No occurrence obstructions in geometric complexity theory

The permanent versus determinant conjecture is a major problem in complexity theory that is equivalent to the separation of the complexity classes VP_{ws} and VNP. Mulmuley and Sohoni (SIAM J. Comput., 2001) suggested to study a strengthened version of this conjecture over the complex numbers that amounts to separating the orbit closures of the determinant and padded permanent polynomials. In that paper it was also proposed to separate these orbit closures by exhibiting occurrence obstructions, which are irreducible representations of GL_{n^2}(C), which occur in one coordinate ring of the orbit closure, but not in the other. We prove that this approach is impossible. However, we do not rule out the general approach to the permanent versus determinant problem via multiplicity obstructions as proposed by Mulmuley and Sohoni.Comment: Substantial revision. This version contains an overview of the proof of the main result. Added material on the model of power sums. Theorem 4.14 in the old version, which had a complicated proof, became the easy Theorem 5.4. To appear in the Journal of the AM

Explicit polynomial sequences with maximal spaces of partial derivatives and a question of K. Mulmuley

We answer a question of K. Mulmuley: In [Efremenko-Landsberg-Schenck-Weyman] it was shown that the method of shifted partial derivatives cannot be used to separate the padded permanent from the determinant. Mulmuley asked if this "no-go" result could be extended to a model without padding. We prove this is indeed the case using the iterated matrix multiplication polynomial. We also provide several examples of polynomials with maximal space of partial derivatives, including the complete symmetric polynomials. We apply Koszul flattenings to these polynomials to have the first explicit sequence of polynomials with symmetric border rank lower bounds higher than the bounds attainable via partial derivatives.Comment: 18 pages - final version to appear in Theory of Computin

Uniform determinantal representations

The problem of expressing a specific polynomial as the determinant of a square matrix of affine-linear forms arises from algebraic geometry, optimisation, complexity theory, and scientific computing. Motivated by recent developments in this last area, we introduce the notion of a uniform determinantal representation, not of a single polynomial but rather of all polynomials in a given number of variables and of a given maximal degree. We derive a lower bound on the size of the matrix, and present a construction achieving that lower bound up to a constant factor as the number of variables is fixed and the degree grows. This construction marks an improvement upon a recent construction due to Plestenjak-Hochstenbach, and we investigate the performance of new representations in their root-finding technique for bivariate systems. Furthermore, we relate uniform determinantal representations to vector spaces of singular matrices, and we conclude with a number of future research directions.Comment: 23 pages, 3 figures, 4 table

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 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

A Polynomial-Time Deterministic Algorithm for An NP-Complete Problem

An NP-complete graph decision problem, the "Multi-stage graph Simple Path" (abbr. MSP) problem, is introduced. The main contribution of this paper is a poly-time algorithm named the ZH algorithm for the problem together with the proof of its correctness, which implies NP=P. (1) A crucial structural property is discovered, whereby all MSP instances are arranged into the sequence $G_{0}$,$G_{1}$,$G_{2}$,... ($G_{k}$ essentially stands for a group of graphs for all $k\geq 0$). For each $G_{j}(j>0)$ in the sequence, there is a graph $G_{i}(0\leq i<j)$ "mathematically homomorphic" to $G_{j}$ which keeps completely accordant with $G_{j}$ on the existence of global solutions. This naturally provides a chance of applying mathematical induction for the proof of an algorithm. In previous attempts, algorithms used for making global decisions were mostly guided by heuristics and intuition. Rather, the ZH algorithm is dedicatedly designed to comply with the proposed proving framework of mathematical induction. (2) Although the ZH algorithm deals with paths, it always regards paths as a collection of edge sets. This is the key to the avoidance of exponential complexity. (3) Any poly-time algorithm that seeks global information can barely avoid the error caused by localized computation. In the ZH algorithm, the proposed reachable-path edge-set $R(e)$ and the computed information for it carry all necessary contextual information, which can be utilized to summarize the "history" and to detect the "future" for searching global solutions. (4) The relation between local strategies and global strategies is discovered and established, wherein preceding decisions can pose constraints to subsequent decisions (and vice versa). This interplay resembles the paradigm of dynamic programming, while much more convoluted. Nevertheless, the computation is always strait forward and decreases monotonically.Comment: 61 pages,14 figure

Splitting Kronecker squares, 2-decomposition numbers, Catalan combinatorics, and the Saxl conjecture

While there has been some progress on the decomposition of Kronecker products of characters of the symmetric groups in recent times, results on the symmetric and alternating part of Kronecker squares are still scarce. Here, new results (and conjectures) are presented on this splitting of the squares that contribute to a refined understanding of the Kronecker squares. Furthermore, connections to 2-modular decomposition numbers, Catalan combinatorics, and to the Saxl conjecture are discussed which further motivate the study of these splittings