The SDP value of random 2CSPs

TL;DR

This paper analyzes the SDP relaxation values of a broad class of sparse random 2-variable Boolean CSPs, establishing high-probability bounds on their optimal values and identifying the typical SDP value for each model.

Contribution

It characterizes the high-probability SDP relaxation value for a wide class of sparse random 2CSP models, including non-regular and spectral-relaxation-distinct cases.

Findings

Identifies the high-probability SDP value for each model

Shows the SDP optimum concentrates around a specific value

Includes models where SDP value is less than spectral relaxation

Abstract

We consider a very wide class of models for sparse random Boolean 2CSPs; equivalently, degree-2 optimization problems over~$\{\pm 1\}^n$. For each model $\mathcal{M}$, we identify the "high-probability value"~$s^*_{\mathcal{M}}$ of the natural SDP relaxation (equivalently, the quantum value). That is, for all $\varepsilon > 0$ we show that the SDP optimum of a random $n$-variable instance is (when normalized by~$n$) in the range $(s^*_{\mathcal{M}}-\varepsilon, s^*_{\mathcal{M}}+\varepsilon)$ with high probability. Our class of models includes non-regular CSPs, and ones where the SDP relaxation value is strictly smaller than the spectral relaxation value.