[Paper Review] Probabilistic Spectral Sparsification In Sublinear Time

Yin Tat Lee|arXiv (Cornell University)|Dec 31, 2013

Complexity and Algorithms in Graphs15 references4 citations

TL;DR

This paper introduces probabilistic $(\varepsilon,\delta)$-spectral sparsification, a sublinear-time method for constructing sparse graphs that preserve cut values within $1\pm\varepsilon$ multiplicative error and $\delta|S|$ additive error for any cut $(S,S^c)$. It achieves $O(n\log n/\varepsilon^2)$ edges in $\tilde{O}(n/\varepsilon^2\delta)$ time, enabling the fastest known sublinear-time algorithms for sparsest cut, balanced separator, and minimum $s$-$t$ cut with $\varepsilon n$ additive error.

ABSTRACT

In this paper, we introduce a variant of spectral sparsification, called probabilistic $(\varepsilon,δ)$-spectral sparsification. Roughly speaking, it preserves the cut value of any cut $(S,S^{c})$ with an $1\pm\varepsilon$ multiplicative error and a $δ\left|S ight|$ additive error. We show how to produce a probabilistic $(\varepsilon,δ)$-spectral sparsifier with $O(n\log n/\varepsilon^{2})$ edges in time $ ilde{O}(n/\varepsilon^{2}δ)$ time for unweighted undirected graph. This gives fastest known sub-linear time algorithms for different cut problems on unweighted undirected graph such as - An $ ilde{O}(n/OPT+n^{3/2+t})$ time $O(\sqrt{\log n/t})$-approximation algorithm for the sparsest cut problem and the balanced separator problem. - A $n^{1+o(1)}/\varepsilon^{4}$ time approximation minimum s-t cut algorithm with an $\varepsilon n$ additive error.

Motivation & Objective

• To address the challenge of solving cut-based optimization problems on dense graphs in sublinear time, where explicit processing is infeasible due to size.
• To develop a new form of spectral sparsification—probabilistic $(\varepsilon,\delta)$-spectral sparsification—that provides both multiplicative and additive error guarantees on cut values.
• To design an algorithm that constructs such sparsifiers in sublinear time, avoiding the $\tilde{O}(m)$ time cost of traditional methods.
• To achieve nearly optimal query complexity and runtime for sublinear-time approximation of fundamental graph problems like sparsest cut and minimum $s$-$t$ cut.

Proposed method

• Proposes a probabilistic spectral sparsifier that satisfies lower and upper bounds on quadratic forms for all real vectors $u$, ensuring cut value preservation with $1\pm\varepsilon$ multiplicative and $\delta\|u\|_2^2$ additive error.
• Uses graph structure and concentration bounds instead of matrix concentration to avoid $\tilde{O}(m)$ time, enabling sublinear-time construction.
• Employs a sampling strategy based on edge probabilities derived from graph structure, avoiding explicit computation of effective resistances.
• Applies a randomized query model where only a sublinear number of edges are probed, and uses binomial concentration to bound error in cut estimates.
• Leverages the general graph query model, where the algorithm can make degree, neighbor, and edge queries to the input graph.
• Uses Yao’s principle and adversarial graph constructions to prove lower bounds on query complexity, showing the algorithm’s near-optimality.

Experimental results

Research questions

• RQ1 Can spectral sparsification be adapted to provide sublinear-time guarantees for dense unweighted graphs?
• RQ2 Is it possible to achieve both $1\pm\varepsilon$ multiplicative and $\delta|S|$ additive error in cut preservation using a sparse random graph in sublinear time?
• RQ3 Can the construction of such a sparsifier be achieved without computing effective resistances or incurring $\tilde{O}(m)$ time?
• RQ4 What is the optimal query complexity for constructing a probabilistic $(\varepsilon,\delta)$-spectral sparsifier in the general graph model?
• RQ5 How does this method compare to existing sublinear-time algorithms in terms of runtime and approximation quality for cut problems?

Key findings

• The algorithm constructs a probabilistic $(\varepsilon,\delta)$-spectral sparsifier with $O(n\log n/\varepsilon^2)$ edges in $\tilde{O}(n/\varepsilon^2\delta)$ time, significantly improving upon previous methods.
• It achieves an $\tilde{O}(n/OPT + n^{3/2+t})$ time $O(\sqrt{\log n/t})$-approximation for the sparsest cut and balanced separator problems.
• It provides an $n^{1+o(1)}/\varepsilon^4$-time approximation algorithm for minimum $s$-$t$ cut with $\varepsilon n$ additive error.
• The query complexity of $\Omega(n/\varepsilon^2 + n/\delta)$ is proven to be tight, showing the algorithm’s near-optimality.
• The method avoids reliance on matrix concentration bounds, which would incur $\delta \sum d(x)u^2(x)$ error, and instead achieves the tighter $\delta\|u\|_2^2$ bound.
• Lower bounds are established for both cut sparsification and sparsest cut problems, confirming the algorithm’s efficiency is nearly optimal in the general graph query model.