[Paper Review] Improved ARV Rounding in Small-set Expanders and Graphs of Bounded Threshold Rank

Shayan Oveis Gharan, Luca Trevisan|arXiv (Cornell University)|Apr 7, 2013

Complexity and Algorithms in Graphs9 references3 citations

TL;DR

This paper presents a structure theorem showing that optimal solutions to the Arora-Rao-Vazirani SDP relaxation on low-threshold-rank graphs and small-set expanders can be nearly covered by a small number of bounded-radius balls. It introduces a rounding scheme achieving an $O(\sqrt{\log k})$ approximation factor for the ARV relaxation and constant-factor approximation via the $k$-th round of the Sherali-Adams hierarchy, where $k$ is the number of balls.

ABSTRACT

We prove a structure theorem for the feasible solutions of the Arora-Rao-Vazirani SDP relaxation on low threshold rank graphs and on small-set expanders. We show that if G is a graph of bounded threshold rank or a small-set expander, then an optimal solution of the Arora-Rao-Vazirani relaxation (or of any stronger version of it) can be almost entirely covered by a small number of balls of bounded radius. Then, we show that, if k is the number of balls, a solution of this form can be rounded with an approximation factor of O(sqrt {log k}) in the case of the Arora-Rao-Vazirani relaxation, and with a constant-factor approximation in the case of the k-th round of the Sherali-Adams hierarchy starting at the Arora-Rao-Vazirani relaxation. The structure theorem and the rounding scheme combine to prove the following result, where G=(V,E) is a graph of expansion ϕ(G), λ_k is the k-th smallest eigenvalue of the normalized Laplacian of G, and ϕ_k(G) = \min_{disjoint S_1,...,S_k} \max_{1 <= i <= k} ϕ(S_i) is the largest expansion of any k disjoint subsets of V: if either λ_k >> log^{2.5} k \cdot phi(G) or ϕ_{k} (G) >> log k \cdot sqrt{log n}\cdot loglog n\cdot ϕ(G), then the Arora-Rao-Vazirani relaxation can be rounded in polynomial time with an approximation ratio O(sqrt{log k}). Stronger approximation guarantees are achievable in time exponential in k via relaxations in the Lasserre hierarchy. Guruswami and Sinop [GS13] and Arora, Ge and Sinop [AGS13] prove that 1+eps approximation is achievable in time 2^{O(k)} poly(n) if either λ_k > ϕ(G)/ poly(eps), or if SSE_{n/k} > sqrt{log k log n} \cdot ϕ(G)/ poly(eps), where SSE_s is the minimal expansion of sets of size at most s.

Motivation & Objective

• To develop a polynomial-time approximation algorithm for the uniform sparsest cut problem in graphs with bounded threshold rank or small-set expansion.
• To establish a structural characterization of optimal ARV relaxation solutions in terms of bounded-radius ball covers.
• To improve the approximation factor of the ARV rounding scheme in restricted graph classes beyond the standard $O(\sqrt{\log n})$ bound.
• To show that stronger approximation guarantees (constant factor) are achievable via the Sherali-Adams hierarchy in time exponential in $k$, the number of covering balls.
• To identify sufficient spectral or expansion conditions under which the improved approximation is provably effective.

Proposed method

• Prove a structure theorem: any optimal solution to the ARV relaxation on a low-threshold-rank graph or small-set expander can be almost entirely covered by $k$ balls of bounded radius.
• Use metric embedding and probabilistic rounding techniques to construct a cut from the ball cover structure.
• Apply a randomized rounding scheme based on a Frechet embedding $f(v) = d(a_1, v)$ to derive a cut with large $\ell_1$-discrepancy on the set $A$.
• Leverage the triangle inequality and concentration bounds to relate distances in the original metric $d^2_{\mathbf{x}}$ to distances in the rounded metric $d(\cdot, \cdot)$.
• Use the $k$-th round of the Sherali-Adams hierarchy to strengthen the relaxation, enabling constant-factor approximation in time $2^{O(k)}$.
• Apply a probabilistic argument to show that with constant probability, the rounded metric preserves a large $\ell_1$ discrepancy on the set $A$, ensuring a good cut.

Experimental results

Research questions

• RQ1 Can the ARV rounding scheme be improved in graphs with bounded threshold rank or small-set expansion?
• RQ2 Under what spectral or expansion conditions can the ARV relaxation be rounded with $O(\sqrt{\log k})$ approximation?
• RQ3 Can the Sherali-Adams hierarchy applied to the ARV relaxation yield constant-factor approximation in time exponential in $k$?
• RQ4 Is there a structural property of ARV solutions in such graphs that enables better rounding?
• RQ5 What is the relationship between $\lambda_k$, $\phi_k(G)$, and the approximation ratio achievable via the ARV relaxation?

Key findings

• An optimal solution to the ARV relaxation on a graph of bounded threshold rank or small-set expansion can be almost entirely covered by $k$ balls of bounded radius.
• For such graphs, the ARV relaxation can be rounded with an approximation factor of $O(\sqrt{\log k})$ in polynomial time.
• If $\lambda_k \gtrsim \log^{2.5}k \cdot \phi(G)$ or $\phi_k(G) \gtrsim \log k \cdot \sqrt{\log n} \cdot \log\log n \cdot \phi(G)$, then the ARV relaxation admits a polynomial-time $O(\sqrt{\log k})$-approximation.
• Using the $k$-th round of the Sherali-Adams hierarchy starting from the ARV relaxation, a constant-factor approximation is achievable in time $2^{O(k)}$.
• The rounding scheme relies on constructing a Frechet embedding based on a center set $C$ of size $2k$, with distance preservation guarantees via triangle inequality and metric concentration.
• A key technical step shows that $\sum_{u,v \in A} |f(u) - f(v)| \gtrsim n^2 / \sqrt{\log k}$, which implies a good cut with high probability.