How Well Do Local Algorithms Solve Semidefinite Programs?

TL;DR

This paper investigates the effectiveness of local algorithms versus semidefinite programming relaxations in solving the minimum graph bisection problem on Erdős-Rényi graphs, revealing conditions where local algorithms succeed or fail.

Contribution

It provides new bounds on SDP relaxation values and demonstrates the performance of simple and sophisticated local algorithms in graph partitioning problems.

Findings

Local algorithms approximate SDP solutions within a factor close to 1 for large degrees.

A dual witness construction bounds the SDP value using the non-backtracking matrix.

Results extend to the planted partition model, showing bounds on SDP-based partial recovery.

Abstract

Several probabilistic models from high-dimensional statistics and machine learning reveal an intriguing --and yet poorly understood-- dichotomy. Either simple local algorithms succeed in estimating the object of interest, or even sophisticated semi-definite programming (SDP) relaxations fail. In order to explore this phenomenon, we study a classical SDP relaxation of the minimum graph bisection problem, when applied to Erd\H{o}s-Renyi random graphs with bounded average degree $d>1$, and obtain several types of results. First, we use a dual witness construction (using the so-called non-backtracking matrix of the graph) to upper bound the SDP value. Second, we prove that a simple local algorithm approximately solves the SDP to within a factor $2d^2/(2d^2+d-1)$ of the upper bound. In particular, the local algorithm is at most $8/9$ suboptimal, and $1+O(1/d)$ suboptimal for large degree. We then analyze a more sophisticated local algorithm, which aggregates information according to the harmonic measure on the limiting Galton-Watson (GW) tree. The resulting lower bound is expressed in terms of the conductance of the GW tree and matches surprisingly well the empirically determined SDP values on large-scale Erd\H{o}s-Renyi graphs. We finally consider the planted partition model. In this case, purely local algorithms are known to fail, but they do succeed if a small amount of side information is available. Our results imply quantitative bounds on the threshold for partial recovery using SDP in this model.