Monotonic and Non-Monotonic Solution Concepts for Generalized Circuits

Steffen Schuldenzucker; Sven Seuken

Home/Archive/arXiv

arXiv · 1907.12854

Monotonic and Non-Monotonic Solution Concepts for Generalized Circuits

Steffen Schuldenzucker·Sven Seuken

View original source ↗Read source document ↗Export metadata ↓

Abstract

Generalized circuits are an important tool in the study of the computational complexity of equilibrium approximation problems. However, in this paper, we reveal that they have a conceptual flaw, namely that the solution concept is not monotonic. By this we mean that if $\varepsilon < \varepsilon'$, then an $\varepsilon$-approximate solution for a certain generalized circuit is not necessarily also an $\varepsilon'$-approximate solution. The reason for this non-monotonicity is the way Boolean operations are modeled. We illustrate that non-monotonicity creates subtle technical issues in prior work that require intricate additional arguments to circumvent. To eliminate this problem, we show that the Boolean gates are a redundant feature: one can simulate stronger, monotonic versions of the Boolean gates using the other gate types. Arguing at the level of these stronger Boolean gates eliminates all of the aforementioned issues in a natural way. We hope that our results will enable new studies of sub-classes of generalized circuits and enabler simpler and more natural reductions from generalized circuits to other equilibrium search problems.

Explore related subjects

cs.CC

Keep this discovery

BibTeX RIS

Steffen Schuldenzucker, Sven Seuken. 2019-07-30. Monotonic and Non-Monotonic Solution Concepts for Generalized Circuits. https://arxiv.org/abs/1907.12854

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↗