On the Relative Power of Reduction Notions in Arithmetic Circuit Complexity

We show that the two main reduction notions in arithmetic circuit complexity, p-projections and c-reductions, differ in power. We do so by showing unconditionally that there are polynomials that are VNP-complete under c-reductions but not under p-projections. We also show that the question of which polynomials are VNP-complete under which type of reductions depends on the underlying field

On the Relative Power of Reduction Notions in Arithmetic Circuit Complexity

We show that the two main reduction notions in arithmetic circuit complexity, p-projections and c-reductions, differ in power. We do so by showing unconditionally that there are polynomials that are VNP-complete under c-reductions but not under p-projections. We also show that the question of which polynomials are VNP-complete under which type of reductions depends on the underlying field

On the Complexity of the Permanent in Various Computational Models

We answer a question in [Landsberg, Ressayre, 2015], showing the regular determinantal complexity of the determinant det_m is O(m^3). We answer questions in, and generalize results of [Aravind, Joglekar, 2015], showing there is no rank one determinantal expression for perm_m or det_m when m >= 3. Finally we state and prove several "folklore" results relating different models of computation

Geometric complexity theory and matrix powering

Valiant's famous determinant versus permanent problem is the flagship problem in algebraic complexity theory. Mulmuley and Sohoni (Siam J Comput 2001, 2008) introduced geometric complexity theory, an approach to study this and related problems via algebraic geometry and representation theory. Their approach works by multiplying the permanent polynomial with a high power of a linear form (a process called padding) and then comparing the orbit closures of the determinant and the padded permanent. This padding was recently used heavily to show no-go results for the method of shifted partial derivatives (Efremenko, Landsberg, Schenck, Weyman, 2016) and for geometric complexity theory (Ikenmeyer Panova, FOCS 2016 and B\"urgisser, Ikenmeyer Panova, FOCS 2016). Following a classical homogenization result of Nisan (STOC 1991) we replace the determinant in geometric complexity theory with the trace of a variable matrix power. This gives an equivalent but much cleaner homogeneous formulation of geometric complexity theory in which the padding is removed. This radically changes the representation theoretic questions involved to prove complexity lower bounds. We prove that in this homogeneous formulation there are no orbit occurrence obstructions that prove even superlinear lower bounds on the complexity of the permanent. This is the first no-go result in geometric complexity theory that rules out superlinear lower bounds in some model. Interestingly---in contrast to the determinant---the trace of a variable matrix power is not uniquely determined by its stabilizer

On Vanishing of {K}ronecker Coefficients

It is shown that: (1) The problem of deciding positivity of Kronecker coefficients is NP-hard. (2) There exists a positive ($\# P$)-formula for a subclass of Kronecker coefficients whose positivity is NP-hard to decide. (3) For any $0 < \epsilon \le 1$, there exists $0<a<1$ such that, for all $m$, there exist $\Omega(2^{m^a})$ partition triples $(\lambda,\mu,\mu)$ in the Kronecker cone such that: (a) the Kronecker coefficient $k^\lambda_{\mu,\mu}$ is zero, (b) the height of $\mu$ is $m$, (c) the height of $\lambda$ is $\le m^\epsilon$, and (d) $|\lambda|= |\mu| \le m^3$. The last result takes a step towards proving the existence of occurrence-based representation-theoretic obstructions in the context of the GCT approach to the permanent vs. determinant problem. Its proof also illustrates the effectiveness of the explicit proof strategy of GCT

On the Complexity of the Permanent in Various Computational Models

We answer a question in [Landsberg, Ressayre, 2015], showing the regular determinantal complexity of the determinant det_m is O(m^3). We answer questions in, and generalize results of [Aravind, Joglekar, 2015], showing there is no rank one determinantal expression for perm_m or det_m when m >= 3. Finally we state and prove several "folklore" results relating different models of computation

On the orbit closure containment problem and slice rank of tensors

We consider the orbit closure containment problem, which, for a given vector and a group orbit, asks if the vector is contained in the closure of the group orbit. Recently, many algorithmic problems related to orbit closures have proved to be quite useful in giving polynomial time algorithms for special cases of the polynomial identity testing problem and several non-convex optimization problems. Answering a question posed by Wigderson, we show that the algorithmic problem corresponding to the orbit closure containment problem is NP-hard. We show this by establishing a computational equivalence between the solvability of homogeneous quadratic equations and a homogeneous version of the matrix completion problem, while showing that the latter is an instance of the orbit closure containment problem. Secondly, we consider the notion of slice rank of tensors, which was recently introduced by Tao, and has subsequently been used for breakthroughs in several combinatorial problems like capsets, sunflower free sets, tri-colored sum-free sets, and progression-free sets. We show that the corresponding algorithmic problem, which can also be phrased as a problem about union of orbit closures, is also NP-hard, hence answering an open question by Bürgisser, Garg, Oliveira, Walter, and Wigderson. We show this by using a connection between the slice rank and the size of a minimum vertex cover of a hypergraph revealed by Tao and Sawin