Related papers: Probabilistically checkable proofs for the Existen…
The Existential Theory of the Reals (ETR) consists of existentially quantified Boolean formulas over equalities and inequalities of polynomial functions of variables in $\mathbb{R}$. In this paper we propose and study the approximate…
This paper investigates $\exists\mathbb{R}(r^{\mathbb{Z}})$, that is the extension of the existential theory of the reals by an additional unary predicate $r^{\mathbb{Z}}$ for the integer powers of a fixed computable real number $r > 0$. If…
A continuous constraint satisfaction problem (CCSP) is a constraint satisfaction problem (CSP) with an interval domain $U \subset \mathbb{R}$. We engage in a systematic study to classify CCSPs that are complete of the Existential Theory of…
Can scientific discovery be made arbitrarily easy by choosing the right representation, collecting enough data, and deploying sufficiently powerful algorithms? This paper argues that the answer is fundamentally negative. We introduce the…
We survey the complexity class $\exists \mathbb{R}$, which captures the complexity of deciding the existential theory of the reals. The class $\exists \mathbb{R}$ has roots in two different traditions, one based on the Blum-Shub-Smale model…
To characterize the computational complexity of satisfiability problems for probabilistic and causal reasoning within the Pearl's Causal Hierarchy, arXiv:2305.09508 [cs.AI] introduce a new natural class, named succ-$\exists$R. This class…
We study algorithmic problems that belong to the complexity class of the existential theory of the reals (ER). A problem is ER-complete if it is as hard as the problem ETR and if it can be written as an ETR formula. Traditionally, these…
In this paper we focus on problems which do not admit a constant-factor approximation in polynomial time and explore how quickly their approximability improves as the allowed running time is gradually increased from polynomial to…
Recently, various natural algorithmic problems have been shown to be $\exists \mathbb{R}$-complete. The reduction relied in many cases on the $\exists \mathbb{R}$-completeness of the problem ETR-INV, which served as a useful intermediate…
We study optimization problems that are neither approximable in polynomial time (at least with a constant factor) nor fixed parameter tractable, under widely believed complexity assumptions. Specifically, we focus on Maximum Independent…
We develop a complexity theory for approximate real computations. We first produce a theory for exact computations but with condition numbers. The input size depends on a condition number, which is not assumed known by the machine. The…
Given any polynomial with real coefficients, the existence of a real quadratic polynomial factor is proven using only basic real analysis. The aim is to provide an approachable proof to anybody who is familiar with the least upper bound…
We study the decision problem Affine Rank Minimization, denoted ARM(k). The input consists of rational matrices A_1,...,A_q in Q^{m x n} and rational scalars b_1,...,b_q in Q. The question is whether there exists a real matrix X in R^{m x…
We prove that there is no fpt-algorithm that can approximate the dominating set problem with any constant ratio, unless FPT= W[1]. Our hardness reduction is built on the second author's recent W[1]-hardness proof of the biclique problem.…
Techniques for the efficient successive under-approximation of the greatest fixpoint in TCTL formulas can be useful in fast refutation of inevitability properties and vacuity checking. We first give an integrated algorithmic framework for…
Techniques for the efficient successive under-approximation of the greatest fixpoint in TCTL formulas can be useful in fast refutation of inevitability properties and vacuity checking. We first give an integrated algorithmic framework for…
We consider the task of properly PAC learning decision trees with queries. Recent work of Koch, Strassle, and Tan showed that the strictest version of this task, where the hypothesis tree $T$ is required to be optimally small, is NP-hard.…
In this article, I provide significant mathematical evidence in support of the existence of short-time approximations of any polynomial order for the computation of density matrices of physical systems described by arbitrarily smooth and…
We consider the hardness of approximation of optimization problems from the point of view of definability. For many NP-hard optimization problems it is known that, unless P = NP, no polynomial-time algorithm can give an approximate solution…
In this paper we propose the PCP-like theorem for sub-linear time inapproximability. Abboud et al. have devised the distributed PCP framework for proving sub-quadratic time inapproximability. Here we try to go further in this direction.…