An Approach to Circuit Lower Bounds via Bounded Width Circuits

Hiroki Morizumi

Home/Archive/arXiv

arXiv · 2305.00794

An Approach to Circuit Lower Bounds via Bounded Width Circuits

Hiroki Morizumi

View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper, we investigate an approach to circuit lower bounds via bounded width circuits. The approach consists of two steps: (i) We convert circuits to (deterministic or nondeterministic) bounded width circuits. (ii) We prove lower bounds for the bounded width circuits. For the second step, we prove that there is an explicit Boolean function $f$ as follows: If a nondeterministic circuit of size $s$ and width $w$ computes $f$, then $w = \Omega(\frac{n^4}{4^{\frac{s}{n}}s^3}) - \frac{\log_2 s}{2}$. For the first step, we give some observations, which include a relation between conversions to bounded width circuits and a standard pebble game.

Explore related subjects

cs.CC

Keep this discovery

BibTeX RIS

Hiroki Morizumi. 2023-05-01. An Approach to Circuit Lower Bounds via Bounded Width Circuits. https://arxiv.org/abs/2305.00794

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↗