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arXiv · 2109.02238

Optimal solutions and ranks in the max-cut SDP

Daniel Hong·Hyunwoo Lee·Alex Wei
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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.

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Daniel Hong, Hyunwoo Lee, Alex Wei. 2021-09-06. Optimal solutions and ranks in the max-cut SDP. https://arxiv.org/abs/2109.02238

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