We survey the area of algebraic complexity theory; with the focus being on the problem of polynomial identity testing (PIT). We discuss the key ideas that have gone into the results of the last few years.

Progress on Polynomial Identity Testing - II.

/pdf/progress-on-polynomial-identity-testing-ii-50emkoimgs.pdf

Progress on Polynomial Identity Testing-II

We show that the bipartite perfect matching problem is in quasi-NC$^2$. That is, it has uniform circuits of quasi-polynomial size $n^{O(\log n)}$, and $O(log^2 n)$ depth. Previously, only an exponential upper bound was known on the size of such circuits with poly-logarithmic depth. 
We obtain our result by an almost complete derandomization of the famous Isolation Lemma when applied to yield an efficient randomized parallel algorithm for the bipartite perfect matching problem.

Bipartite Perfect Matching is in quasi-NC

Kayal has recently introduced the method of shifted partial derivatives as a way to give the first exponential lower bound for computing an explicit polynomial as a sum of powers of constant-degree polynomials. This method has garnered further attention because of the work of Gupta, Kamath, Kayal and Saptharishi who used this method to obtain lower bounds that approach the "chasm at depth-4". In this work, we investigate to what extent this method can be used to obtain deterministic polynomial identity testing (PIT) algorithms, which are algorithms that decide whether a given algebraic circuit computes the zero polynomial. In particular, we give a poly(s)(log s)-time deterministic black-box PIT algorithm for a size-s sum of powers of constant-degree polynomials. This is the first sub-exponential deterministic PIT algorithm for this model of computation and the first PIT algorithm based on the method of shifted partial derivatives. We also study the problem of divisibility testing, which when given multivariate polynomials f and g (as algebraic circuits) asks to decide whether f divides g. Using Strassen's technique for the elimination of divisions, we show that one can obtain deterministic divisibility testing algorithms via deterministic PIT algorithms, and this reduction does not dramatically increase the complexity of the underlying algebraic circuits. Using this reduction, we show that deciding divisibility of a constant-degree polynomial f into a sparse polynomial g reduces to PIT of sums of a monomial multiplied by a power of constant-degree polynomials. We then extend the method of shifted partial derivatives to give a poly(s)(log s)-time deterministic black-box PIT algorithm for this model of computation, and thus derive a corresponding deterministic divisibility algorithm. This is the first non-trivial deterministic algorithm for this problem. Previous work on multivariate divisibility testing due to Saha, Saptharishi and Saxena gave algorithms for when f is linear and g is sparse, and essentially worked via PIT algorithms for read-once (oblivious) algebraic branching programs (roABPs). We give explicit sums of powers of quadratic polynomials that require exponentially-large roABPs in a strong sense, showing that techniques known for roABPs have limited applicability in our regime. Finally, by combining our results with the algorithm of Forbes, Shpilka and Saptharishi we obtain poly(s)(log log s)-time deterministic black-box PIT algorithms for various models (translations of sparse polynomials, and sums of monomials multiplied by a power of a linear polynomial) where only poly(s) (Theta(log s))-time such algorithms were previously known.

Deterministic Divisibility Testing via Shifted Partial Derivatives

Mathematical Foundations of Computer Science 2012

We give an $n^{O(\log n)}$-time ($n$ is the input size) blackbox polynomial identity testing algorithm for unknown-order read-once oblivious arithmetic branching programs (ROABPs). The best time complexity known for blackbox polynomial identity testing (PIT) for this class was $n^{O(\log^2 n)}$ due to Forbes, Saptharishi, and Shpilka [Proceedings of the 2014 ACM Symposium on Theory of Computing, 2014, pp. 867--875]. Moreover, their result holds only when the individual degree is small, while we do not need any such assumption. With this, we match the time complexity for the unknown-order ROABP with the known-order ROABP (due to Forbes and Shpilka [Proceedings of the 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, 2013, pp. 243--252]) and also with the depth-3 set-multilinear circuits (due to Agrawal, Saha, and Saxena [Proceedings of the 2013 ACM Symposium on Theory of Computing, 2013, pp. 321--330]). Our proof is simpler and involves a new technique called basis isolation. The depth-3 ...

Hitting-sets for ROABP and Sum of Set-Multilinear circuits.

A read-once oblivious arithmetic branching program (ROABP) is an arithmetic branching program (ABP) where each variable occurs in at most one layer. We give the first polynomial time whitebox identity test for a polynomial computed by a sum of constantly many ROABPs. We also give a corresponding blackbox algorithm with quasi-polynomial time complexity $n^{O(\log n)}$. In both the cases, our time complexity is double exponential in the number of ROABPs. 
ROABPs are a generalization of set-multilinear depth-$3$ circuits. The prior results for the sum of constantly many set-multilinear depth-$3$ circuits were only slightly better than brute-force, i.e. exponential-time. 
Our techniques are a new interplay of three concepts for ROABP: low evaluation dimension, basis isolating weight assignment and low-support rank concentration. We relate basis isolation to rank concentration and extend it to a sum of two ROABPs using evaluation dimension (or partial derivatives).

Deterministic Identity Testing for Sum of Read-Once Oblivious Arithmetic Branching Programs

A read-once oblivious arithmetic branching program (ROABP) is an arithmetic branching program (ABP) where each variable occurs in at most one layer. We give the first polynomial time whitebox identity test for a polynomial computed by a sum of constantly many ROABPs. We also give a corresponding blackbox algorithm with quasi-polynomial time complexity nO(log n). In both the cases, our time complexity is double exponential in the number of ROABPs.ROABPs are a generalization of set-multilinear depth-3 circuits. The prior results for the sum of constantly many set-multilinear depth-3 circuits were only slightly better than brute-force, i.e. exponential-time.Our techniques are a new interplay of three concepts for ROABP: low evaluation dimension, basis isolating weight assignment and low-support rank concentration. We relate basis isolation to rank concentration and extend it to a sum of two ROABPs using evaluation dimension (or partial derivatives).

/pdf/deterministic-identity-testing-for-sum-of-read-once-148enfkyto.pdf

Deterministic identity testing for sum of read-once oblivious arithmetic branching programs

We give improved hitting sets for two special cases of Read-once Oblivious Arithmetic Branching Programs (ROABP) First is the case of an ROABP with known order of the variables The best previously known hitting set for this case had size $(nw)^{O(\log n)}$ where $n$ is the number of variables and $w$ is the width of the ROABP Even for a constant-width ROABP, nothing better than a quasi-polynomial bound was known We improve the hitting-set size for the known-order case to $n^{O(\log w)}$ In particular, this gives the first polynomial-size hitting set for constant-width ROABP (known-order) However, our hitting set only works when the characteristic of the field is zero or large enough To construct the hitting set, we use the concept of the rank of the partial derivative matrix Unlike previous approaches which build up from mapping variables to monomials, we map variables to polynomials directly 
The second case we consider is that of polynomials computable by width-$w$ ROABPs in any order of the variables The best previously known hitting set for this case had size $d^{O(\log w)}(nw)^{O(\log \log w)}$, where $d$ is the individual degree We improve the hitting-set size to $(ndw)^{O(\log \log w)}$

/pdf/identity-testing-for-constant-width-and-any-order-read-once-5018leh6zp.pdf

Identity Testing for Constant-Width, and Any-Order, Read-Once Oblivious Arithmetic Branching Programs

A red-blue graph is a graph where every edge is colored either red or blue. The exact perfect matching problem asks for a perfect matching in a red-blue graph that has exactly a given number of red edges. We show that for complete and bipartite complete graphs, the exact perfect matching problem is logspace equivalent to the perfect matching problem. Hence, an efficient parallel algorithm for perfect matching would carry over to the exact perfect matching problem for this class of graphs. We also report some progress in extending the result to arbitrary graphs.

Exact Perfect Matching in Complete Graphs

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