[Paper Review] Sampling Arbitrary Subgraphs Exactly Uniformly in Sublinear Time

Hendrik Fichtenberger, Mingze Gao+1 more|arXiv (Cornell University)|May 4, 2020

Markov Chains and Monte Carlo Methods21 references4 citations

TL;DR

This paper presents a sublinear-time algorithm that samples any arbitrary subgraph $H$ exactly uniformly from a large graph $G$ using degree, neighbor, pair, and edge-sampling queries. The algorithm achieves query and time complexity of $\tilde{O}(\min\{m, m^{\rho(H)}/\#H\})$ and $\tilde{O}(m^{\rho(H)}/\#H)$, respectively, where $\rho(H)$ is the fractional edge cover of $H$ and $\#H$ is the number of copies of $H$ in $G$, making it nearly optimal for cliques.

ABSTRACT

We present a simple sublinear-time algorithm for sampling an arbitrary subgraph $H$ \emph{exactly uniformly} from a graph $G$ with $m$ edges, to which the algorithm has access by performing the following types of queries: (1) degree queries, (2) neighbor queries, (3) pair queries and (4) edge sampling queries. The query complexity and running time of our algorithm are $ ilde{O}(\min\{m, \frac{m^{ρ(H)}}{\# H}\})$ and $ ilde{O}(\frac{m^{ρ(H)}}{\# H})$, respectively, where $ρ(H)$ is the fractional edge-cover of $H$ and $\# H$ is the number of copies of $H$ in $G$. For any clique on $r$ vertices, i.e., $H=K_r$, our algorithm is almost optimal as any algorithm that samples an $H$ from any distribution that has $Ω(1)$ total probability mass on the set of all copies of $H$ must perform $Ω(\min\{m, \frac{m^{ρ(H)}}{\# H\cdot (cr)^r}\})$ queries. Together with the query and time complexities of the $(1\pm \varepsilon)$-approximation algorithm for the number of subgraphs $H$ by Assadi, Kapralov and Khanna [ITCS 2018] and the lower bound by Eden and Rosenbaum [APPROX 2018] for approximately counting cliques, our results suggest that in our query model, approximately counting cliques is "equivalent to" exactly uniformly sampling cliques, in the sense that the query and time complexities of exactly uniform sampling and randomized approximate counting are within a polylogarithmic factor of each other. This stands in interesting contrast to an analogous relation between approximate counting and almost uniformly sampling for self-reducible problems in the polynomial-time regime by Jerrum, Valiant and Vazirani [TCS 1986].

Motivation & Objective

• To develop a sublinear-time algorithm that samples any fixed subgraph $H$ exactly uniformly from a large graph $G$.
• To address the challenge of uniform sampling for arbitrary subgraphs beyond simple structures like edges or triangles.
• To establish tight query and time complexity bounds for exact uniform sampling in the general graph query model.
• To explore the relationship between exact uniform sampling and approximate counting of subgraphs, particularly cliques.

Proposed method

• The algorithm uses a recursive decomposition based on fractional edge covers of the target subgraph $H$, leveraging the concept of fractional edge cover $\rho(H)$ to guide sampling.
• It employs a three-stage process: sampling a 'wedge' or 'star' structure, then extending it to form a full copy of $H$, using isomorphism checks on constant-sized subgraphs.
• The algorithm performs edge-sampling queries to sample potential edges and uses degree and neighbor queries to explore local neighborhoods efficiently.
• A rejection sampling framework ensures that each copy of $H$ is selected with exactly uniform probability $1/\#H$.
• The method uses a randomized procedure that bounds query complexity via the fractional edge cover density $\rho(H)$, ensuring sublinear performance.
• The algorithm assumes knowledge of the total number of edges $m$, but this can be relaxed with a $(1+\varepsilon)$-approximation of $m$ at a small cost in accuracy.

Experimental results

Research questions

• RQ1 Can we design a sublinear-time algorithm that samples any arbitrary subgraph $H$ exactly uniformly from a large graph $G$ using standard query types?
• RQ2 What is the optimal query and time complexity for exact uniform sampling of a subgraph $H$ in the general graph query model?
• RQ3 How does the complexity of exact uniform sampling relate to the problem of approximate counting of subgraphs, especially cliques?
• RQ4 Is there a fundamental equivalence between exact uniform sampling and approximate counting in sublinear time?

Key findings

• The algorithm achieves $\tilde{O}(\min\{m, m^{\rho(H)}/\#H\})$ query complexity and $\tilde{O}(m^{\rho(H)}/\#H)$ time complexity for exact uniform sampling of any subgraph $H$.
• For $r$-cliques ($H = K_r$), the algorithm's complexity is nearly optimal, matching the lower bound for any algorithm sampling from a distribution with $\Omega(1)$ mass on $K_r$ copies.
• The results suggest that in the given query model, exact uniform sampling of cliques and randomized approximate counting of cliques have query and time complexities within polylogarithmic factors of each other.
• This implies a strong equivalence between exact uniform sampling and approximate counting for cliques in sublinear time, contrasting with the classical JVV framework in polynomial time.
• The algorithm's performance is robust to uncertainty in $m$, as a $(1+\varepsilon)$-approximation of $m$ suffices to maintain $\varepsilon$-approximate uniformity.
• The analysis establishes a tight connection between the fractional edge cover number $\rho(H)$ and the sampling complexity, showing that $\rho(H)$ is the key structural parameter governing efficiency.