Optimal solutions and ranks in the max-cut SDP

TL;DR

This paper investigates the conditions under which the max-cut semidefinite programming relaxation yields rank-1 solutions for specific graph classes, enhancing understanding of SDP solution structure in combinatorial optimization.

Contribution

It characterizes when max-cut SDP solutions are rank-1 for cycle graphs and analyzes solution behavior under graph operations like vertex sum and edge sum.

Findings

Cycle graphs have rank-1 SDP solutions under certain conditions.

Solutions to max-cut SDP for combined graphs can be explicitly characterized.

The study provides insights into SDP solution structure for graph operations.

Abstract

The max-cut problem is a classical graph theory problem which is NP-complete. The best polynomial time approximation scheme relies on \emph{semidefinite programming} (SDP). We study the conditions under which graphs of certain classes have rank~1 solutions to the max-cut SDP. We apply these findings to look at how solutions to the max-cut SDP behave under simple combinatorial constructions. Our results determine when solutions to the max-cut SDP for cycle graphs are rank~1. We find the solutions to the max-cut SDP of the vertex~sum of two graphs. We then characterize the SDP solutions upon joining two triangle graphs by an edge~sum.