SDP-based bounds for graph partition via extended ADMM

TL;DR

This paper introduces SDP relaxations solved by extended ADMM to efficiently compute tight bounds for large graph partition problems, outperforming traditional interior point methods in speed and memory usage.

Contribution

The paper presents a novel SDP relaxation with nonnegativity constraints and an extended ADMM approach for large-scale graph partition problems, providing high-quality bounds efficiently.

Findings

High-quality lower bounds for k-equipartition on large graphs within minutes.

Effective SDP-based heuristics produce tighter upper bounds than existing methods.

Overcomes memory limitations of interior point methods for large instances.

Abstract

We study two NP-complete graph partition problems, $k$-equipartition problems and graph partition problems with knapsack constraints (GPKC). We introduce tight SDP relaxations with nonnegativity constraints to get lower bounds, the SDP relaxations are solved by an extended alternating direction method of multipliers (ADMM). In this way, we obtain high quality lower bounds for $k$-equipartition on large instances up to $n =1000$ vertices within as few as five minutes and for GPKC problems up to $n=500$ vertices within as little as one hour. On the other hand, interior point methods fail to solve instances from $n=300$ due to memory requirements. We also design heuristics to generate upper bounds from the SDP solutions, giving us tighter upper bounds than other methods proposed in the literature with low computational expense.