On Explicit Branching Programs for the Rectangular Determinant and Permanent Polynomials
V. Arvind, Abhranil Chatterjee, Rajit Datta, Partha Mukhopadhyay
aa r X i v : . [ c s . CC ] A ug On Explicit Branching Programs for theRectangular Determinant and PermanentPolynomials
V. Arvind
Institute of Mathematical Sciences (HBNI), Chennai, Indiaemail: [email protected]
Abhranil Chatterjee
Institute of Mathematical Sciences (HBNI), Chennai, Indiaemail: [email protected]
Rajit Datta
Chennai Mathematical Institute, Chennai, Indiaemail: [email protected]
Partha Mukhopadhyay
Chennai Mathematical Institute, Chennai, Indiaemail: [email protected]
Abstract
We study the arithmetic circuit complexity of some well-known family of polynomials throughthe lens of parameterized complexity. Our main focus is on the construction of explicit algebraicbranching programs (ABP) for determinant and permanent polynomials of the rectangular symbolicmatrix in both commutative and noncommutative settings. The main results are:We show an explicit O ∗ ( (cid:0) n ↓ k/ (cid:1) )-size ABP construction for noncommutative permanent polyno-mial of k × n symbolic matrix. We obtain this via an explicit ABP construction of size O ∗ ( (cid:0) n ↓ k/ (cid:1) )for S ∗ n,k , noncommutative symmetrized version of the elementary symmetric polynomial S n,k .We obtain an explicit O ∗ (2 k )-size ABP construction for the commutative rectangular determin-ant polynomial of the k × n symbolic matrix.In contrast, we show that evaluating the rectangular noncommutative determinant over rationalmatrices is Theory of Computation → Computational Complexity; CircuitComplexity
Keywords and phrases
Determinant, Permanent, Parameterized Complexity, Branching Programs
The complexity of arithmetic computations is usually studied in the model of arithmeticcircuits and its various restrictions. An arithmetic circuit is a directed acyclic graph witheach indegree-0 node (called an input gate) labeled by either a variable in { x , x , . . . , x n } or a scalar from the field F , and all other nodes (called gates) labeled as either + or × gate. At a special node (designated the output gate), the circuit computes a multivariatepolynomial in F [ x , x , . . . , x n ]. Usually we use the notation F [ X ] to denote the polynomialring F [ x , x , . . . , x n ].Arithmetic computations are also considered in the noncommutative setting. The freenoncommutative ring F h y , y , . . . , y n i is usually denoted by F h Y i . In the ring F h Y i , Throughout the paper, we use X to denote the set of commuting variables and Y to denote the set ofnoncommuting variables. On Explicit Branching Programs monomials are words in Y ∗ and polynomials in F h Y i are F -linear combinations of words. Wedefine noncommutative arithmetic circuits essentially as their commutative counterparts.The only difference is that at each product gate in a noncommutative circuit there is aprescribed left to right ordering of its inputs.A more restricted model than arithmetic circuits are algebraic branching programs. An algebraic branching program (ABP) is a directed acyclic graph with one in-degree-0 vertexcalled source , and one out-degree-0 vertex called sink . The vertex set of the graph is par-titioned into layers 0 , , . . . , ℓ , with directed edges only only between adjacent layers ( i to i + 1). The source and the sink are at layers zero and ℓ respectively. Each edge is labeledby a linear form over variables x , x , . . . , x n . The polynomial computed by the ABP is thesum over all source-to-sink directed paths of the product of linear forms that label the edgesof the path. An ABP is homogeneous if all edge labels are homogeneous linear forms. ABPscan be defined in both commutative and noncommutative settings.The main purpose of the current paper is to present new arithmetic complexity upperbound results, in the form of “optimal” algebraic branching programs, for some importantpolynomials in both the commutative and noncommutative domains. These results aremotivated by our recent work on an algebraic approach to designing efficient parameterizedalgorithms for various combinatorial problems [1].We now proceed to define the polynomials and explain the results obtained. The Elementary Symmetric Polynomial
We first recall the definition of k th elementary symmetric polynomial S n,k ∈ F [ X ], over the n variables X = { x , x , . . . , x n } , S n,k ( X ) = X S ⊆ [ n ]: | S | = k Y i ∈ S x i . It is well-known that S n,k ( X ) can be computed by an algebraic branching program of size O ( nk ). In this paper, we consider the noncommutative symmetrized version S ∗ n,k , in thering F h Y i , defined as: S ∗ n,k ( Y ) = X T ⊆ [ n ]: | T | = k X σ ∈ S k Y i ∈ T y σ ( i ) . The complexity of the polynomial S ∗ n,k is first considered by Nisan in his seminal work innoncommutative computation [11]. Nisan shows that any ABP for S ∗ n,k is of size Ω( (cid:0) n ↓ k/ (cid:1) ) .Furthermore, Nisan also shows the existence of ABP of size O ( (cid:0) n ↓ k/ (cid:1) ) for S ∗ n,k . However,it is not clear how to construct such an ABP in time O ( (cid:0) n ↓ k/ (cid:1) ). Note that an ABP of size O ∗ ( n k ) for S ∗ n,k can be directly constructed in O ∗ ( n k ) time by opening up the expressioncompletely . The main upper bound question is whether we can achieve any constant factorsaving of the parameter k in terms of size and run time of the construction. In this paper, wegive such an explicit construction. Note that Nisan’s result also rules out any FPT ( k )-sizeABP for S ∗ n,k . That also justifies the problem from an exact computation point of view. We use (cid:0) n ↓ r (cid:1) to denote P ri =0 (cid:0) ni (cid:1) . In this paper we use the notation O ∗ ( · ) freely to suppress the terms asymptotically smaller than themain term. rvind et. al. 3 Rectangular Permanent and Rectangular Determinant Polynomial
The next polynomial of interest in the current paper is rectangular permanent polynomial.Given a k × n rectangular matrix X = ( x i,j ) ≤ i ≤ k, ≤ j ≤ n of commuting variables and a k × n rectangular matrix Y = ( y i,j ) ≤ i ≤ k, ≤ j ≤ n of noncommuting variables, the rectangularpermanent polynomial in commutative and noncommutative domains are defined as followsrPer( X ) = X σ ∈ I k,n k Y i =1 x i,σ ( i ) , rPer( Y ) = X σ ∈ I k,n k Y i =1 y i,σ ( i ) . Here, I k,n is the set of all injections from [ k ] → [ n ]. An alternative view is that rPer( X ) = P S ⊂ [ n ]: | S | = k Per( X S ) where X S is the k × k submatrix where the columns are indexed by theset S . Of course, such a polynomial can be computed in time O ∗ ( n k ) using a circuit of similarsize, the main interesting issue is to understand whether the dependence on the parameter k can be improved. It is implicit in the work of Vassilevska and Williams [12] that therPer( X ) polynomial in the commutative setting can be computed by an algebraic branchingprogram of size O ∗ (2 k ). This problem originates from its connection with combinatorialproblems studied in the context of exact algorithm design [12]. In the noncommutativesetting, set-multilinearizing S ∗ n,k ( Y ) polynomial (i.e. replacing each y i at position j by y j,i ),we obtain rPer( Y ) where Y is a k × n symbolic matrix of noncommuting variables. Usingthis connection with the explicit construction of S ∗ n,k ( Y ) polynomial, we provide an ABPfor rPer( Y ) in the noncommutative setting of size O ∗ ( (cid:0) n ↓ k/ (cid:1) ). The construction time is alsosimilar.As in the usual commutative case, the noncommutative determinant polynomial of asymbolic matrix Y = ( y i,j ) ≤ i,j ≤ k is defined as follows (the variables in the monomials areordered from left to right): Det ( Y ) = X σ ∈ S k y ,σ (1) . . . y k,σ ( k ) . Nisan [11] has also shown that any algebraic branching program for the noncommutativedeterminant of a k × k symbolic matrix must be of size Ω(2 k ). In this paper we give anexplicit construction of such an ABP in time O ∗ (2 k ). Here too, the main point is that Nisanhas also shown that the lower bound is tight, but we provide an explicit construction.Moreover, motivated by the result of Vassilevska and Williams [12], we study the complex-ity of the rectangular determinant polynomial (in commutative domain) defined as follows.rDet( X ) = X S ∈ ( [ n ] k ) Det ( X S ) . The above definition is well-known in mathematics. It is often referred to as the Cullisdeterminant [10]. We prove that the rectangular determinant polynomial can be computedusing O ∗ (2 k )-size explicit ABP. Whether one can explicitly obtain a branching program ofsimilar size for rectangular determinant polynomial in the noncommutative domain, remainsas an open problem.Finally, we consider the problem of evaluating the noncommutative rectangular determ-inant over matrix algebras and show that it is O ∗ ( n o ( k ) )-size ABP. Recently, we have shown the On Explicit Branching Programs n × n determinant over matrix algebras is well-studied, and computing it remains P -hardeven over 2 × P -hardness of noncommutative determinant, authors in [3]reduce the evaluation of commutative permanent to this case; whereas, r × r matrices over any field, can be computedin time O ∗ (2 k r k ). Our Results
We first formally define what we mean by explicit circuit upper bounds. ◮ Definition 1 (Explicit Circuit Upper Bound) . A family { f n } n> of degree- k polynomialsin the commutative ring F [ x , x , . . . , x n ] (or the noncommutative ring F h y , y , . . . , y n i ) has q ( n, k )-explicit upper bounds if there is an O ∗ ( q ( n, k )) time-bounded algorithm A that oninput h n , k i outputs a circuit C n of size O ∗ ( q ( n, k )) computing f n . We show the following explicit upper bound results. ◮ Theorem 2. 1.
The family of symmetrized elementary polynomials { S ∗ n,k ( Y ) } n> has (cid:0) n ↓ k/ (cid:1) -explicit ABPs over any field. The noncommutative rectangular permanent family { rPer( Y ) } n> , where Y is a k × n symbolic matrix of variables has (cid:0) n ↓ k/ (cid:1) -explicit ABPs. ◮ Remark 3.
We note here that there is an algorithm of run time O ∗ ( (cid:0) n ↓ k/ (cid:1) ) for computingthe rectangular permanent over rings and semirings [6]. Our contribution in Theorem 2.2 isthat we obtain an (cid:0) n ↓ k/ (cid:1) -explicit ABP for it. ◮ Theorem 4. 1.
The family of noncommutative determinants { Det ( Y ) } k> has k -explicitABPs over any field. There is a family { f n } of noncommutative degree- k polynomials f n such that f n has thesame support as S ∗ n,k , and it has k -explicit ABPs. This result holds over any field thathas at least n distinct elements. The commutative rectangular determinant family { rDet( X ) } k> , where X is a k × n matrixof variables has k -explicit ABPs. We stress here that the constructive aspect of the above upper bounds is new. The existence of the ABPs claimed in the first two parts of Theorem 2 and the first part ofTheorem 4 follows from Nisan’s work [11] which shows a tight connection between optimalABP-size for some f ∈ F h X i and ranks of the matrices M r whose rows are labeled by degree r monomials, columns by degree k − r monomials and the ( m , m ) th entry is the coefficientof m m in f . But constructing an ABP for f would be substantially slower in general (forexample, we could adapt the Beimel et al. algorithm for learning multiplicity automata [4]to solve this problem).Next we describe the parameterized hardness result for rectangular determinant polyno-mial when we evaluate over matrix algebras. ◮ Theorem 5.
For any fixed ǫ > , evaluating the k × n rectangular determinant polynomialover n ǫ × n ǫ rational matrices is -hard, treating k as fixed parameter. rvind et. al. 5 However, we can easily design an algorithm of run time O ∗ (2 k r k ) for computing therectangular permanent and determinant polynomials with r × r matrix entries over anyfield. Organization
The paper is organized as follows. In Section 2, we provide the necessary background. Theproofs of Theorem 2 and Theorem 4 are given in Section 3 and Section 4 repectively. Weprove Theorem 5 in Section 5.
We provide some background results from noncommutative computation. Given a commut-ative circuit C , we can naturally associate a noncommutative circuit C nc by prescribing aninput order at each multiplication gate. This is captured in the following definition. ◮ Definition 6.
Given a commutative circuit C computing a polynomial in F [ x , x , ..., x n ] ,the noncommutative version of C , C nc is the noncommutative circuit obtained from C by fix-ing an ordering of the inputs to each product gate in C and replacing x i by the noncommutingvariable y i : 1 ≤ i ≤ n . Let f ∈ F [ X ] be a homogenous degree- k polynomial computed by a circuit C , and letˆ f ( Y ) ∈ F h Y i be the polynomial computed by C nc . Let X k denote the set of all degree- k monomials over X . As usual, Y k denotes all degree- k noncommutative monomials (i.e.,words) over Y . Each monomial m ∈ X k can appear as different noncommutative monomialsˆ m in ˆ f . We use the notation ˆ m → m to denote that ˆ m ∈ Y k will be transformed to m ∈ X k by substituting x i for y i , ≤ i ≤ n . Then, we observe the following, [ m ] f = P ˆ m → m [ ˆ m ] ˆ f . For each monomial ˆ m = y i y i · · · y i k , the permutation σ ∈ S k maps ˆ m to the monomialˆ m σ defined as ˆ m σ = y i σ (1) y i σ (2) · · · y i σ ( k ) . By linearity, ˆ f = P ˆ m ∈ Y k [ ˆ m ] ˆ f · ˆ m is mapped by σ to the polynomial, ˆ f σ = P ˆ m ∈ Y k [ ˆ m ] ˆ f · ˆ m σ . This gives the following definition. ◮ Definition 7.
The symmetrized polynomial of f , f ∗ is degree- k homogeneous polynomial f ∗ = P σ ∈ S k ˆ f σ . Next, we recall the definition of Hadamard product of two polynomials. ◮ Definition 8.
Given polynomials f, g , their Hadamard product is defined as f ◦ g = X m ([ m ] f · [ m ] g ) · m, where [ m ] f denotes the coefficient of monomial m in f . In the commutative setting, computing the Hadamard product is intractable in general.This is readily seen as the Hadamard product of the determinant polynomial with itselfyields the permanent polynomial. However, in the noncommutative setting the Hadamardproduct of two ABPs can be computed efficiently [2]. ◮ Theorem 9. [2]
Given a noncommutative ABP of size S ′ for degree k polynomial f ∈ F h y , y , . . . , y n i and a noncommutative ABP of size S for another degree k polynomial g ∈ F h y , y , ..., y n i , we can compute a noncommutative ABP of size SS ′ for f ◦ g in deterministic SS ′ · poly ( n, k ) time. On Explicit Branching Programs
Let C be a circuit and B an ABP computing homogeneous degree- k polynomials f, g ∈ F h Y i respectively. Then their Hadamard product f ◦ g has a noncommutative circuit ofpolynomially bounded size which can be computed efficiently [2].Furthermore, if C is given by black-box access then f ◦ g ( a , a , . . . , a n ) for a i ∈ F , ≤ i ≤ n can be evaluated by evaluating C on matrices defined by the ABP B [3] as follows:For each i ∈ [ n ], the transition matrix M i ∈ M s ( F ) are computed from the noncommutativeABP B (which is of size s ) that encode layers. We define M i [ k, ℓ ] = [ x i ] L k,ℓ , where L k,ℓ isthe linear form on the edge ( k, ℓ ). Now to compute ( f ◦ g )( a , a , . . . , a n ) where a i ∈ F foreach 1 ≤ i ≤ n , we compute C ( a M , a M , . . . a n M n ). The value ( f ◦ g )( a , a , . . . , a n ) isthe (1 , s ) th entry of the matrix f ( a M , a M , . . . , a n M n ). ◮ Lemma 10. [3]
Given a circuit C and a ABP B computing homogeneous noncommutativepolynomials f and g in F h Y i , the Hadamard product f ◦ g can be evaluated at any point ( a , . . . , a n ) ∈ F n by evaluating C ( a M , . . . , a n M n ) where M , . . . , M n are the transitionmatrices of B , and the dimension of each M i is the size of B . In this section, we present the construction of explicit ABPs for S ∗ n,k ( Y ) and noncommutativerPer( Y ). S ∗ n,k ( Y ) The construction of the ABP for S ∗ n,k ( Y ) is inspired by a inclusion-exclusion based dynamicprogramming algorithm for the disjoint sum problem [5]. The main result of this section isthe following. Proof of Theorem 2.1.
Let us denote by F the family of subsets of [ n ] of size exactly k/ ↓ F denote the family of subsets of [ n ] of size at most k/
2. For a subset S ⊂ [ n ], wedefine m S = Q j ∈ S y j . Let us define f S = X σ ∈ S k/ k/ Y j =1 y i σ ( j ) where S ∈ F and S = { i , i , . . . , i k/ } , otherwise for subsets S / ∈ F , we define f S = 0. Notethat, for each S ∈ F , f S is the symmetrization of the monomial m S which we denote by m ∗ S (notice Definition 7).For each S ∈↓ F , let us define ˆ f S = P S ⊆ A f A where A ∈ F . We now show, using theinclusion-exclusion principle, that we can express S ∗ n,k using an appropriate combination ofthese symmetrized polynomials for different subsets. ◮ Lemma 11. S ∗ n,k = X S ∈↓ F ( − | S | ˆ f S . Proof.
Let us first note that, S ∗ n,k = P A ∈ F P B ∈ F [ A ∩ B = ∅ ] f A f B , where we use [ P ] to rvind et. al. 7 denote that the proposition P is true. By the inclusion-exclusion principle: S ∗ n,k = X A ∈ F X B ∈ F [ A ∩ B = ∅ ] f A f B = X A ∈ F X B ∈ F X S ∈↓ F ( − | S | [ S ⊆ A ∩ B ] f A f B = X S ∈↓ F ( − | S | X A ∈ F X B ∈ F [ S ⊆ A ][ S ⊆ B ] f A f B = X S ∈↓ F ( − | S | X A ∈ F [ S ⊆ A ] f A ! = X S ∈↓ F ( − | S | ˆ f S . ◭ Now we describe two ABPs where the first ABP simultaneously computes f A for each A ∈ F and the second one simultaneously computes ˆ f S for each S ∈↓ F . ◮ Lemma 12.
There is an (cid:0) n ↓ k/ (cid:1) -explicit multi-output ABP B that outputs the collection { f A } for each A ∈ F . Proof.
First note that, m ∗ S = P j ∈ S m ∗ S \{ j } · x j . Now, the construction of the ABP isobvious. It consists of ( k/ ℓ ∈ { , , . . . , k/ } has (cid:0) nℓ (cid:1) many nodesindexed by ℓ size subsets of [ n ]. In ( ℓ + 1) th layer, the node indexed by S is connected tothe nodes S \ { j } in the previous layer with an edge label x j for each j ∈ S . Clearly, in thelast layer, the S th sink node computes f S . ◭◮ Lemma 13.
There is an (cid:0) n ↓ k/ (cid:1) -explicit multi-output ABP B that outputs the collection { ˆ f S } for each S ∈↓ F . Proof.
To construct such an ABP, we use ideas from [5]. We define ˆ f i,S = P S ⊆ A f A where S ⊆ A and A ∩ [ i ] = S ∩ [ i ]. Note that, ˆ f n,S = f S and ˆ f ,S = ˆ f S . From the definition, it isclear that ˆ f i − ,S = ˆ f i,S + ˆ f i,S ∪{ i } if i / ∈ S and ˆ f i − ,S = ˆ f i,S if i ∈ S . Hence, we can takea copy of ABP B from Lemma 12, and then simultaneously compute ˆ f i,S for each S ∈↓ F and i ranging from n to 0. Clearly, the new ABP B consists of ( n + k/ (cid:0) n ↓ k/ (cid:1) nodes at each layer. The number of edges in the ABP is also linear inthe number of nodes. ◭ Let f = P m ∈ Y k [ m ] f · m be a noncommutative polynomial of degree k in F h Y i . The reverse of f is defined as the polynomial f R = X m ∈ Y k [ m ] f · m R , where m R is the reverse of the word m . ◮ Lemma 14. [Reversing an ABP] Suppose B is a multi-output ABP with r sink nodeswhere the i th sink node computes f i ∈ F h Y i for each i ∈ [ r ] . We can construct an ABP oftwice the size of B that computes the polynomial P ri =1 f i · L i · f Ri where L i are affine linearforms. Proof.
Suppose B has ℓ layers, then we construct an ABP of 2 ℓ + 1 layers where the first ℓ layers are the copy of ABP B and the last ℓ layers are the “mirror image" of the ABP B ,call it B R . In the ( ℓ + 1) th layer we connect the i th sink node of ABP B to the i th sourcenode of B R by an edge with edge label L i . Note that, B R has r source nodes and one sinknode and the polynomial computed between i th source node and sink is f Ri . ◭ On Explicit Branching Programs
Now, applying the construction of Lemma 14 to the multi-output ABP B of Lemma 13with L S = ( − | S | we obtain an ABP that computes the polynomial P S ( − | S | ˆ f S · ˆ f RS . Sinceˆ f S is a symmetrized polynomial, we note that ˆ f RS = ˆ f S and using Lemma 11 we concludethat this ABP computes S ∗ n,k . The ABP size is O ( k (cid:0) n ↓ k/ (cid:1) ). ◭ rPer( Y ) Proof of Theorem 2.2. A (cid:0) n ↓ k/ (cid:1) -explicit ABP for the rectangular permanent polynomial canbe obtained easily from the (cid:0) n ↓ k/ (cid:1) -explicit ABP for S ∗ n,k ( Y ) by careful set-multilinearization.This can be done by simply renaming the variables y i : 1 ≤ i ≤ n at the position 1 ≤ j ≤ k by y j,i . ◭ We divide the proof in three subsections. k -explicit ABP for k × k noncommutative determinant In this section, we present an optimal explicit ABP construction for the noncommutativedeterminant polynomial for the square symbolic matrix. .
Proof of Theorem 4.1.
The ABP B has k + 1 layers with (cid:0) kℓ (cid:1) nodes at the layer ℓ for each0 ≤ ℓ ≤ k . The source of the ABP is labeled ∅ and the nodes in layer ℓ are labeled by thedistinct size ℓ subsets S ⊆ [ k ], 1 ≤ ℓ ≤ k , hence the sink is labeled [ k ]. From the nodelabeled S in layer ℓ , there are k − ℓ outgoing edges ( S, S ∪ { j } ), j ∈ [ k ] \ S .Define the sign sgn ( S, j ) as sgn ( S, j ) = ( − t j , where t j is the number of elements in S larger than j . Equivalently, t j is the number of swaps required to insert j in the correctposition, treating S as a sorted list.For noncommutative determinant polynomial, we connect the set S in the i th layer to aset S ∪ { j } in the ( i + 1) th layer with the edge label sgn ( S, j ) · x i +1 ,j The source to sink pathsin this ABP are in 1-1 correspondence to the node labels on the paths which give subsetchains ∅ ⊂ T ⊂ T ⊂ · · · ⊂ T k = [ k ] such that | T i \ T i − | = 1 for all i ≤ k . Such subsetchains are clearly in 1-1 correspondence with permutations σ ∈ S k listed as a sequence: σ (1) , σ (2) , . . . , σ ( k ), where T i = { σ (1) , σ (2) , . . . , σ ( i ) } . The following claim spells out theconnection between the sign sgn ( σ ) of σ and the sgn ( S, j ) function defined above. ⊲ Claim 15.
For each σ ∈ S k and T i = { σ (1) , σ (2) , . . . , σ ( i ) } , we have sgn ( σ ) = k Y i =1 sgn ( T i − , σ ( i )) . Proof.
We first note that sgn ( σ ) = ( − t , if there are t transpositions ( r i s i ) , ≤ i ≤ t suchthat σ · ( r s ) · ( r s ) · · · ( r t s t ) = 1. Equivalently, interpreting this as sorting the list σ (1) , σ (2) , . . . , σ ( k ) by swaps ( r i s i ), applying these t swaps will sort the list into 1 , , . . . , k .As already noted, sgn ( T i − , σ ( i )) = ( − t i , where t i is the number of swaps required to insert σ ( i ) in the correct position into the sorted order of T i − (where σ ( i ) is initially placed to theright of T i − ). Hence, P ki =1 t i is the total number of swaps required for this insertion sortprocedure to sort σ (1) , σ (2) , . . . , σ ( k ). It follows that Q ki =1 sgn ( T i − , σ ( i )) = ( − P i t i = sgn ( σ ), which proves the claim. ⊳ The fact that the ABP computes the noncommutative determinant polynomial followsdirectly from Claim 15 and the edge labels. ⊳ rvind et. al. 9 k -explicit ABP weakly equivalent to S ∗ n,k A polynomial f ∈ F [ X ] (resp. F h Y i ) is said to be weakly equivalent to a polynomial g ∈ F [ X ](resp. F h Y i ), if for each monomial m over X , [ m ] f = 0 if and only if [ m ] g = 0. For theconstruction of an ABP computing a polynomial weakly equivalent to S ∗ n,k , we will suitablymodify the ABP construction described above. Proof of Theorem 4.2.
Let α i , ≤ i ≤ n be distinct elements from F . For each j ∈ [ k ] \ S ,the edge ( S, S ∪ { j } ) is labeled by the linear form sgn ( S, j ) · P ni =1 α ji x i , where x i , ≤ i ≤ n are noncommuting variables. This gives an ABP B of size O ∗ (2 k ).We show that the polynomial computed by ABP B is weakly equivalent to S ∗ n,k . Clearly, B computes a homogeneous degree k polynomial in the variables x i , ≤ i ≤ n . We determ-ine the coefficient of a monomial x i x i · · · x i k . As noted, each source to sink path in B corresponds to a permutation σ ∈ S k . Along that path the ABP compute the product oflinear forms sgn ( σ ) L σ (1) L σ (2) · · · L σ ( k ) , where L σ ( q ) = n X i =1 α σ ( q ) i x i , where the sign is given by the previous claim. The coefficient of monomial x i x i · · · x i k in the above product is given by sgn ( σ ) Q kq =1 α σ ( q ) i q . Thus, the coefficient of x i x i · · · x i k in the ABP is given by P σ ∈ S k sgn ( σ ) Q kq =1 α σ ( q ) i q , which is the determinant of the k × k Vandermonde matrix whose q th column is ( α i q , α i q , . . . , α ki q ) T . Clearly, that determinant isnon-zero if and only if the monomial x i x i · · · x i k is multilinear. Clearly the proof worksfor any field that contains at least n distinct elements. ◭◮ Remark 16.
A polynomial f ∈ F h Y i is positively weakly equivalent to S ∗ n,k , if for eachmultilinear monomial m ∈ Y k , [ m ] f >
0. In the above proof, let g be the polynomialcomputed by ABP B that is weakly equivalent to S ∗ n,k . Clearly, f = g ◦ g is positively weaklyequivalent to S ∗ n,k , and f has a 4 k -explicit ABP, since B is 2 k -explicit. This follows fromTheorem 9. We leave open the problem of finding a 2 k -explicit ABP for some polynomialthat is positively weakly equivalent to S ∗ n,k . Such an explicit construction would imply adeterministic O ∗ (2 k ) time algorithm for k -path which is a long-standing open problem [9]. k -explicit ABP for k × n commutative rectangular determinant In this section, we present the ABP construction for commutative determinant polynomialfor k × n symbolic matrix. Proof of Theorem 4.3.
We adapt the ABP presented in Subsection 4.1. The main differenceis that, for the edge (
S, S ∪{ j } ), the linear form is sgn ( S, j ) · ( P ni =1 x j,i z i ), where z i : 1 ≤ i ≤ n are fresh noncommuting variables, and the x j,i : 1 ≤ j ≤ k, ≤ i ≤ n are commutingvariables.Then with a similar argument as before, the coefficient of the monomial z i z i . . . z i k where i < i < . . . < i k is given by P σ ∈ S k sgn ( σ ) x σ (1) ,i . . . x σ ( k ) ,i k Now for a fixed σ ∈ S k ,let τ σ be the injection [ k ] → [ n ] such that τ σ ( j ) = i σ − ( j ) : 1 ≤ j ≤ k .Let ( j , j ) be an index pair that is an inversion in σ , i.e. j < j and σ ( j ) > σ ( j ).Let ℓ = σ ( j ) and ℓ = σ ( j ). So i τ σ ( ℓ ) = i σ − ( ℓ ) and i τ σ ( ℓ ) = i σ − ( ℓ ) . Clearly, i τ σ ( ℓ ) < i τ σ ( ℓ ) . Hence: X σ ∈ S k sgn ( σ ) x σ (1) ,i . . . x σ ( k ) ,i k = X τ σ ∈ I k,n sgn ( τ σ ) x ,τ σ (1) . . . x k,τ σ ( k ) . Now the idea is to filter out only the good monomials z i z i . . . z i k where i < i < . . .
0, then it is A ). We show this by areduction from the k -paths indirected graphs.However, there is a simple algorithm of run time O ∗ (2 k r k ) to evaluate rectangularpermanent or rectangular determinant of size k × n over matrix algebras of dimension r .The proof is given in the appendix.For the proof of Theorem 5, we also use the notion of Graph Polynomial . Let G ( V, E ) bea directed graph with n vertices where V ( G ) = { v , v , . . . , v n } . A k -walk is a sequence of k vertices v i , v i , . . . , v i k where ( v i j , v i j +1 ) ∈ E for each 1 ≤ j ≤ k −
1. A k -path is a k -walkwhere no vertex is repeated. Let A be the adjacency matrix of G , and let z , z , . . . , z n benoncommuting variables. Define an n × n matrix BB [ i, j ] = A [ i, j ] · z i , ≤ i, j ≤ n. Let ~ n . Let ~z be the length n vector defined by ~z [ i ] = z i .The graph polynomial C G ∈ F h Z i is defined as C G ( z , z , . . . , z n ) = ~ T · B k − · ~z. Let W be the set of all k -walks in G . The following observation is folklore. ⊲ Observation 1. C G ( z , z , . . . , z n ) = X v i v i ...v ik ∈ W z i z i · · · z i k . Hence, G contains a k -path if and only if the graph polynomial C G contains a multilinearterm. Let I k,n be the set of injections from [ k ] → [ n ]. Define S := { f ∈ I k, n |∃ g ∈ I k,n such that ∀ i ∈ [ k ] , f (2 i −
1) = g ( i ); f (2 i ) = n + g ( i ) } . Clearly, there is a bijection between S and I k,n . We denote each f ∈ S as f g where g ∈ I k,n is the corresponding injection. By a simple counting argument, we observe thefollowing. rvind et. al. 11 ⊲ Observation 2.
For each f ∈ S, sgn ( f ) = ( − k ( k − .Consider a set of noncommuting variables Y = { y , , y , , . . . , y k, n } corresponding tothe entries of a 2 k × n symbolic matrix Y . Given f ∈ I k, n , define m f = Q ki =1 y i,f ( i ) . ◮ Lemma 17.
There is an ABP B of poly ( n, k ) size that computes a polynomial F ∈ F h Y i such that for each f ∈ I k, n , [ m f ] F = 1 if f ∈ S and otherwise [ m f ] F = 0 . Proof.
The ABP B consists of 2 k + 1 layers, labelled { , , . . . , k } . For each even i ∈ [0 , k ], there is exactly one node q i at level i . For each odd i ∈ [0 , k ], there are n nodes p i, , p i, , . . . , p i,n at level i . We now describe the edges of B . For each even i ∈ [0 , k − j ∈ [ n ], there is an edge from q i to p i +1 ,j labelled x i +1 ,j . For each odd i ∈ [0 , k − j ∈ [ n ], there is an edge from p i,j to q i +1 labelled x i +1 ,n + j . For an injection f ∈ I k, n , B contributes a monomial m f if and only if f ∈ S and B can be computed in poly ( n, k )time. ◭ Suppose, Y is a 2 k × n matrix where the ( i, j ) th entry is y i,j . By Observation 2 andLemma 17,rDet( Y ) ◦ F ( Y ) = X f g ∈ S sgn ( f g ) m f g = ( − k ( k − X g ∈ I k,n m f g . Let Z = { z , . . . , z n } be a set of noncommuting variables. Define for each g ∈ I k,n , m ′ g = Q ki =1 z g ( i ) . Define a map τ such that τ : y i,j z j if i is odd, and τ : y i,j i . In other words, τ ( m f g ) = m ′ g . Notice that,rDet( Y ) ◦ F ( Y ) | τ = ( − k ( k − X g ∈ I k,n m f g | τ = ( − k ( k − X g ∈ I k,n m ′ g = ( − k ( k − S ∗ n,k ( Z ) . Given a directed graph G on n vertices, we first construct an ABP for the noncommut-ative graph polynomial C G over rationals. From the definition, it follows that C G has apolynomial size ABP. Notice that, ((rDet( Y ) ◦ F ( Y ) | τ ) ◦ C G ( Z ))( ~
1) = S ∗ n,k ( Z ) ◦ C G ( Z )( ~ k -paths in the graph G , and hence evaluating this termis C G ( Z ) by replacing eachedge labeled by z j at i th layer by two edges where the first edge is labeled by y i − ,j and second one is labeled by y i,n + j . Let C ′ G ( Y ) is the new polynomial computed by theABP. Notice that, each monomial of the modified graph polynomial looks like Q ki =1 y i,f ( i ) for some f : [2 k ] [2 n ]. More importantly, for each k -path v i v i . . . v i k , if g ∈ I k,n isthe corresponding injection, then Q ki =1 z g ( i ) is converted to Q ki =1 y i,f g ( i ) for f g ∈ S . No-tice that, (rDet( Y ) ◦ F ( Y ) | τ ) ◦ C G ( Z ) = (rDet( Y ) ◦ F ( Y ) ◦ C ′ G ( Y )) | τ and hence, evaluating(rDet( Y ) ◦ F ( Y ) ◦ C ′ G ( Y ))( ~
1) is A to evaluate rDet( Y ) overmatrix inputs. As, C ′ G ( Y ) and F ( Y ) are computed by ABPs, we obtain an ABP B ′ comput-ing C ′ G ◦ F ( Y ). From ABP B ′ , we construct the t × t transition matrices M , , . . . , M k, n where t is the size of the ABP B ′ . From Lemma 10 we know that, we are interested tocompute rDet( Y ) over the matrix tuple ( M , , . . . , M k, n ) which is same as invoking thealgorithm A on the following 2 k × n matrix A : a i,j = M i,j . By a simple reduction we geta similar hardness over n ǫ × n ǫ dimensional matrix algebras for any fixed ǫ > ◭ References Vikraman Arvind, Abhranil Chatterjee, Rajit Datta, and Partha Mukhopadhyay. Fast exactalgorithms using hadamard product of polynomials.
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The main result of the section is as follows. ◮ Theorem 18.
Let F be any field and A be an r dimensional algebra over F with basis e , e , . . . , e r . Let { A ij } ≤ i ≤ k ≤ j ≤ n be a k × n matrix with A ij ∈ A . Then rPer( A ) and rDet( A ) can be computed in deterministic O ∗ (2 k r k ) time. Proof.
We present the proof for rectangular permanent. The proof for rectangular determ-inant is identical. The proof follows easily from expressing each entry A i,j in the standardbasis and then rearranging terms. Let e , e , . . . , e r be the standard basis for A over F . Firstwe note thatrPer( A ) = X f ∈ I k,n k Y i =1 A if ( i ) = X f ∈ I k,n k Y i =1 r X ℓ =1 A ( ℓ ) if ( i ) e ℓ = X f ∈ I k,n X ( t ,t ,...,t k ) ∈ [ r ] k k Y i =1 A ( t i ) if ( i ) k Y i =1 e t i = X ( t ,t ,...,t k ) ∈ [ r ] k ( X f ∈ I k,n k Y i =1 A ( t i ) if ( i ) ) k Y i =1 e t i . (1)Now we observe that X f ∈ I k,n k Y i =1 A ( t i ) if ( i ) = rPer( A ( t ,t ,...,t k ) ) , where A ( t ,t ,...,t k ) is the k × n matrix defined as A ( t ,t ,...,t k ) ij = A t i ij . Thus we haverPer( A ) = X ( t ,t ,...,t k ) ∈ [ r ] k rPer( A ( t ,t ,...,t k ) ) k Y i =1 e t i . (2)For a fixed ( t , t , . . . , t k ) ∈ [ r ] k the value rPer( A ( t ,t ,...,t k ) ) can be computed in O ∗ (2 k )time using the rectangular permanent algorithm [12]. Now we can compute rPer( A ) bycomputing r k many such rectangular permanents and putting them together according toequation 2. This gives a deterministic O ∗ (2 k r k ) time algorithm for computing rPer( A ). ◭ As a direct corollary we get the following. ◮ Corollary 19.
Let F be any field and let A be a k × n matrix with A ij ∈ M r × r ( F ) . Then rPer( A ) and rDet( A ) can be computed in O ∗ (2 k r k ))