scholarly journals On the Structure of Valiant's Complexity Classes

1999 ◽  
Vol Vol. 3 no. 3 ◽  
Author(s):  
Peter Bürgisser
Keyword(s):  
Finite Fields ◽  
Structural Complexity ◽  
Polynomial Hierarchy ◽  
Complexity Classes ◽  
Np Completeness ◽  
Minimal Pairs ◽  
International Audience

International audience In Valiant developed an algebraic analogue of the theory of NP-completeness for computations of polynomials over a field. We further develop this theory in the spirit of structural complexity and obtain analogues of well-known results by Baker, Gill, and Solovay, Ladner, and Schöning.\par We show that if Valiant's hypothesis is true, then there is a p-definable family, which is neither p-computable nor \textitVNP-complete. More generally, we define the posets of p-degrees and c-degrees of p-definable families and prove that any countable poset can be embedded in either of them, provided Valiant's hypothesis is true. Moreover, we establish the existence of minimal pairs for \textitVP in \textitVNP.\par Over finite fields, we give a \emphspecific example of a family of polynomials which is neither \textitVNP-complete nor p-computable, provided the polynomial hierarchy does not collapse.\par We define relativized complexity classes VP^h and VNP^h and construct complete families in these classes. Moreover, we prove that there is a p-family h satisfying VP^h = VNP^h.

Logical Characterizations of Bounded Query Classes I: Logspace Oracle Machines

1993 ◽  
Vol 18 (1) ◽  
pp. 65-92
Author(s):  
Iain A. Stewart
Keyword(s):  
Truth Table ◽  
Complexity Class ◽  
Polynomial Hierarchy ◽  
Complexity Classes ◽  
Turing Machines ◽  
Complete Problems

We consider three sub-logics of the logic (±HP)*[FOs] and show that these sub-logics capture the complexity classes obtained by considering logspace deterministic oracle Turing machines with oracles in NP where the number of oracle calls is unrestricted and constant, respectively; that is, the classes LNP and LNP[O(1)]. We conclude that if certain logics are of the same expressibility then the Polynomial Hierarchy collapses. We also exhibit some new complete problems for the complexity class LNP via projection translations (the first to be discovered: projection translations are extremely weak logical reductions between problems) and characterize the complexity class LNP[O(1)] as the closure of NP under a new, extremely strict truth-table reduction (which we introduce in this paper).

scholarly journals Examining conifer canopy structural complexity across forest ages and elevations with LiDAR data

2010 ◽  
Vol 40 (4) ◽  
pp. 774-787 ◽  
Author(s):  
Van R. Kane ◽  
Jonathan D. Bakker ◽  
Robert J. McGaughey ◽  
James A. Lutz ◽  
Rolf F. Gersonde ◽  
...  
Keyword(s):  
Pacific Northwest ◽  
Canopy Structure ◽  
Structural Complexity ◽  
Low Complexity ◽  
Primary Forest ◽  
Complexity Classes ◽  
High Complexity ◽  
Development Models ◽  
Management Actions ◽  
The Pacific

LiDAR measurements of canopy structure can be used to classify forest stands into structural stages to study spatial patterns of canopy structure, identify habitat, or plan management actions. A key assumption in this process is that differences in canopy structure based on forest age and elevation are consistent with predictions from models of stand development. Three LiDAR metrics (95th percentile height, rumple, and canopy density) were computed for 59 secondary and 35 primary forest plots in the Pacific Northwest, USA. Hierarchical clustering identified two precanopy closure classes, two low-complexity postcanopy closure classes, and four high-complexity postcanopy closure classes. Forest development models suggest that secondary plots should be characterized by low-complexity classes and primary plots characterized by high-complexity classes. While the most and least complex classes largely confirmed this relationship, intermediate-complexity classes were unexpectedly composed of both secondary and primary forest types. Complexity classes were not associated with elevation, except that primary Tsuga mertensiana (Bong.) Carrière (mountain hemlock) plots were complex. These results suggest that canopy structure does not develop in a linear fashion and emphasize the importance of measuring structural conditions rather than relying on development models to estimate structural complexity across forested landscapes.

scholarly journals Counting Quiver Representations over Finite Fields Via Graph Enumeration

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Geir Helleloid ◽  
Fernando Rodriguez-Villegas
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Quiver Representations ◽  
Dimension Vector ◽  
Compte Rendu ◽  
Isomorphism Classes ◽  
International Audience ◽  
Derivatives Of ◽  
Nous Utilisons ◽  
Complete Account

International audience Let $\Gamma$ be a quiver on $n$ vertices $v_1, v_2, \ldots , v_n$ with $g_{ij}$ edges between $v_i$ and $v_j$, and let $\boldsymbol{\alpha} \in \mathbb{N}^n$. Hua gave a formula for $A_{\Gamma}(\boldsymbol{\alpha}, q)$, the number of isomorphism classes of absolutely indecomposable representations of $\Gamma$ over the finite field $\mathbb{F}_q$ with dimension vector $\boldsymbol{\alpha}$. We use Hua's formula to show that the derivatives of $A_{\Gamma}(\boldsymbol{\alpha}, q)$ with respect to $q$, when evaluated at $q = 1$, are polynomials in the variables $g_{ij}$, and we can compute the highest degree terms in these polynomials. The formulas for these coefficients depend on the enumeration of certain families of connected graphs. This note simply gives an overview of these results; a complete account of this research is available on the arXiv and has been submitted for publication. Soit $\Gamma$ un carquois sur $n$ sommets $ v_1, v_2, \ldots , v_n$ avec $g_{ij}$ arêtes entre $v_i$ et $v_j$, et soit $\boldsymbol{\alpha} \in \mathbb{N}^n$. Hua a donné une formule pour $A_{\Gamma}(\boldsymbol{\alpha}, q)$, le nombre de classes d'isomorphisme absolument indécomposables de représentations de $\Gamma$ sur le corps fini $\mathbb{F}_q$ avec vecteur de dimension $\boldsymbol{\alpha}$. Nous utilisons la formule de Hua pour montrer que les dérivées de $A_{\Gamma}(\boldsymbol{\alpha}, q)$ par rapport à $q$, alors évaluée à $q=1$, sont des polynômes dans les variables $g_{ij}$, et on peut calculer les termes de plus haut degré de ces polynômes. Les formules pour ces coefficients dépendent de l'énumération de certaines familles de graphes connectés. Cette note donne simplement un aperçu de ces résultats, un compte rendu complet de cette recherche est disponible sur arXiv et a été soumis pour publication.

scholarly journals Parameterized Problems Related to Seidel's Switching

2011 ◽  
Vol Vol. 13 no. 2 (Graph and Algorithms) ◽  
Author(s):  
Eva Jelinkova ◽  
Ondrej Suchy ◽  
Petr Hlineny ◽  
Jan Kratochvil
Keyword(s):  
Minimum Degree ◽  
The Other ◽  
Complete Graphs ◽  
Maximum Likelihood Decoding ◽  
Induced Subgraph ◽  
Fixed Parameter Tractable ◽  
Graph Theoretic ◽  
Np Completeness ◽  
Fixed Parameter ◽  
International Audience

Graphs and Algorithms International audience Seidel's switching is a graph operation which makes a given vertex adjacent to precisely those vertices to which it was non-adjacent before, while keeping the rest of the graph unchanged. Two graphs are called switching-equivalent if one can be made isomorphic to the other by a sequence of switches. In this paper, we continue the study of computational complexity aspects of Seidel's switching, concentrating on Fixed Parameter Complexity. Among other results we show that switching to a graph with at most k edges, to a graph of maximum degree at most k, to a k-regular graph, or to a graph with minimum degree at least k are fixed parameter tractable problems, where k is the parameter. On the other hand, switching to a graph that contains a given fixed graph as an induced subgraph is W [1]-complete. We also show the NP-completeness of switching to a graph with a clique of linear size, and of switching to a graph with small number of edges. A consequence of the latter result is the NP-completeness of Maximum Likelihood Decoding of graph theoretic codes based on complete graphs.

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.

scholarly journals On $k$-simplexes in $(2k-1)$-dimensional vector spaces over finite fields

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Le Anh Vinh
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Finite Fields ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Nous Montrons ◽  
International Audience

International audience We show that if the cardinality of a subset of the $(2k-1)$-dimensional vector space over a finite field with $q$ elements is $\gg q^{2k-1-\frac{1}{ 2k}}$, then it contains a positive proportional of all $k$-simplexes up to congruence. Nous montrons que si la cardinalité d'un sous-ensemble de l'espace vectoriel à $(2k-1)$ dimensions sur un corps fini à $q$ éléments est $\gg q^{2k-1-\frac{1}{ 2k}}$, alors il contient une proportion non-nulle de tous les $k$-simplexes de congruence.

scholarly journals Crooked Maps in Finite Fields

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Gohar Kyureghyan
Keyword(s):  
Finite Fields ◽  
Geometrical Characterization ◽  
Power Maps ◽  
International Audience ◽  
Almost All

International audience We consider the maps $f:\mathbb{F}_{2^n} →\mathbb{F}_{2^n}$ with the property that the set $\{ f(x+a)+ f(x): x ∈F_{2^n}\}$ is a hyperplane or a complement of hyperplane for every $a ∈\mathbb{F}_{2^n}^*$. The main goal of the talk is to show that almost all maps $f(x) = Σ_{b ∈B}c_b(x+b)^d$, where $B ⊂\mathbb{F}_{2^n}$ and $Σ_{b ∈B}c_b ≠0$, are not of that type. In particular, the only such power maps have exponents $2^i+2^j$ with $gcd(n, i-j)=1$. We give also a geometrical characterization of this maps.

THE OPERATORS MIN AND MAX ON THE POLYNOMIAL HIERARCHY

2000 ◽  
Vol 11 (02) ◽  
pp. 315-342 ◽  
Author(s):  
HARALD HEMPEL ◽  
GERD WECHSUNG
Keyword(s):  
Polynomial Hierarchy ◽  
Complexity Classes ◽  
Optimization Function ◽  
Inclusion Structure

By defining a general max and a general min operator for complexity classes we obtain that there are other interesting classes of optimization functions besides Krentel's class OptP. We investigate the behavior of these operators on the polynomial hierarchy, in particular we study the inclusion structure of the classes max · P, max · NP, max · coNP, min · P, min · NP, and min · coNP. It turns out that our operators when applied to the polynomial hierarchy yield a refinement of Krentel's hierarchy of optimization functions. We prove that this refinement is strict unless the polynomial hierarchy collapses and show that the refinement is useful to exactly classify optimization functions. Moreover, our investigations shed new light on Krentel's result that every function from some level of the polynomial hierarchy can be characterized in terms of an optimization function.

scholarly journals Logarithmic Space Verifiers on NP-complete

2019 ◽  
Author(s):  
Frank Vega
Keyword(s):  
Computer Science ◽  
Complexity Class ◽  
Complexity Classes ◽  
Input Tape ◽  
Open Problems ◽  
Logarithmic Space ◽  
Np Completeness ◽  
Np Complete ◽  
Definition Of ◽  
Np Problem

P versus NP is considered as one of the most important open problems in computer science. This consists in knowing the answer of the following question: Is P equal to NP? A precise statement of the P versus NP problem was introduced independently by Stephen Cook and Leonid Levin. Since that date, all efforts to find a proof for this problem have f ailed. NP is the complexity class of languages defined b y p olynomial t ime v erifiers M su ch th at wh en th e in put is an el ement of the language with its certificate, then M outputs a string which belongs to a single language in P. Another major complexity classes are L and NL. The certificate-based definition of NL is based on logarithmic space Turing machine with an additional special read-once input tape: This is called a logarithmic space verifier. NL is the complexity class of languages defined by logarithmic space verifiers M s uch t hat when t he i nput i s a n e lement o f t he l anguage with i ts c ertificate, th en M outputs 1. To attack the P versus NP problem, the NP-completeness is a useful concept. We demonstrate there is an NP-complete language defined by a logarithmic space verifier M such that when the input is an element of the language with its certificate, then M outputs a s tring which belongs to a single language in L. In this way, we obtain if L is not equal to NL, then P = NP. In addition, we show that L is not equal to NL. Hence, we prove the complexity class P is equal to NP.

scholarly journals The complexity of deciding whether a graph admits an orientation with fixed weak diameter

2016 ◽  
Vol Vol. 17 no. 3 (Graph Theory) ◽  
Author(s):  
Julien Bensmail ◽  
Romaric Duvignau ◽  
Sergey Kirgizov
Keyword(s):  
Undirected Graph ◽  
Oriented Graph ◽  
Extremal Graph ◽  
Np Completeness ◽  
Directed Paths ◽  
International Audience ◽  
Np Complete ◽  
Distance Properties ◽  
Vertex Colouring

International audience An oriented graph $\overrightarrow{G}$ is said weak (resp. strong) if, for every pair $\{ u,v \}$ of vertices of $\overrightarrow{G}$, there are directed paths joining $u$ and $v$ in either direction (resp. both directions). In case, for every pair of vertices, some of these directed paths have length at most $k$, we call $\overrightarrow{G}$ $k$-weak (resp. $k$-strong). We consider several problems asking whether an undirected graph $G$ admits orientations satisfying some connectivity and distance properties. As a main result, we show that deciding whether $G$ admits a $k$-weak orientation is NP-complete for every $k \geq 2$. This notably implies the NP-completeness of several problems asking whether $G$ is an extremal graph (in terms of needed colours) for some vertex-colouring problems.

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