The authors study the lift-and-project rank of the stable set polytopes of graphs with respect to the Lovász-Schrijver SDP operator LS+. They focus on finding relatively small graphs with high LS+-rank, as this indicates that the stable set polytope is difficult to obtain using the LS+ operator.
The key contributions are:
The authors define a family of graphs {Hk} and show that the LS+-rank of the stable set polytope of Hk is at least 1/16 times the number of vertices in Hk. This is the first known family of graphs whose LS+-rank grows linearly with the number of vertices, which is the best possible up to a constant factor.
The authors exploit the rich symmetries of the graphs Hk to simplify the analysis. They introduce the notion of A-balancing automorphisms and use this to focus on points in the LS+-relaxations that have at most two distinct entries. This allows them to work with a 2-dimensional "shadow" of the LS+-relaxations, which greatly simplifies the technical analysis.
The authors characterize the stable set polytope and the first LS+-relaxation of the graphs Hk, and then construct a compact convex set that is a strict superset of the first LS+-relaxation but a subset of the second LS+-relaxation. This enables them to establish a lower bound of 2 on the LS+-rank of Hk for k ≥ 4.
The authors develop a recursive approach to construct certificate matrices for points in the higher-level LS+-relaxations of Hk. This allows them to find points in the LS+-relaxations that violate certain valid inequalities for the stable set polytope, thereby establishing the main result.
To Another Language
from source content
arxiv.org
Deeper Inquiries