On Lifting Integrality Gaps to SSEH Hardness for Globally Constrained CSPs

TL;DR

This paper establishes NP-hardness of approximating constrained Boolean Max-CSPs within a factor related to the integrality gap of Lasserre relaxations, assuming the Small-Set Expansion Hypothesis, advancing the understanding of CSP hardness.

Contribution

It connects the integrality gap of SDP relaxations to approximation hardness for constrained Max-CSPs under SSEH, introducing a novel composition technique for dictatorship tests.

Findings

NP-hardness of approximation tied to integrality gaps

New reduction framework combining Raghavendra's approach and SSE

A novel bias-dependent dictatorship test composition

Abstract

A $\mu$-constrained Boolean Max-CSP$(\psi)$ instance is a Boolean Max-CSP instance on predicate $\psi:\{0,1\}^r \to \{0,1\}$ where the objective is to find a labeling of relative weight exactly $\mu$ that maximizes the fraction of satisfied constraints. In this work, we study the approximability of constrained Boolean Max-CSPs via SDP hierarchies by relating the integrality gap of Max-CSP $(\psi)$ to its $\mu$-dependent approximation curve. Formally, assuming the Small-Set Expansion Hypothesis, we show that it is NP-hard to approximate $\mu$-constrained instances of Max-CSP($\psi$) up to factor ${\sf Gap}_{\ell,\mu}(\psi)/\log(1/\mu)^2$ (ignoring factors depending on $r$) for any $\ell \geq \ell(\mu,r)$. Here, ${\sf Gap}_{\ell,\mu}(\psi)$ is the optimal integrality gap of $\ell$-round Lasserre relaxation for $\mu$-constrained Max-CSP($\psi$) instances. Our results are derived by combining the framework of Raghavendra [STOC 2008] along with more recent advances in rounding Lasserre relaxations and reductions from the Small-Set Expansion (SSE) problem. A crucial component of our reduction is a novel way of composing generic bias-dependent dictatorship tests with SSE, which could be of independent interest.