Counting independent terms in big-oh notation

Fabiano de S. Oliveira; Valmir C. Barbosa

doi:10.1007/s11590-016-1092-7

Home/Archive/arXiv

arXiv · 1512.09363

Counting independent terms in big-oh notation

Fabiano de S. Oliveira·Valmir C. Barbosa

View original source ↗Read source document ↗Export metadata ↓

Abstract

The field of computational complexity is concerned both with the intrinsic hardness of computational problems and with the efficiency of algorithms to solve them. Given such a problem, normally one designs an algorithm to solve it and sets about establishing bounds on its performance as functions of the algorithm's variables, particularly upper bounds expressed via the big-oh notation. But if we were given some inscrutable code and were asked to figure out its big-oh profile from performance data on a given set of inputs, how hard would we have to grapple with the various possibilities before zooming in on a reasonably small set of candidates? Here we show that, even if we restricted our search to upper bounds given by polynomials, the number of possibilities could be arbitrarily large for two or more variables. This is unexpected, given the available body of examples on algorithmic efficiency, and serves to illustrate the many facets of the big-oh notation, as well as its counter-intuitive twists.

Explore related subjects

cs.CC

Keep this discovery

BibTeX RIS

Fabiano de S. Oliveira, Valmir C. Barbosa. 2015-12-31. Counting independent terms in big-oh notation. https://doi.org/10.1007/s11590-016-1092-7

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↗