The parameterized space complexity of embedding along a path

Hubie Chen; Moritz Müller

Home/Archive/arXiv

arXiv · 1602.09022

The parameterized space complexity of embedding along a path

Hubie Chen·Moritz Müller

View original source ↗Read source document ↗Export metadata ↓

Abstract

The embedding problem is to decide, given an ordered pair of structures, whether or not there is an injective homomorphism from the first structure to the second. We study this problem using an established perspective in parameterized complexity theory: the universe size of the first structure is taken to be the parameter, and we define the embedding problem relative to a class ${\cl A}$ of structures to be the restricted version of the general problem where the first structure must come from ${\cl A}$. We initiate a systematic complexity study of this problem family, by considering classes whose structures are what we call rooted path structures; these structures have paths as Gaifman graphs. Our main theorem is a dichotomy theorem on classes of rooted path structures.

Explore related subjects

cs.CC

Keep this discovery

BibTeX RIS

Hubie Chen, Moritz Müller. 2016-02-29. The parameterized space complexity of embedding along a path. https://arxiv.org/abs/1602.09022

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

Max Independent Set Remains NP-hard when Excluding a Planar Induced Minor

We show that there is a fixed planar graph $H$, namely the $5 \times 5$ grid, such that Max Independent Set remains NP-hard in $H$-induced-minor-free graphs. This refutes the Dallard--Milani\v{c}--\v{S}torgel conjecture and a weakening of it by Gartland and Lokshtanov, and by Korhonen.

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↗