On the complexity of the clone membership problem

Emil Jeřábek

doi:10.1007/s00224-020-10016-7

Home/Archive/arXiv

arXiv · 1909.12211

On the complexity of the clone membership problem

Emil Jeřábek

View original source ↗Read source document ↗Export metadata ↓

Abstract

We investigate the complexity of the Boolean clone membership problem (CMP): given a set of Boolean functions $F$ and a Boolean function $f$, determine if $f$ is in the clone generated by $F$, i.e., if it can be expressed by a circuit with $F$-gates. Here, $f$ and elements of $F$ are given as circuits or formulas over the usual De Morgan basis. B\"ohler and Schnoor (2007) proved that for any fixed $F$, the problem is coNP-complete, with a few exceptions where it is in P. Vollmer (2009) incorrectly claimed that the full problem CMP is also coNP-complete. We prove that CMP is in fact $\Theta^P_2$-complete, and we complement B\"ohler and Schnoor's results by showing that for fixed $f$, the problem is NP-complete unless $f$ is a projection. More generally, we study the problem $B$-CMP where $F$ and $f$ are given by circuits using gates from $B$. For most choices of $B$, we classify the complexity of $B$-CMP as being $\Theta^P_2$-complete (possibly under randomized reductions), coDP-complete, or in P.

Explore related subjects

cs.CC cs.LO

Keep this discovery

BibTeX RIS

Emil Jeřábek. 2019-09-26. On the complexity of the clone membership problem. https://doi.org/10.1007/s00224-020-10016-7

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↗