scholarly journals The trouble with the second quantifier

4OR ◽  
2021 ◽  
Author(s):  
Gerhard J. Woeginger
Keyword(s):  
Computational Complexity ◽  
Robust Optimization ◽  
Complexity Theory ◽  
Operational Research ◽  
Optimization Problems ◽  
Bilevel Optimization ◽  
Theoretical Background ◽  
Research Community ◽  
Universal Quantifier ◽  
Computational Complexity Theory

AbstractWe survey optimization problems that allow natural simple formulations with one existential and one universal quantifier. We summarize the theoretical background from computational complexity theory, and we present a multitude of illustrating examples. We discuss the connections to robust optimization and to bilevel optimization, and we explain the reasons why the operational research community should be interested in the theoretical aspects of this area.

The future of computational complexity theory: part II

1996 ◽  
Vol 27 (4) ◽  
pp. 3-7
Author(s):  
E. Allender ◽  
J. Feigenbaum ◽  
J. Goldsmith ◽  
T. Pitassi ◽  
S. Rudich
Keyword(s):  
Computational Complexity ◽  
Complexity Theory ◽  
Computational Complexity Theory ◽  
The Future

scholarly journals Computational Complexity Theory and the Philosophy of Mathematics†

2019 ◽  
Vol 27 (3) ◽  
pp. 381-439
Author(s):  
Walter Dean
Keyword(s):  
Computational Complexity ◽  
Computer Science ◽  
Complexity Theory ◽  
Philosophy Of Mathematics ◽  
Computability Theory ◽  
Computational Complexity Theory ◽  
Mathematical Problems ◽  
Np Problem

Abstract Computational complexity theory is a subfield of computer science originating in computability theory and the study of algorithms for solving practical mathematical problems. Amongst its aims is classifying problems by their degree of difficulty — i.e., how hard they are to solve computationally. This paper highlights the significance of complexity theory relative to questions traditionally asked by philosophers of mathematics while also attempting to isolate some new ones — e.g., about the notion of feasibility in mathematics, the $\mathbf{P} \neq \mathbf{NP}$ problem and why it has proven hard to resolve, and the role of non-classical modes of computation and proof.

scholarly journals Incremental FPT Delay

Algorithms ◽  
2020 ◽  
Vol 13 (5) ◽  
pp. 122
Author(s):  
Arne Meier
Keyword(s):  
Computational Complexity ◽  
Complexity Theory ◽  
Complexity Classes ◽  
Computational Complexity Theory ◽  
Function Classes ◽  
Classical Setting ◽  
Classical Function ◽  
Multivalued Functions ◽  
Relationship Of ◽  
The Relationship

In this paper, we study the relationship of parameterized enumeration complexity classes defined by Creignou et al. (MFCS 2013). Specifically, we introduce two hierarchies (IncFPTa and CapIncFPTa) of enumeration complexity classes for incremental fpt-time in terms of exponent slices and show how they interleave. Furthermore, we define several parameterized function classes and, in particular, introduce the parameterized counterpart of the class of nondeterministic multivalued functions with values that are polynomially verifiable and guaranteed to exist, TFNP, known from Megiddo and Papadimitriou (TCS 1991). We show that this class TF(para-NP), the restriction of the function variant of NP to total functions, collapsing to F(FPT), the function variant of FPT, is equivalent to the result that OutputFPT coincides with IncFPT. In addition, these collapses are shown to be equivalent to TFNP = FP, and also equivalent to P equals NP intersected with coNP. Finally, we show that these two collapses are equivalent to the collapse of IncP and OutputP in the classical setting. These results are the first direct connections of collapses in parameterized enumeration complexity to collapses in classical enumeration complexity, parameterized function complexity, classical function complexity, and computational complexity theory.

scholarly journals The pervasive reach of resource-bounded Kolmogorov complexity in computational complexity theory

2011 ◽  
Vol 77 (1) ◽  
pp. 14-40 ◽  
Author(s):  
Eric Allender ◽  
Michal Koucký ◽  
Detlef Ronneburger ◽  
Sambuddha Roy
Keyword(s):  
Computational Complexity ◽  
Complexity Theory ◽  
Kolmogorov Complexity ◽  
Computational Complexity Theory

scholarly journals Bosons vs. Fermions – A computational complexity perspective

2021 ◽  
Vol 43 (suppl 1) ◽  
Author(s):  
Daniel Jost Brod
Keyword(s):  
Computational Complexity ◽  
Complexity Theory ◽  
Undergraduate Students ◽  
Quantum Systems ◽  
Polynomial Hierarchy ◽  
Complexity Classes ◽  
Computational Power ◽  
Linear Optics ◽  
Computational Complexity Theory ◽  
Classical Computer

Recent years have seen a flurry of activity in the fields of quantum computing and quantum complexity theory, which aim to understand the computational capabilities of quantum systems by applying the toolbox of computational complexity theory. This paper explores the conceptually rich and technologically useful connection between the dynamics of free quantum particles and complexity theory. I review results on the computational power of two simple quantum systems, built out of noninteracting bosons (linear optics) or noninteracting fermions. These rudimentary quantum computers display radically different capabilities—while free fermions are easy to simulate on a classical computer, and therefore devoid of nontrivial computational power, a free-boson computer can perform tasks expected to be classically intractable. To build the argument for these results, I introduce concepts from computational complexity theory. I describe some complexity classes, starting with P and NP and building up to the less common #P and polynomial hierarchy, and the relations between them. I identify how probabilities in free-bosonic and free-fermionic systems fit within this classification, which then underpins their difference in computational power. This paper is aimed at graduate or advanced undergraduate students with a Physics background, hopefully serving as a soft introduction to this exciting and highly evolving field.

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