β. Then G = G 1 + G 2 1, o G i odd, and if m 1 and m 2 are the repective ize of C 1 and C 2 (m = m 1 + m 2 + 1), then G 2m m = 2m 1. A cycle in G i either a cycle in G 1 or a cycle in G 2 and i therefore even. The reuable gate of C are α and the reuable gate of C 2 (by definition, C 1 i cloed and in particular t β i not reuable). A path (in G) from to a reuable gate of G 2 cannot enter G 1 o the reuable gate of G 2 atify the firt and the third condition in the lemma. Furthermore, if uch a path P i removed from G, the only cycle cover of G \ P ha to be made of a cycle cover of G 2 \ P and a cycle cover of G 1 \ 1. Indeed, the vertex 1 = t γ ha to be either in a cycle cover of G 1 or in a cycle cover of G 2. But G 2 \ (P {t γ }) i a graph of odd ize and cannot be covered by cycle of even ize and G 1 i alo of odd ize. Thu, the reuable gate in G 2 alo atify the econd condition of the lemma. It remain to prove that the reuable gate α atifie the condition of the lemma: (1) A -t α -path P i a -t γ -path P γ followed by a 1 -t β -path P β. Thu P = P γ + P β 1 a t γ = 1 and P i odd. (2) The graph G \ P i the dijoint union of (G 2 \ P γ ) and (G 1 \ P β ), o by induction G \ P i either empty or ha a unique cycle cover which i a perfect matching of weight 1.

19 SYMMETRIC DETERMINANTAL REPRESENTATION 19 (3) A w(p ) = w(p γ )w(p β ), we have P P () 2 w(p ) = () P γ + P β 2 2 w(p γ )w(p β ) = () P γ 2 w(p γ ) () P β 2 w(p β ), whence P () 2 w(p ) = () P γ 2 w(p γ ) () P β 2 w(p β ) Pγ Pβ = f γ f β = f α. Finally, the only way to cover G \ {} i to cover G 2 \ { 2 } on one hand and G 1 \ { 1 } on the other hand for parity reaon a before. The weight of thi cover i the product of the weight of the cover of G 1 and G 2, that i 1. Proof of Theorem 4. Let C be a weakly-kew circuit computing the polynomial f, and G be the graph built from C in Lemma 4. The circuit C ha a unique output, and there exit in G a vertex t correponding to thi output. Let G be the graph obtained from G by adding an edge between t and of weight 1 G 2 () 2. There i no cycle cover of G containing the 2-cycle t. Indeed, G \ {, t} i odd and G contain only even cycle. Thi mean that a cycle cover of G contain a cycle made of a -t-path plu (t, ) or a t--path plu (, t). Let P be uch a path. Then G \ P = G \ P. Hence, by Lemma 4, there i exactly one cycle cover of G \ P and it i a perfect matching of weight 1. Thi mean that there i a one-to-one correpondence between the cycle cover of G and the path from to t or from t to. There i alo a one-to-one correpondence between the path from to t and the path from t to. Let u recall that the ign of a cycle cover i the ign of the underlying permutation and it igned weight i the product of it ign and weight. Let C be a cycle cover of G involving the -t-path P. The previou paragraph how that the weight of C equal 1 G 2 () 2 w(p ). A C ha an odd cycle and a perfect matching, it ign i () G\P /2, that i the number of couple in the perfect matching. The invere cycle cover C of G ha the ame igned weight a C. Hence the um of the igned weight of all cycle cover of G equal twice the um over all -t-path P of 1 G 2 () 2 () G\P 2 w(p ) = 1 P 2 () 2 w(p ). By Lemma 4, thi equal f and Lemma 1 conclude the proof Symmetric determinantal repreentation of the determinant. Let u denote by DET n the formal determinant of the n n matrix (x i,j ). Thi polynomial ha a weakly-kew circuit of ize-o(n 5 ) ([2, 22]) or even O(n 4 ) if we ue algebraic branching program contructed by Mahajan and Vinay [20]. Thi weaklykew circuit can be repreented by a determinant of a ymmetric matrix a proved in thi paper in Theorem 4. After a talk from one of u preenting our reult, Meena Mahajan and Prajakta Nimbhorkar have communicated u the following theorem, which how that for the determinant polynomial, the ymmetrization can be done more efficiently that in the general cae. A thi reult i not publihed, we find intereting to give here it proof.

20 20 B. GRENET, E. L. KALTOFEN, P. KOIRAN, AND N. PORTIER Theorem 5 (Meena Mahajan and Prajakta Nimbhorkar). For every n there i a ymmetric matrix M of dimenion 4n and entrie in {x i,j 1 i, j n} {0; 1; ; 1/2} uch that DET n = det M. Proof. Contruct the weighted graph G computing DET n with the method ued in Section 3 of [20]. It i a directed acyclic weighted graph with three ditinguihed vertice, t + and t. Every weight i 0, 1 or a variable x i,j. The graph atifie DET n = w(p ) w(p ) t + path P in G t path P in G Moreover, thi graph ha 2n vertice, at mot 4n 4 edge and the following nice tructure: it i made of n + 1 layer, the firt layer being and the lat one being {t +, t }. Every edge i from a layer i to a layer i + 1. A a conequence, every -t + -path ha n + 1 vertice, and o ha every -t -path. From the graph G we can eaily obtain an algebraic branching program G for computing DET n : add a vertex t, an edge (t +, t) of weight 1 and an edge (t, t) of weight. We could then proceed to built from thi algebraic branching program a kew-circuit of ize O(n 4 ) (ee for example proof of Propoition 1 in [14]) and then a ymmetric determinantal repreentation of dimenion O(n 4 ) with the method decribed in Theorem 4. But ymmetrizing directly the algebraic branching program G allow u to achieve a better bound a we are going to ee. Let V be the et of vertice of G \ {, t} and E be the et of edge of G. The ymmetric weighted graph G i defined a follow by duplicating vertice in graph G. The et of it vertice i { out, t in } {u in, u out u V }. The et of it edge i {u out v in (u, v) E} {u in u out u V }. Weight on edge are defined by w(u out v in ) = w(u, v) and w(u in u out ) = 1. The graph G ha 4n vertice arranged in 2n + 2 layer and atifie the following property: (3) DET n = w(p ) acceptable out-t in-path P in G Recall that a path P in a graph G i called acceptable if G \ P admit a cycle cover. To prove Property 3, let u have a look at ome acceptable out -t in -path P in G and at ome cycle cover C of G \ P. We prove that for every u V, the vertice u in and u out are both in P or together in a length-2 cycle of C. The firt vertex of the path P i out. The econd vertex i ome u 1,in where u 1 i a vertex of the econd layer of G. The third vertex i u 1,out a u 1,in i only linked to out and u 1,out. Let u now conider another vertex v in where v belong to the econd layer of G. It i only linked to out and v out, and o it i not in P but belong to the weight 1 and length-2 cycle v in v out in C. The ame reaoning applie to the following layer. Thu we jut proved that there i a weight-preerving bijection between acceptable out -t in -path in G and -t-path in G. Moreover, for every acceptable out -t in -path P in G, the graph G \ P ha only one cycle cover, which i of weight 1 and ign () G\P /2 = () 2n3 n+2 = () n. Becaue of the ymmetry of the graph we alo have: (4) DET n = w(p ) acceptable t in - out -path P in G