 I am Avril from Institute of Mathematical Sciences, Chennai. I am going to present my work on first exact algorithm using Haraband product of polynomial. This is a joint work with V. Irvind Rajidhath and Parthamukh Bhatia. So in this talk we will study mainly two problems on arithmetic circuits. In the last talk already we have seen the definition. So let me start with the basics still. What is an arithmetic circuit? It is a DAG where the input nodes are labeled by some symbol x1 to xn and some field elements. The internal nodes are either some addition gates or multiplication gates. In every nodes it is computing some polynomial and we said that arithmetic circuit is computing some polynomial that is computed at the root. So here for example it is computing this expression. So as a polynomial this is the polynomial computed by this arithmetic circuit. So in general a polynomial in general from is just a sum of monomials where with each monomials some coefficients from the underlying field is associated. A monomial is a form x1 power e1, x2 power e2 to xn power en. If it is a degree k all the sum of ei is bounded by k and we say that the monomial is multilinear if each ei is either 0 or 1 ok. So here in this talk we will look at mainly two problems first studied by Kottis and Williams. The first one is multilinear monomials detection. Given a polynomial as a circuit as an arithmetic circuit we want to know does there exist a degree k multilinear term we want to decide. And the second one is the counting version we want to compute the sum of the coefficients of all the degree k multilinear terms ok. So for this problem what kind of running time do we expect? This is a parameterist problem. So what is the parameterist problem? In parameterist problem in your input some part of the input that is fixed. So in general when we talk about tractable problems we say that ok we have some algorithm which is polynomial we have a polynomial time algorithm polynomial in the size of the input instance. In parameterist problems we do not talk about the we do not need it to be polynomial in the entire input instance. We need f of k polynomial so that for the parameter that is fixed for that it can be any computable function and for the remaining part we expect a polynomial time. So for example both the problems we defined this is a parameterist problem where k is also part of the input which is fixed and we expect some f of k polynomial time algorithm. We say that it is fixed parameter tractable if we have such algorithm and also there is some negative evidence just like NP hardness that ok if it is W1 hard then it is unlikely to have some f of k polynomial kind of algorithm. For counting problems similarly like sharp p we have sharp W1 if a problem is sharp W1 then it is unlikely to have some f of k polynomial time algorithm ok. So the question we have here is given a problem parameterist problem is it fixed parameterist tractable or not. Like for our case the KMMD problem multiliner monomial detection this is fixed parameter tractable and there we ask what is the best f of k dependence that we can get. And for the multiliner monomial counting that problem counting the sum there as it is known to be sharp W1 hard k will be in the exponent we want to improve the dependence of k in the exponent ok. Now why these problems are interesting? So there are so many combinational problems to name a few k path k free t dominating set m dimensional k matching. For all these combinational problems Kutis and Williams they came up with that Oste 2 power k randomized algorithm designing a design an algorithm for KMMD problem and they reduced all these problems to this multiliner monomial detection ok. So whatever the multiliner monomial counting for the counting version we do not have anything better than the brute force and alone and Guttner they showed that using some popular techniques like color coding you cannot get anything better than N power k by 2. So the challenge here was to come up with something for KMMC between the brute force. So what do you want? You want to improve the exponent improve the factor dependence on k in the exponent. So what is the consequence of it? It will improve all the counting versions of all those problems in one shot. So Kutis and Williams posed it as an open problem that can you improve it. So the main result that I am going to present so there are two results mainly in our papers the result I am going to present in my talk that for KMMC problem actually we can get deterministic Oste N power k by 2 plus extra C log k factor time algorithm ok. So what is our approach? So we take a different approach from the previous techniques for that let me define two notions ok first a polynomial elementary symmetric polynomial. What is this polynomial? So as I have told you what is multilinear monomial and over x1 to xn suppose of degree k you want to define elementary symmetric polynomial this is the sum of all the degree k multilinear terms this is elementary symmetric polynomial. Let me define Hadamard product of two polynomials. So Hadamard product is an operation between two polynomials that will again give you another new polynomial where with each monomial the coefficient associated is the product of their coefficient. So box Mf this is just like coefficient of the monomial in the polynomial f. So it is clear that if some monomial is not there in either f and g that will not be there in that resulting polynomial ok. So after defining so now probably it is quite obvious how are we going to use for KNMLC problem. So for KNMLC suppose given a polynomial I want to take Hadamard product with symmetric polynomial. So what will it do? It will just filter out the multilinear part from the polynomial ok. So the KNMLC problem actually we can reduce it to computing commutative Hadamard product with symmetric elementary symmetric polynomial of degree k. So given a polynomial as a circuit you want to compute the sum of coefficients of degree k multilinear terms we are saying that ok reduce that problem computing this C Hadamard product SNK and evaluate it at all 1 ok. So what do you want? We want to compute Hadamard product with a often arbitrary polynomial with this elementary symmetric polynomial between the brute force. So that is the problem what we want to solve ok. Now one thing that we can do we can go to some known cases where Hadamard product is easy to compute for commutative Hadamard product it is difficult to compute as you can see for determinant if you can compute Hadamard product it will give you permanent. So it is difficult to compute so we want to go to reduce this problem to some known cases where the Hadamard product is easy to compute. So where is it easy to compute? So for that I will make a detour to non commutative computation. What is non commutative computation? Just like we have defined those polynomials now suppose in a circuit you put an ordering in your multiplication gates. So when we talk about a monomial x 1 into x 2 we do not differentiate between whether it is x 1 x 2 or x 2 x 1 but in non commutative case we will treat them as a string and we will treat them separately whether it is x 1 x 2 or x 2 x 1. So for in this case here this x 2 x 1 it is coming or x 1 x 2 we are treating them separately. So what is the non commutative polynomial? Instead of sum of monomials it is like sum of words or string over x 1 to x n and with each word some coefficient is associated. So let me define another computational model just like arithmetic circuit another model algebraic branching program what is it? So suppose the polynomial is computed by an ABP of width w of d many layers that means you have some matrices m 1 m 2 up to m d d many matrices where entry of each matrix is some linear forms of this form and your polynomial is computed at the top right corner of their product then we say that this polynomial is computed by an ABP. So this is known to be a weaker model than arithmetic circuit. So suppose your polynomial is given by this model that will be useful for us in this algorithm and if the polynomial is homogeneous given by some homogeneous ABP then you can as well assume that if it is of degree k there are k many such linear matrices and the polynomial is computed at the top right corner of this product. So what was known from the previous works by Irvind, Pushkar, Srikant it is known that Hadamard product is easy to compute if one of the polynomial is given by this ABP model I will come back to that but assuming that is true now the question is can we reduce computation of commutative Hadamard product to this non commutative computation because we know it is easy to compute. So what do you want we want to reduce that commutative computation to this non commutative computation. So for that the idea we use is symmetrization what is symmetrization? So let me start with a small example. Suppose what is the easiest case of computing Hadamard product given two polynomials suppose instead of a polynomial we just have a monomial we want to compute Hadamard product with a monomial. So what is this just checking the coefficient of that monomial in this polynomial ok. So here suppose in this commutative polynomial I want to check the coefficient of x 1, x 2, x 3 what is the coefficient it is 2. Now I want to treat that polynomial treat this circuit as non commutative circuit and still I want to know what is the coefficient of this monomial x 1, x 2, x 3. So when you are treating it as non commutative what can happen you want to check all the possible strings that can give you that monomial x 1, x 2, x 3 and you want to check their coefficients separately. So what you will do you will check the coefficient of x 1, x 2, x 3 it is 1 coefficient of x 3, x 2, x 1 it is 1 and the other possible strings that can give you that monomial that is 0 and you will sum all the coefficients of it and you will get 2 correctly ok. So this symmetrization idea for monomial which was now we want to generalize it for general polynomial. So what we do we give a transformation theorem from non commutative Hadamard product computation to non commutative computation. Because CNC is just treating a circuit as non commutative circuit and SNK star we defining a non commutative polynomial what is this. So for a monomial what we told that all the possible strings that can give you this monomial. So for SNK for this commutative polynomial what is the corresponding non commutative polynomial you want to treat them as non commutative and you want to see what are all the strings that can give you this non this commutative polynomial. So SNK if you recall it was all the sum of all the multilinear monomials. So what is the possible non commutative strings that can give you this polynomial that is this all K-length strings and you are defining all the sum of all the K-length strings as SNK star ok that is what we call symmetrized elementary symmetric polynomial. And the main result that we are going to use we show that commutative Hadamard product with a polynomial and SNK and evaluating it at any point is same as treating that circuit as non commutative taking non commutative Hadamard product with SNK star and evaluating at any point. So actually in the paper we prove it in more generality for any polynomial we define a star version. But for this algorithm it is enough to prove that C Hadamard for SNK at all one is same as C and C Hadamard for SNK star at all one. So just again take a step back so what we showed that actually that KMLC problem we can reduce it to computing Hadamard product with SNK and evaluate it at all one. Now we are saying we are actually can reduce this problem to taking non commutative Hadamard product which is known to be easy when a polynomial is given by an ABP model and take Hadamard product with SNK star and evaluate it at all one ok. So now what we can do suppose we take up special case of our problem but the polynomial is not given by a circuit given by an ABP and we will try to solve that problem KMLC problem for that case. So what we need we need suppose it is given by an ABP so we need to look at that how to compute that Hadamard product when one of the polynomial is given by an ABP and we want some algorithm for SNK star ok. So this is the result by Erwin and Srekan when one of the polynomial is given by an ABP suppose the polynomial is computed as top right corner of this matrix M1 to Mk then you can actually define from that ABP B you can define some n many matrices A1B to NB and this Hadamard product is computed at the top right corner of the resulting matrix. So what is this matrix I am not going into the details. So this is this will be a super diagonal matrix where all this these are again some matrices this is a block matrix where this entry is telling you this is again a matrix where the PQA entry of that matrix is exactly first you look at the PQA entry of that matrix that is a linear matrix and you look at the coefficient of YI variable in that matrix you put there. So you can check actually that if you take this product. So the Hadamard product will be computed at the top right corner. So to just give you an example take a monomial M1 M2 some Y1 Y2 Y3 if you take a monomial. So what will happen at the top right corner Y1 Y2 Y3 that will be computed with the coefficient of that Y1 Y2 Y3 in the polynomial G ok. So as you can see that this F Hadamard G at all 1 it will be computed at the all 1 at the top right corner of this matrix ok. So this was already known. So what we now need here we said that ok this polynomial F is given by an ABP and for G we just need some evaluation algorithm. We just need some algorithm that given some matrices if we input some matrices it will give me the resulting matrix. So we need some evaluation algorithm for SNK star then we will be able to compute this C Hadamard product SNK star at all 1 ok. So what we see actually this algorithm was already there just recall this is SNK star and from the previous works of Berklons et al actually you can get an evaluation algorithm for SNK star. So what it does if you plug some matrices I mean suppose the you want to evaluate SNK star in some matrices A1 to An. So if these matrices are of dimension w by w matrices then actually you can evaluate SNK star in time k into n choose down arrow k by 2 is just like sum of n choose the i goes from 0 to k by 2 n into n choose down arrow k by 2 into poly nw. In this running time you can actually evaluate SNK star. So let us just take a step back. So what we have we have an evaluation algorithm for SNK star we are dealing with a special case where the polynomial is given by an ABP and we want to compute the sum of the coefficients. So just go back to this slide. So KNMLs of that problem reduces to taking Hadamard product with SNK evaluating at all 1 which reduces to this problem. Now we have an algorithm for this we know how to take Hadamard product and evaluate it at all 1 ok. So what do we do? First we construct those matrices as defined from the ABP model and we evaluate those matrices and we output the top right corner of the entry. The running time is bounded by n choose down arrow k by 2 poly nw if w is the size of each matrices ok. So it solves for the special case when the polynomial is given by an ABP it beats the brute force and it give you k choose n choose down arrow k by 2 poly nw algorithm. In general the polynomial is given by a circuit. So what you can do? A circuit of size s you can convert it to an ABP of s for order log k. So we are starting with some poly n size circuit. So we can convert it to an ABP where w is some n for order log k and we can use the previous algorithm to compute the sum and the entire running time now it will be bounded by some n to the power k by 2 plus c log k ok. So that is how now we are able to get exponent better than k in exponent we are able to save this now it is like n power k by 2 ok. So the other results we have is for KMMD problem as I mentioned that what Cortes and Williams did they reduced all those set of problems for KMMD for monotone circuits where all the coefficients are non-negative. Here for general circuits we give a 4.32 power k algorithm but in poly space ok. So I should mention a related result by Pratt just after our paper came Pratt also considered these two problems and for KNMLC problem he actually comes up with N choose down arrow k by 2 algorithm for general circuits whereas for only ABP we got this running time he gets the running time for general circuits also and 4.08 power k time algorithm for multilayer and monomal detection. But he uses apolarity and taking inner product so his algorithm depends on the characteristic of the ground field whereas our algorithm it is independent of the characteristic of the ground field it also works for small finite field ok that is thank you.