Exotic Quantifiers, Complexity Classes, and Complete Problems

Bürgisser, Peter; Cucker, Felipe

doi:10.1007/978-3-540-73420-8_20

Cited by 8 publications

(10 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…When a ∈ [1,4), the requirements (10) that ensure a relative error bounded by ε are independent of a. This is consistent with the fact that, for all x > 0, the condition number cond √ (x) given by (8) is constant (and equal to 1 2 ). For arbitrary a > 0, in contrast, the scaling process, i.e., the computation of b and q above, depends on the magnitude of a, both in terms of complexity (the value of k grows linearly with log q) and accuracy (the log of u −1 mach also grows linearly with log q).…”

Section: Stability and Conditionsupporting

confidence: 66%

“…Unlike the situation in the discrete setting, however, there was no avalanche of NP Rcomplete problems after the publication of [6]. We won't delve into the reasons of this contrast (the interested reader may find a possible cause in [8]). Also, we note here that the inclusion NP R ⊂ EXP R was not proved in [6] and that it is certainly non-trivial (see, e.g., [33,45]).…”

Section: Introductionmentioning

confidence: 98%

See 1 more Smart Citation

A Theory of Complexity, Condition, and Roundoff

Cucker

2015

Forum math. Sigma

Self Cite

View full text Add to dashboard Cite

We develop a theory of complexity for numerical computations that takes into account the condition of the input data and allows for roundoff in the computations. We follow the lines of the theory developed by Blum, Shub and Smale for computations over R (which in turn followed those of the classical, discrete, complexity theory as laid down by Cook, Karp, and Levin, among others). In particular, we focus on complexity classes of decision problems and, paramount among them, on appropriate versions of the classes P, NP, and EXP of polynomial, nondeterministic polynomial, and exponential time, respectively. We prove some basic relationships between these complexity classes, and provide natural NP-complete problems.

show abstract

Section: Stability and Conditionsupporting

confidence: 66%

Section: Introductionmentioning

confidence: 98%

A Theory of Complexity, Condition, and Roundoff

Cucker

2015

Forum math. Sigma

Self Cite

View full text Add to dashboard Cite

show abstract

“…The geometry of the sets of zeros of polynomials equalities, or more generally solutions of polynomial inequalities, is strongly tied to complexity theory. The problem of deciding whether such a set is nonempty is the paramount NP R -complete problem (i.e., NP-complete over the reals) [7]; deciding whether it is unbounded is H∃-complete and whether a point is isolated on it is H∀-complete [9]; computing its Euler characteristic, or counting its points (in the zero dimensional case), #P R -complete [8], . .…”

Section: Introductionmentioning

confidence: 99%

Computing the Homology of Real Projective Sets

2017

Self Cite

View full text Add to dashboard Cite

We describe and analyze a numerical algorithm for computing the homology (Betti numbers and torsion coefficients) of real projective varieties. Here numerical means that the algorithm is numerically stable (in a sense to be made precise). Its cost depends on the condition of the input as well as on its size and is singly exponential in the number of variables (the dimension of the ambient space) and polynomial in the condition and the degrees of the defining polynomials. In addition, we show that outside of an exceptional set of measure exponentially small in the size of the data, the algorithm takes exponential time.

show abstract

“…The view of the decision problem of the existential theory of the reals as a complexity class existed only implicitly in the literature, e.g. [8,37], until being studied more extensively under the name NPR by Bürgisser and Cucker [7] and then under the name ∃R by Schaefer and Štefankovič [33,35]. In this paper we shall adopt the naming ∃R.…”

Section: The Existential Theory Of the Realsmentioning

confidence: 99%

The Real Computational Complexity of Minmax Value and Equilibrium Refinements in Multi-player Games

Hansen

2018

Theory Comput Syst

View full text Add to dashboard Cite

We show that for several solution concepts for finite n-player games, where n ≥ 3, the task of simply verifying its conditions is computationally equivalent to the decision problem of the existential theory of the reals. This holds for trembling hand perfect equilibrium, proper equilibrium, and CURB sets in strategic form games and for (the strategy part of) sequential equilibrium, trembling hand perfect equilibrium, and quasi-perfect equilibrium in extensive form games of perfect recall. For obtaining these results we first show that the decision problem for the minmax value in n-player games, where n ≥ 3, is also equivalent to the decision problem for the existential theory of the reals.Our results thus improve previous results of NP-hardness as well as Sqrt-Sum-hardness of the decision problems to completeness for ∃R, the complexity class corresponding to the decision problem of the existential theory of the reals. As a byproduct we also obtain a simpler proof of a result by Schaefer and Štefankovič giving ∃R-completeness for the problem of deciding existence of a probability constrained Nash equilibrium.

show abstract

Exotic Quantifiers, Complexity Classes, and Complete Problems

Cited by 8 publications

References 19 publications

A Theory of Complexity, Condition, and Roundoff

A Theory of Complexity, Condition, and Roundoff

Computing the Homology of Real Projective Sets

The Real Computational Complexity of Minmax Value and Equilibrium Refinements in Multi-player Games

Contact Info

Product

Resources

About