Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 1311.7229

Hierarchies

Friedhelm Meyer auf der Heide·Kamil Swierkot
View original source ↗Read source document ↗Export metadata ↓

Abstract

We study the complexity theory for the local distributed setting introduced by Korman, Peleg and Fraigniaud. They have defined three complexity classes LD (Local Decision), NLD (Nondeterministic Local Decision) and NLD^#n. The class LD consists of all languages which can be decided with a constant number of communication rounds. The class NLD consists of all languages which can be verified by a nondeterministic algorithm with a constant number of communication rounds. In order to define the nondeterministic classes, they have transferred the notation of nondeterminism into the distributed setting by the use of certificates and verifiers. The class NLD^#n consists of all languages which can be verified by a nondeterministic algorithm where each node has access to an oracle for the number of nodes. They have shown the hierarchy LD subset NLD subset NLD^#n. Our main contributions are strict hierarchies within the classes defined by Korman, Peleg and Fraigniaud. We define additional complexity classes: the class LD(t) consists of all languages which can be decided with at most t communication rounds. The class NLD-O(f) consists of all languages which can be verified by a local verifier such that the size of the certificates that are needed to verify the language are bounded by a function from O(f). Our main results are refined strict hierarchies within these nondeterministic classes.

Explore related subjects

cs.CC

Keep this discovery

BibTeXRIS

Friedhelm Meyer auf der Heide, Kamil Swierkot. 2013-11-28. Hierarchies. https://arxiv.org/abs/1311.7229

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The Computational Complexity of Holant Problems on 4-regular Graphs from the Stable Subgroup Sequence of $SL(2,\mathbb{C})$

The Holant framework provides a general setting for studying counting problems and includes graph homomorphisms (\#GH) and counting constraint satisfaction problems (\#CSP) as special cases. Over the past twenty years, a series of computational complexity dichotomies have been established for Holant problems, but the classification for complex-valued signatures is still open. The main obstacle is the case in which all signatures have even arity. In this paper, we establish a dichotomy for Holant problems with a complex-valued 4-ary signature, which is a key base case for the full classification of Holant problems. We present a new strategy by introducing Schur's theorem, the classification of finite subgroups of $\mathrm{SL}(2,\mathbb{C})$ and stable subgroup sequences into the proof. These new techniques are of independent interest.

cs.CC↗
arXiv2026

Topology inside NC$^1$

We show that ACC$^0$ is precisely what can be computed with constant-width circuits of polynomial size and polylogarithmic genus. This extends a characterization given by Hansen, showing that planar constant-width circuits also characterize ACC$^0$. Thus polylogarithmic genus provides no additional computational power in this model. We consider other generalizations of planarity, including crossing number and thickness. We show that constant-width circuits of polynomial size and thickness two already suffice to capture all of NC$^1$.

cs.CC↗