Linear Projections of the Vandermonde Polynomial

An n-variate Vandermonde polynomial is the determinant of the n x n matrix where the ith column is the vector (1, x_i, x_i^2, ...., x_i^{n-1})^T. Vandermonde polynomials play a crucial role in the theory of alternating polynomials and occur in Lagrangian polynomial interpolation as well as in the theory of error correcting codes. In this work we study structural and computational aspects of linear projections of Vandermonde polynomials. Firstly, we consider the problem of testing if a given polynomial is linearly equivalent to the Vandermonde polynomial. We obtain a deterministic polynomial time algorithm to test if the polynomial f is linearly equivalent to the Vandermonde polynomial when f is given as product of linear factors. In the case when the polynomial f is given as a black-box our algorithm runs in randomized polynomial time. Exploring the structure of projections of Vandermonde polynomials further, we describe the group of symmetries of a Vandermonde polynomial and show that the associated Lie algebra is simple.Comment: Submitted to a conferenc

Large Deviations and Random Energy Models

A unified treatment for the existence of free energy in several random energy models is presented. If the sequence of distributions associated with the particle systems obeys a large deviation principle, then the free energy exists almost surely. This includes all the known cases as well as some heavy-tailed distributions.Comment: 10 page

Lower bounds for multilinear bounded order ABPs

Proving super-polynomial size lower bounds for syntactic multilinear Algebraic Branching Programs(smABPs) computing an explicit polynomial is a challenging problem in Algebraic Complexity Theory. The order in which variables in $\{x_1,\ldots,x_n\}$ appear along source to sink paths in any smABP can be viewed as a permutation in $S_n$. In this article, we consider the following special classes of smABPs where the order of occurrence of variables along a source to sink path is restricted: Strict circular-interval ABPs: For every subprogram the index set of variables occurring in it is contained in some circular interval of $\{1,\ldots,n\}$. L-ordered ABPs: There is a set of L permutations of variables such that every source to sink path in the ABP reads variables in one of the L orders. We prove exponential lower bound for the size of a strict circular-interval ABP computing an explicit n-variate multilinear polynomial in VP. For the same polynomial, we show that any sum of L-ordered ABPs of small size will require exponential ($2^{n^{\Omega(1)}}$) many summands, when $L \leq 2^{n^{1/2-\epsilon}}, \epsilon>0$. At the heart of above lower bound arguments is a new decomposition theorem for smABPs: We show that any polynomial computable by an smABP of size S can be written as a sum of O(S) many multilinear polynomials where each summand is a product of two polynomials in at most 2n/3 variables computable by smABPs. As a corollary, we obtain a low bottom fan-in version of the depth reduction by Tavenas [MFCS 2013] in the case of smABPs. In particular, we show that a polynomial having size S smABPs can be expressed as a sum of products of multilinear polynomials on $O(\sqrt{n})$ variables, where the total number of summands is bounded by $2^{O(\sqrt{n}\log n \log S)}$. Additionally, we show that L-ordered ABPs can be transformed into L-pass smABPs with a polynomial blowup in size

New Algorithms and Hard Instances for Non-Commutative Computation

Motivated by the recent developments on the complexity of non-com\-mu\-ta\-tive determinant and permanent [Chien et al.\ STOC 2011, Bl\"aser ICALP 2013, Gentry CCC 2014] we attempt at obtaining a tight characterization of hard instances of non-commutative permanent. We show that computing Cayley permanent and determinant on weight\-ed adjacency matrices of graphs of component size six is $\#{\sf P}$ complete on algebras that contain $2\times 2$ matrices and the permutation group $S_3$. Also, we prove a lower bound of $2^{\Omega(n)}$ on the size of branching programs computing the Cayley permanent on adjacency matrices of graphs with component size bounded by two. Further, we observe that the lower bound holds for almost all graphs of component size two. On the positive side, we show that the Cayley permanent on graphs of component size $c$ can be computed in time $n^{c{\sf poly}(t)}$, where $t$ is a parameter depending on the labels of the vertices. Finally, we exhibit polynomials that are equivalent to the Cayley permanent polynomial but are easy to compute over commutative domains.Comment: Submitted to a conferenc

Parameterized Analogues of Probabilistic Computation

We study structural aspects of randomized parameterized computation. We introduce a new class ${\sf W[P]}$-${\sf PFPT}$ as a natural parameterized analogue of ${\sf PP}$. Our definition uses the machine based characterization of the parameterized complexity class ${\sf W[P]}$ obtained by Chen et.al [TCS 2005]. We translate most of the structural properties and characterizations of the class ${\sf PP}$ to the new class ${W[P]}$-${\sf PFPT}$. We study a parameterization of the polynomial identity testing problem based on the degree of the polynomial computed by the arithmetic circuit. We obtain a parameterized analogue of the well known Schwartz-Zippel lemma [Schwartz, JACM 80 and Zippel, EUROSAM 79]. Additionally, we introduce a parameterized variant of permanent, and prove its $\#W[1]$ completeness.Comment: Submitted to a conferenc

Limitations of sum of products of Read-Once Polynomials

We study limitations of polynomials computed by depth two circuits built over read-once polynomials (ROPs) and depth three syntactically multi-linear formulas. We prove an exponential lower bound for the size of the $\Sigma\Pi^{[N^{1/30}]}$ arithmetic circuits built over syntactically multi-linear $\Sigma\Pi\Sigma^{[N^{8/15}]}$ arithmetic circuits computing a product of variable disjoint linear forms on $N$ variables. We extend the result to the case of $\Sigma\Pi^{[N^{1/30}]}$ arithmetic circuits built over ROPs of unbounded depth, where the number of variables with $+$ gates as a parent in an proper sub formula is bounded by $N^{1/2+1/30}$. We show that the same lower bound holds for the permanent polynomial. Finally we obtain an exponential lower bound for the sum of ROPs computing a polynomial in ${\sf VP}$ defined by Raz and Yehudayoff. Our results demonstrate a class of formulas of unbounded depth with exponential size lower bound against the permanent and can be seen as an exponential improvement over the multilinear formula size lower bounds given by Raz for a sub-class of multi-linear and non-multi-linear formulas. Our proof techniques are built on the one developed by Raz and later extended by Kumar et. al.\cite{KMS13} and are based on non-trivial analysis of ROPs under random partitions. Further, our results exhibit strengths and limitations of the lower bound techniques introduced by Raz\cite{Raz04a}.Comment: Submitted to a conferenc

Regularity of Binomial Edge Ideals of Certain Block Graphs

We obtain an improved lower bound for the regularity of the binomial edge ideals of trees. We prove an upper bound for the regularity of the binomial edge ideals of certain subclass of block-graphs. As a consequence we obtain sharp upper and lower bounds for the regularity of binomial edge ideals of a class of trees called lobsters. We also obtain precise expressions for the regularities of binomial edge ideals of certain classes of trees and block graphs.Comment: Some more minor changes don

Initial stages of cavitation damage and erosion on copper and brass tested in a rotating disk device

In view of the differences in flow and experimental conditions, there has been a continuing debate as to whether or not the ultrasonic method of producing cavitation damage is similar to the damage occurring in cavitating flow systems, namely, venturi and rotating disk devices. In this paper, the progress of cavitation damage during incubation periods on polycrystalline copper and brass tested in a rotating disk device is presented. The results indicate several similarities and differences in the damage mechanism encountered in a rotating disk device (which simulates field rotary devices) and a magnetostriction apparatus. The macroscopic erosion appears similar to that in the vibratory device except for nonuniform erosion and apparent plastic flow during the initial damage phase

Size scale effect in cavitation erosion

An overview and data analyses pertaining to cavitation erosion size scale effects are presented. The exponents n in the power law relationship are found to vary from 1.7 to 4.9 for venturi and rotating disk devices supporting the values reported in the literature. Suggestions for future studies were made to arrive at further true scale effects

On the Complexity of Matroid Isomorphism Problem

We study the complexity of testing if two given matroids are isomorphic. The problem is easily seen to be in $\Sigma_2^p$. In the case of linear matroids, which are represented over polynomially growing fields, we note that the problem is unlikely to be $\Sigma_2^p$-complete and is \co\NP-hard. We show that when the rank of the matroid is bounded by a constant, linear matroid isomorphism, and matroid isomorphism are both polynomial time many-one equivalent to graph isomorphism. We give a polynomial time Turing reduction from graphic matroid isomorphism problem to the graph isomorphism problem. Using this, we are able to show that graphic matroid isomorphism testing for planar graphs can be done in deterministic polynomial time. We then give a polynomial time many-one reduction from bounded rank matroid isomorphism problem to graphic matroid isomorphism, thus showing that all the above problems are polynomial time equivalent. Further, for linear and graphic matroids, we prove that the automorphism problem is polynomial time equivalent to the corresponding isomorphism problems. In addition, we give a polynomial time membership test algorithm for the automorphism group of a graphic matroid