Kernelization—a mathematical key concept for provably effective polynomial-time preprocessing of NP-hard problems—plays a central role in parameterized complexity and has triggered an extensive line of research. This is in part due to a lower bounds framework that allows to exclude polynomial-size kernels under the assumption of NP * coNP/poly. In this paper we consider a restricted yet natural...

Given a graph G and an integer k, the Π Edge Completion/Editing/Deletion problem asks whether it is possible to add, edit, or delete at most k edges in G such that one obtains a graph that fulfills the property Π . Edge modification problems have received considerable interest from a parameterized point of view. When parameterized by k, many of these problems turned out to be fixed-parameter tr...

In this paper, we investigate condition numbers of eigenvalue problems of matrix polynomials with nonsingular leading coefficients, generalizing classical results of matrix perturbation theory. We provide a relation between the condition numbers of eigenvalues and the pseudospectral growth rate. We obtain that if a simple eigenvalue of a matrix polynomial is ill-conditioned in some respects, th...

  Semidefinite optimization relaxations are among the widely used approaches to find global optimal or approximate solutions for many nonconvex problems. Here, we consider a specific quadratically constrained quadratic problem with an additional linear constraint. We prove that under certain conditions the semidefinite relaxation approach enables us to find a global optimal solution of the unde...

In this article, we consider a collection of geometric problems involving points colored by two colors (red and blue), referred to as bichromatic problems. The motivation behind studying these problems is two fold; (i) these problems appear naturally and frequently in the fields like Machine learning, Data mining, and so on, and (ii) we are interested in extending the algorithms and techniques ...

In this work we present a new method to compute the delays of delay differential equations (DDEs), such that the DDE has a purely imaginary eigenvalue. For delay differential equations with multiple delays, the critical curves or critical surfaces in delay space (that is, the set of delays where the DDE has a purely imaginary eigenvalue) are parameterized. We show how the method is related to o...

We discuss the explicit construction of the Schrödinger equations admitting representation through some family of general polynomials. Almost all solvable quantum potentials are shown to be generated by this approach. Some generalization has been performed also in higher dimensional problems. PACS numbers: 03.65.-w, 03.65.Ge, 03.65.Ca, 02.90.+p Published in Journal of Physics A: Mathematical an...

In this paper we study constrained eigenvalue optimization of noncommutative (nc) polynomials, focusing on the polydisc and the ball. Our three main results are as follows: (1) an nc polynomial is nonnegative if and only if it admits a weighted sum of hermitian squares decomposition; (2) (eigenvalue) optima for nc polynomials can be computed using a single semidefinite program (SDP) – this shar...

The Isomorphism of Polynomial (IP) [27], which is the main concern of this paper, originally corresponds to the problem of recovering the secret key of a C∗ scheme [26]. Besides, the security of various other schemes (e.g. signature, authentication [27], traitor tracing [5], etc. . . ) also depends on the practical hardness of IP. Due to its numerous applications, the Isomorphism of Polynomial ...

We consider polynomial eigenvalue problems P(A, α, β)x = 0 in which the matrix polynomial is homogeneous in the eigenvalue (α, β) ∈ C2. In this framework infinite eigenvalues are on the same footing as finite eigenvalues. We view the problem in projective spaces to avoid normalization of the eigenpairs. We show that a polynomial eigenvalue problem is wellposed when its eigenvalues are simple. W...