Graphs with Stable Set Polytopes Requiring High Lift-and-Project Ranks for the Lovász-Schrijver SDP Operator

Belangrijkste concepten

The authors construct a family of graphs {Hk} whose stable set polytopes require a linear number of iterations of the Lovász-Schrijver SDP operator LS+ to compute, which is the best possible up to a constant factor.

Samenvatting

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:

1. 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.

2. 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.

3. 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.

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.

Statistieken

The following sentences contain key metrics or important figures used to support the author's key logics:
The LS+-rank of the fractional stable set polytope of Hk is at least 1/16 times the number of vertices in Hk.
The SDP described by LS+ that fails to exactly represent the stable set polytope of Hk has size nΩ(n), where n is the number of vertices in Hk.

Citaten

None.

Belangrijkste Inzichten Gedestilleerd Uit

Stable Set Polytopes with High Lift-and-Project Ranks for the Lovász-Schrijver SDP Operator

by Yu H... om arxiv.org 04-26-2024

https://arxiv.org/pdf/2303.08971.pdf

Diepere vragen

How can the techniques developed in this paper be extended to analyze the lift-and-project ranks of stable set polytopes of other families of graphs

The techniques developed in this paper can be extended to analyze the lift-and-project ranks of stable set polytopes of other families of graphs by identifying symmetries and A-balancing automorphisms in those graphs. By characterizing the shadow of the LS+-relaxations in a lower-dimensional space, similar to how it was done for the graphs Hk, one can simplify the analysis and potentially derive lower bounds on the lift-and-project ranks of stable set polytopes for those graphs. Additionally, constructing compact convex sets based on linear and quadratic inequalities can help in proving the existence of points in the LS+-relaxations that violate certain constraints, leading to insights into the lift-and-project ranks of stable set polytopes for other graph families.

Can the authors' approach of working with 2-dimensional "shadows" of the LS+-relaxations be generalized to study lift-and-project ranks of other combinatorial optimization problems

The authors' approach of working with 2-dimensional "shadows" of the LS+-relaxations can be generalized to study lift-and-project ranks of other combinatorial optimization problems by adapting the concept of A-balancing automorphisms and identifying symmetries in those problems. By focusing on lower-dimensional representations of the LS+-relaxations, one can simplify the analysis and potentially derive insights into the lift-and-project ranks of stable set polytopes for various optimization problems. This approach can be applied to a wide range of combinatorial optimization problems to study the complexity of approximating their solutions using semidefinite programming techniques.

What are the implications of the authors' results on the complexity of approximating the stable set problem using semidefinite programming techniques

The authors' results have implications on the complexity of approximating the stable set problem using semidefinite programming techniques by providing insights into the lift-and-project ranks of stable set polytopes for graphs with specific symmetries. The lower bounds on the LS+-rank obtained in the paper suggest that certain families of graphs require a significant number of iterations of the LS+ operator to compute the stable set polytope accurately. This highlights the challenges in efficiently approximating stable set polytopes for graphs with complex structures and symmetries. The techniques developed in the paper can aid in understanding the computational complexity of solving the stable set problem using semidefinite programming and provide valuable insights for developing more efficient algorithms for this problem.