Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 1506.01083

Dependent Random Graphs and Multiparty Pointer Jumping

Joshua Brody·Mario Sanchez
View original source ↗Read source document ↗Export metadata ↓

Abstract

We initiate a study of a relaxed version of the standard Erdos-Renyi random graph model, where each edge may depend on a few other edges. We call such graphs "dependent random graphs". Our main result in this direction is a thorough understanding of the clique number of dependent random graphs. We also obtain bounds for the chromatic number. Surprisingly, many of the standard properties of random graphs also hold in this relaxed setting. We show that with high probability, a dependent random graph will contain a clique of size $\frac{(1-o(1))\log n}{\log(1/p)}$, and the chromatic number will be at most $\frac{n \log(1/1-p)}{\log n}$. As an application and second main result, we give a new communication protocol for the k-player Multiparty Pointer Jumping (MPJ_k) problem in the number-on-the-forehead (NOF) model. Multiparty Pointer Jumping is one of the canonical NOF communication problems, yet even for three players, its communication complexity is not well understood. Our protocol for MPJ_3 costs $O(\frac{n\log\log n}{\log n})$ communication, improving on a bound of Brody and Chakrabarti [BC08]. We extend our protocol to the non-Boolean pointer jumping problem $\widehat{MPJ}_k$, achieving an upper bound which is o(n) for any $k >= 4$ players. This is the first o(n) bound for $\widehat{MPJ}_k$ and improves on a bound of Damm, Jukna, and Sgall [DJS98] which has stood for almost twenty years.

Explore related subjects

cs.CC

Keep this discovery

BibTeXRIS

Joshua Brody, Mario Sanchez. 2015-06-02. Dependent Random Graphs and Multiparty Pointer Jumping. https://arxiv.org/abs/1506.01083

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↗