[Paper Review] Hypergraph Two-Coloring in the Streaming Model

Jaikumar Radhakrishnan, Saswata Shannigrahi+1 more|arXiv (Cornell University)|Dec 14, 2015

Limits and Structures in Graph Theory3 citations

TL;DR

This paper studies space-efficient streaming algorithms for two-coloring $n$-uniform hypergraphs, where hyperedges arrive sequentially. It proves that deterministic streaming algorithms require $\Omega(q/n)$ or $\Omega(\sqrt{1/(n\ln n)}\cdot 2^n)$ space depending on hyperedge count, but a randomized algorithm achieves $O(v\log v)$ space for up to $\frac{1}{10}\sqrt{\frac{n}{\ln n}}2^n$ hyperedges, matching known existential bounds.

ABSTRACT

We consider space-efficient algorithms for two-coloring $n$-uniform hypergraphs $H=(V,E)$ in the streaming model, when the hyperedges arrive one at a time. It is known that any such hypergraph with at most $0.7 \sqrt{\frac{n}{\ln n}} 2^n$ hyperedges has a two-coloring [Radhakrishnan & Srinivasan, RSA, 2000], which can be found deterministically in polynomial time, if allowed full access to the input. 1. Let $s^D(v, q, n)$ be the minimum space used by a deterministic one-pass streaming algorithm that on receiving an $n$-uniform hypergraph $H$ on $v$ vertices and $q$ hyperedges produces a proper two-coloring of $H$. We show that $s^D(n^2, q, n) = Ω(q/n)$ when $q \leq 0.7 \sqrt{\frac{n}{\ln n}} 2^n$, and $s^D(n^2, q, n) = Ω(\sqrt{\frac{1}{n\ln n}} 2^n)$ otherwise. 2. Let $s^R(v, q,n)$ be the minimum space used by a randomized one-pass streaming algorithm that on receiving an $n$-uniform hypergraph $H$ on $v$ vertices and $q$ hyperedges with high probability produces a proper two-coloring of $H$ (or declares failure). We show that $s^R(v, \frac{1}{10}\sqrt{\frac{n}{\ln n}} 2^n, n) = O(v \log v)$ by giving an efficient randomized streaming algorithm. The above results are inspired by the study of the number $q(n)$, the minimum possible number of hyperedges in a $n$-uniform hypergraph that is not two-colorable. It is known that $q(n) = Ω(\sqrt{\frac{n}{\ln n}})$ [Radhakrishnan-Srinivasan] and $ q(n)= O(n^2 2^n)$ [Erdos, 1963]. Our first result shows that no space-efficient deterministic streaming algorithm can match the performance of the offline algorithm of Radhakrishnan and Srinivasan; the second result shows that there is, however, a space-efficient randomized streaming algorithm for the task.

Motivation & Objective

• To understand the space complexity of one-pass streaming algorithms for two-coloring $n$-uniform hypergraphs.
• To determine whether deterministic or randomized streaming algorithms can efficiently solve the problem under space constraints.
• To establish tight lower bounds on space usage for deterministic streaming algorithms in the one-pass model.
• To design a space-efficient randomized streaming algorithm that matches the existential threshold for two-colorability.
• To explore the gap between deterministic and randomized streaming complexity for hypergraph two-coloring.

Proposed method

• Leverages the Moser-Tardos algorithm for constructive $\text{P}\neq\text{NP}$-type problems, adapted to the streaming setting.
• Uses a randomized re-coloring strategy: upon seeing a monochromatic hyperedge, re-randomize its vertices if it doesn’t conflict with previously marked hyperedges.
• Applies the Lovász Local Lemma (LLL) condition to bound the number of passes: ensures termination in $O(\log |V|)$ expected passes.
• Implements a maximal independent set (MIS) over monochromatic hyperedges to avoid conflicts during re-coloring.
• Employs a pass-based approach: each pass processes all hyperedges, marks and re-colors conflicting monochromatic hyperedges.
• Uses Markov’s inequality to bound failure probability, ensuring success with high probability in $O(\log |V|)$ passes.

Experimental results

Research questions

• RQ1 What is the minimum space required by a deterministic one-pass streaming algorithm to two-color $n$-uniform hypergraphs with $q$ hyperedges?
• RQ2 Can a randomized streaming algorithm achieve sublinear space for hypergraphs with up to $\Omega(\sqrt{n/\ln n} \cdot 2^n)$ hyperedges?
• RQ3 Is there a gap between the space complexity of deterministic and randomized streaming algorithms for hypergraph two-coloring?
• RQ4 Can the Moser-Tardos algorithm be adapted to the streaming model with limited memory and one-pass access?
• RQ5 What is the optimal number of passes for a randomized streaming algorithm to achieve a valid two-coloring with high probability?

Key findings

• For deterministic streaming algorithms, the space complexity is $\Omega(q/n)$ when $q \leq 0.7\sqrt{\frac{n}{\ln n}}2^n$, and $\Omega(\sqrt{1/(n\ln n)}\cdot 2^n)$ otherwise.
• A randomized streaming algorithm achieves $O(v\log v)$ space for hypergraphs with up to $\frac{1}{10}\sqrt{\frac{n}{\ln n}}2^n$ hyperedges.
• The randomized algorithm terminates in $O(\log |V|)$ expected passes, with success probability at least $3/4$ using $O(\log |V|)$ passes.
• The lower bound for deterministic algorithms is tight up to $\mathrm{poly}(n)$ factors, showing a strong separation from randomized counterparts.
• The Moser-Tardos algorithm can be adapted to the streaming model, preserving expected termination time under space constraints.
• The algorithm's performance matches the existential threshold for two-colorability established by Radhakrishnan and Srinivasan (2000), showing optimality in space for randomized streaming.