Related papers: VPSPACE and a transfer theorem over the complex fi…
The main purpose of this paper is to study the NP-complete subset-sum problem, not in the usual context of time-complexity-based classification of the algorithms (exponential/polynomial), but through a new kind of algorithmic classification…
Many natural optimization problems derived from $\sf NP$ admit bilevel and multilevel extensions in which decisions are made sequentially by multiple players with conflicting objectives, as in interdiction, adversarial selection, and…
We study the complexity of infinite-domain constraint satisfaction problems: our basic setting is that a complexity classification for the CSPs of first-order expansions of a structure $\mathfrak A$ can be transferred to a classification of…
We provide a list of new natural $\mathsf{VNP}$-intermediate polynomial families, based on basic (combinatorial) $\mathsf{NP}$-complete problems that are complete under parsimonious reductions. Over finite fields, these families are in…
We define counting classes #P_R and #P_C in the Blum-Shub-Smale setting of computations over the real or complex numbers, respectively. The problems of counting the number of solutions of systems of polynomial inequalities over R, or of…
Models of computations over the integers are equivalent from a computability and complexity theory point of view by the Church-Turing thesis. It is not possible to unify discrete-time models over the reals. The situation is unclear but…
By considering a discrete tape where each cell corresponds to an integer, thus to a possible sum, a pseudo-polynomial solution can be given to subset sum problem, which is an NP-complete problem and a cornerstone application for this study,…
The subject logic in computer science should entail proof theoretic applications. So the question arises whether open problems in computational complexity can be solved by advanced proof theoretic techniques. In particular, consider the…
Vertically parametrised polynomial systems are a particular nice class of parametrised polynomial systems for which a lot of interesting algebraic information is encoded in its combinatorics. Given a fixed polynomial system, we empirically…
Memcomputing is a novel non-Turing paradigm of computation that uses interacting memory cells (memprocessors for short) to store and process information on the same physical platform. It was recently proved mathematically that memcomputing…
The planted clique problem is well-studied in the context of observing, explaining, and predicting interesting computational phenomena associated with statistical problems. When equating computational efficiency with the existence of…
We propose a new complexity measure of space for the BSS model of computation. We define LOGSPACE\_W and PSPACE\_W complexity classes over the reals. We prove that LOGSPACE\_W is included in NC^2\_R and in P\_W, i.e. is small enough for…
We describe a classification of degree n complex coefficient polynomials with respect to combinatorial patterns that arise from the two real algebraic curves obtained as the zero sets for their real and imaginary part. In particular, we…
The constraint satisfaction problem (CSP) involves deciding, given a set of variables and a set of constraints on the variables, whether or not there is an assignment to the variables satisfying all of the constraints. One formulation of…
We study the pullback of the apolarity invariant of complex polynomials in one variable under a polynomial map on the complex plane. As a consequence, we obtain variations of the classical results of Grace and Walsh in which the unit disk,…
Constraint satisfaction problems (CSPs) for first-order reducts of finitely bounded homogeneous structures form a large class of computational problems that might exhibit a complexity dichotomy, P versus NP-complete. A powerful method to…
In 1990 Subramanian defined the complexity class CC as the set of problems log-space reducible to the comparator circuit value problem (CCV). He and Mayr showed that NL \subseteq CC \subseteq P, and proved that in addition to CCV several…
We study the problem of minimizing a multivariate polynomial function over the unit hypercube. By representing the polynomial through a hypergraph and exploiting its sparsity structure, we establish a new sufficient condition under which…
In this paper we explore the noncommutative analogues, $\mathrm{VP}_{nc}$ and $\mathrm{VNP}_{nc}$, of Valiant's algebraic complexity classes and show some striking connections to classical formal language theory. Our main results are the…
We show that the problem of determining the feasibility of quadratic systems over $\mathbb{C}$, $\mathbb{R}$, and $\mathbb{Z}$ requires exponential time. This separates P and NP over these fields/rings in the BCSS model of computation.