Rounding Large Independent Sets on Expanders

TL;DR

A new clustering property of large independent sets in expanding graphs - every large independent set has a larger-than-expected intersection with some member of a small list - and its formalization in the low-degree sum-of-squares proof system is formalized.

Abstract

We develop a new approach for approximating large independent sets when the input graph is a one-sided spectral expander - that is, the uniform random walk matrix of the graph has its second eigenvalue bounded away from 1. Consequently, we obtain a polynomial time algorithm to find linear-sized independent sets in one-sided expanders that are almost $3$-colorable or are promised to contain an independent set of size $(1/2-ε)n$. Our second result above can be refined to require only a weaker vertex expansion property with an efficient certificate. In a surprising contrast to our algorithmic result, we observe that the analogous task of finding a linear-sized independent set in almost $4$-colorable one-sided expanders (even when the second eigenvalue is $o_n(1)$) is NP-hard, assuming the Unique Games Conjecture. All prior algorithms that beat the worst-case guarantees for this problem rely on bottom eigenspace enumeration techniques (following the classical spectral methods of Alon and Kahale) and require two-sided expansion, meaning a bounded number of negative eigenvalues of magnitude $Ω(1)$. Such techniques naturally extend to almost $k$-colorable graphs for any constant $k$, in contrast to analogous guarantees on one-sided expanders, which are Unique Games-hard to achieve for $k \geq 4$. Our rounding builds on the method of simulating multiple samples from a pseudo-distribution introduced by Bafna et. al. for rounding Unique Games instances. The key to our analysis is a new clustering property of large independent sets in expanding graphs - every large independent set has a larger-than-expected intersection with some member of a small list - and its formalization in the low-degree sum-of-squares proof system.

Figures (3)

• Figure 1: The gadget for $2$ independent sets.
• Figure 2: The triangle gadget for $2$ valid $3$-colorings. There are $2$ ways to partition the $9$ vertices into $3$ disjoint triangles. The highlighted triangles show the partition $\{S_{\pi}\}_{\pi\in \mathbb{S}_3^+}$.
• Figure 3: $S_{\pi^+} = \{11,22,33\}$ and $S_{\pi^-} = \{11,23,32\}$, and $\mathsf{wt}(S_{\pi^+})$, $\mathsf{wt}(S_{\pi^-}) \leqslant O(\varepsilon)$, which means that $\mathsf{wt}(\{12,13,21,23\}) \geqslant 1-O(\varepsilon)$. Here $\{12,13,21,23\}$ forms a bipartite structure.

Theorems & Definitions (127)

• Proposition 1.1: See \ref{['prop:hardness-formal']}
• Theorem 1
• Theorem 2
• Definition 1.2: Small-set vertex expansion
• Theorem 3: Informal \ref{['thm:ssve-main']}
• Lemma 2.1
• Claim 2.2
• proof
• Claim 2.3
• proof