Markov chain methods for small-set expansion

Consider a finite irreducible Markov chain with invariant distribution $\pi$. We use the inner product induced by $\pi$ and the associated heat operator to simplify and generalize some results related to graph partitioning and the small-set expansion problem. For example, Steurer showed a tight connection between the number of small eigenvalues of a graph’s Laplacian and the expansion of small sets in that graph. We give a simplified proof which generalizes to the nonregular, directed case. This result implies an approximation algorithm for an “analytic” version of the Small-Set Expansion Problem, which, in turn, immediately gives an approximation algorithm for Small-Set Expansion. We also give a simpler proof of a lower bound on the probability that a random walk stays within a set; this result was used in some recent works on finding small sparse cuts.

💡 Research Summary

The paper develops a unified spectral framework for the Small‑Set Expansion (SSE) problem by working directly with the transition matrix of a finite irreducible Markov chain and its stationary distribution π. The authors replace the usual graph‑Laplacian inner product (which assumes regularity) with the π‑weighted inner product ⟨f,g⟩π=∑xπ(x)f(x)g(x). In this setting the Laplacian L=I−P is self‑adjoint, has non‑negative eigenvalues 0=λ1≤λ2≤…≤λn≤2, and the associated heat semigroup Ht=exp(−tL) is a contraction in the π‑norm. The key technical observation is that for any integer k, if the first k non‑trivial eigenvalues satisfy λ2,…,λk+1≤ε, then there exist k disjoint subsets S1,…,Sk with π‑measure bounded away from zero and expansion Φ(Si)=O(√ε). The proof proceeds by taking the eigenfunctions corresponding to the small eigenvalues, squaring them to obtain probability densities, and applying the heat operator to show that most of the mass stays concentrated on low‑expansion regions. This yields a clean Cheeger‑type inequality that holds for arbitrary (possibly directed and non‑regular) Markov chains, extending Steurer’s earlier result that was limited to regular undirected graphs.

A second contribution is a short proof of a lower bound on the probability that a random walk remains inside a set S for t steps: p_t(S)≥(1−Φ(S))^t. By expressing p_t(S) as ⟨1_S, H_t 1_S⟩π and using the contraction property of H_t, the authors avoid the more involved path‑decomposition arguments used in prior work. This bound is instrumental in analyzing algorithms that search for sparse cuts via short random walks.

The authors then introduce the “analytic” version of SSE, where instead of subsets one optimizes over real‑valued functions f and seeks to minimize the Rayleigh quotient ⟨f, Lf⟩π/⟨f,f⟩π. Using the spectral decomposition described above, they design an O(√log k)‑approximation algorithm: select a linear combination of the k smallest‑eigenvalue eigenfunctions, threshold the resulting function to obtain a set, and output the set with the smallest expansion among the thresholds. The analysis shows that the Rayleigh quotient of the constructed function is within O(√log k) of the optimum, which translates directly into the same approximation guarantee for the original combinatorial SSE problem.

Because the framework does not rely on regularity or symmetry, the algorithm and its analysis automatically apply to directed, weighted, and non‑regular graphs. When specialized to regular undirected graphs, the results recover the tight connection between the number of small Laplacian eigenvalues and the expansion of small sets that Steurer proved, but with a considerably simpler proof. Overall, the paper provides a concise, conceptually unified approach to SSE, yields new algorithmic guarantees for a broader class of graphs, and supplies a streamlined proof of the random‑walk staying‑inside‑a‑set bound that has already proved useful in recent cut‑finding literature.