scholarly journals The Polynomial Complexity of Vector Addition Systems with States

2020 ◽  
pp. 622-641 ◽  
Author(s):  
Florian Zuleger
Keyword(s):  
Computer Science ◽  
Time Complexity ◽  
Theoretical Computer Science ◽  
Initial Configuration ◽  
Computational Time ◽  
Theoretical Computer ◽  
Vector Addition ◽  
Standard Notion ◽  
Vector Addition System ◽  
Asymptotic Complexity

AbstractVector addition systems are an important model in theoretical computer science and have been used in a variety of areas. In this paper, we consider vector addition systems with states over a parameterized initial configuration. For these systems, we are interested in the standard notion of computational time complexity, i.e., we want to understand the length of the longest trace for a fixed vector addition system with states depending on the size of the initial configuration. We show that the asymptotic complexity of a given vector addition system with states is either $$\varTheta (N^k)$$ Θ ( N k ) for some computable integer k, where N is the size of the initial configuration, or at least exponential. We further show that k can be computed in polynomial time in the size of the considered vector addition system. Finally, we show that $$1 \le k \le 2^n$$ 1 ≤ k ≤ 2 n , where n is the dimension of the considered vector addition system.

scholarly journals Optimization Over the Boolean Hypercube Via Sums of Nonnegative Circuit Polynomials

2021 ◽  
Author(s):  
Mareike Dressler ◽  
Adam Kurpisz ◽  
Timo de Wolff
Keyword(s):  
Computer Science ◽  
Affine Transformation ◽  
Optimization Problems ◽  
Polynomial Optimization ◽  
Theoretical Computer Science ◽  
Sums Of Squares ◽  
Theoretical Computer ◽  
Complexity Bounds ◽  
Polynomial Optimization Problems ◽  
Transformation Of Variables

AbstractVarious key problems from theoretical computer science can be expressed as polynomial optimization problems over the boolean hypercube. One particularly successful way to prove complexity bounds for these types of problems is based on sums of squares (SOS) as nonnegativity certificates. In this article, we initiate optimization problems over the boolean hypercube via a recent, alternative certificate called sums of nonnegative circuit polynomials (SONC). We show that key results for SOS-based certificates remain valid: First, for polynomials, which are nonnegative over the n-variate boolean hypercube with constraints of degree d there exists a SONC certificate of degree at most $$n+d$$ n + d . Second, if there exists a degree d SONC certificate for nonnegativity of a polynomial over the boolean hypercube, then there also exists a short degree d SONC certificate that includes at most $$n^{O(d)}$$ n O ( d ) nonnegative circuit polynomials. Moreover, we prove that, in opposite to SOS, the SONC cone is not closed under taking affine transformation of variables and that for SONC there does not exist an equivalent to Putinar’s Positivstellensatz for SOS. We discuss these results from both the algebraic and the optimization perspective.

scholarly journals Introduction

2018 ◽  
Vol 27 (4) ◽  
pp. 441-441
Author(s):  
PAUL BALISTER ◽  
BÉLA BOLLOBÁS ◽  
IMRE LEADER ◽  
ROB MORRIS ◽  
OLIVER RIORDAN
Keyword(s):  
Computer Science ◽  
Theoretical Computer Science ◽  
Special Issue ◽  
Open Problems ◽  
Theoretical Computer ◽  
Discrete Probability ◽  
The Common

This special issue is devoted to papers from the meeting on Combinatorics and Probability, held at the Mathematisches Forschungsinstitut in Oberwolfach from the 17th to the 23rd April 2016. The lectures at this meeting focused on the common themes of Combinatorics and Discrete Probability, with many of the problems studied originating in Theoretical Computer Science. The lectures, many of which were given by young participants, stimulated fruitful discussions. The fact that the participants work in different and yet related topics, and the open problems session held during the meeting, encouraged interesting discussions and collaborations.

Sign in / Sign up

Export Citation Format

Share Document