Unbounded-width CSPs are Untestable in a Sublinear Number of Queries

Yumou Fei

Home/Archive/arXiv

arXiv · 2510.27012

Unbounded-width CSPs are Untestable in a Sublinear Number of Queries

Yumou Fei

View original source ↗Read source document ↗Export metadata ↓

Abstract

The bounded-degree query model, introduced by Goldreich and Ron (\textit{Algorithmica, 2002}), is a standard framework in graph property testing and sublinear-time algorithms. Many properties studied in this model, such as bipartiteness and 3-colorability of graphs, can be expressed as satisfiability of constraint satisfaction problems (CSPs). We prove that for the entire class of \emph{unbounded-width} CSPs, testing satisfiability requires $\Omega(n)$ queries in the bounded-degree model. This result unifies and generalizes several previous lower bounds. In particular, it applies to all CSPs that are known to be $\mathbf{NP}$-hard to solve, including $k$-colorability of $\ell$-uniform hypergraphs for any $k,\ell \ge 2$ with $(k,\ell) \neq (2,2)$. Our proof combines the techniques from Bogdanov, Obata, and Trevisan (\textit{FOCS, 2002}), who established the first $\Omega(n)$ query lower bound for CSP testing in the bounded-degree model, with known results from universal algebra.

Explore related subjects

cs.CC cs.DS

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Yumou Fei. 2025-10-30. Unbounded-width CSPs are Untestable in a Sublinear Number of Queries. https://arxiv.org/abs/2510.27012

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↗