Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2004.12002

Finding Planted Cliques in Sublinear Time

Jay Mardia·Hilal Asi·Kabir Aladin Chandrasekher
View original source ↗Read source document ↗Export metadata ↓

Abstract

We study the planted clique problem in which a clique of size k is planted in an Erdos-Renyi graph G(n,1/2) and one is interested in recovering this planted clique. It is widely believed that it exhibits a statistical-computational gap when computational efficiency is equated with the existence of polynomial time algorithms. We study this problem under a more fine-grained computational lens and consider the following two questions. 1. Do there exist sublinear time algorithms for recovering the planted clique? 2. What is the smallest running time any algorithm can hope to have? We show that because of a well known clique-completion property, very elementary sublinear time recovery algorithms do indeed exist for clique sizes k = {\omega}(\sqrt{n}). This points to a qualitatively stronger statistical-computational gap. The planted clique recovery problem can be solved without even looking at most of the input above the {\Theta}(\sqrt{n}) threshold and cannot be solved by any efficient algorithm below it. A running time lower bound for the recovery problem follows easily from the results of [RS19], and this implies our recovery algorithms are optimal whenever k = {\Omega}(n^{2/3}). However, for k = o(n^{2/3}) there is a gap between our algorithmic upper bound and the information-theoretic lower bound implied by [RS19]. With some caveats, we show stronger detection lower bounds based on the Planted Clique Conjecture for a natural but restricted class of algorithms. The key idea is to relate very fast sublinear time algorithms for detecting large planted cliques to polynomial time algorithms for detecting small planted cliques.

Explore related subjects

cs.CCcs.DScs.ITmath.IT

Keep this discovery

BibTeXRIS

Jay Mardia, Hilal Asi, Kabir Aladin Chandrasekher. 2020-04-24. Finding Planted Cliques in Sublinear Time. https://arxiv.org/abs/2004.12002

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

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↗