Related papers: Polynomial and analytic methods for classifying co…
Polynomial systems occur in many areas of science and engineering. Unlike general nonlinear systems, the algebraic structure enables to compute all solutions of a polynomial system. We describe our massive parallel predictor-corrector…
Hamiltonian matrices appear in a variety or problems in physics and engineering, mostly related to the time evolution of linear dynamical systems as for instance in ion beam optics. The time evolution is given by symplectic transfer…
Although NP-Complete problems are the most difficult decisional problems, it is possible to discover in them polynomial (or easy) observables. We study the Graph Partitioning Problem showing that it is possible to recognize in it two…
Let $G$ be a graph on $n$ vertices with adjacency matrix $A$, and let $\mathbf{1}$ be the all-ones vector. We call $G$ controllable if the set of vectors $\mathbf{1}, A\mathbf{1}, \dots, A^{n-1}\mathbf{1}$ spans the whole space…
In the deletion version of the list homomorphism problem, we are given graphs G and H, a list L(v) that is a subset of V(H) for each vertex v of G, and an integer k. The task is to decide whether there exists a subset W of V(G) of size at…
Laman graphs model planar frameworks that are rigid for a general choice of distances between the vertices. There are finitely many ways, up to isometries, to realize a Laman graph in the plane. Such realizations can be seen as solutions of…
We generalize some homotopy calculation techniques such as splittings and matching trees that are introduced for the computations in the case of the independence complexes of graphs to arbitrary simplicial complexes, and exemplify their…
We provide simple criteria and algorithms for expressing homogeneous polynomials as sums of powers of independent linear forms, or equivalently, for decomposing symmetric tensors into sums of rank-1 symmetric tensors of linearly independent…
We consider the problem of classifying the dynamics of complex polynomials $f: \mathbb{C} \to \mathbb{C}$ restricted to their basins of infinity. We synthesize existing combinatorial tools --- tableaux, trees, and laminations --- into a new…
The paper deals with the analytic complexity of solutions to bivariate holonomic hypergeometric systems of the Horn type. We obtain estimates on the analytic complexity of Puiseux polynomial solutions to the hypergeometric systems defined…
The P versus NP problem asks whether every language verifiable in polynomial time can also be decided in deterministic polynomial time. In this paper, we present a constructive proof that P = NP by introducing a universal, graph-based…
Let $\mathbb C$ be the set of complex numbers, and let $\mathcal P$ be a collection of complex polynomial maps in several variables. Assuming at least one $P\in\mathcal P$ depends on at least two variables, we classify all possibilities for…
We prove a combinatorial version of Thom's Isotopy Lemma for projection maps applied to any complex or real toric variety. Our results are constructive and give rise to a method for associating the Whitney strata of the projection to the…
We study here several variants of the covariates fine balance problem where we generalize some of these problems and introduce a number of others. We present here a comprehensive complexity study of the covariates problems providing…
We prove a complexity dichotomy theorem for all non-negative weighted counting Constraint Satisfaction Problems (CSP). This caps a long series of important results on counting problems including unweighted and weighted graph homomorphisms…
A homomorphism from a graph $G$ to a graph $H$ is an edge-preserving mapping from $V(G)$ to $V(H)$. For a fixed graph $H$, in the list homomorphism problem, denoted by LHom($H$), we are given a graph $G$, whose every vertex $v$ is equipped…
Holant problems capture a class of Sum-of-Product computations such as counting matchings. It is inspired by holographic algorithms and is equivalent to tensor networks, with counting CSP being a special case. A classification for Holant…
Trigraph list homomorphism problems (also known as list matrix partition problems) have generated recent interest, partly because there are concrete problems that are not known to be polynomial time solvable or NP-complete. Thus while…
In this paper we investigate the intrinsic sequential time complexity of universal elimination procedures for arbitrary continuous data structures encoding input and output objects of elimination theory (i.e. polynomial equation systems)…
We give a complexity dichotomy theorem for the counting Constraint Satisfaction Problem (#CSP in short) with complex weights. To this end, we give three conditions for its tractability. Let F be any finite set of complex-valued functions,…