Application of the Lovász-Schrijver Lift-and-Project Operator to Compact Stable Set Integer Programs (2407.19290v2)

Abstract: The Lov\'asz theta function $\theta(G)$ provides a very good upper bound on the stability number of a graph $G$. It can be computed in polynomial time by solving a semidefinite program (SDP), which also turns out to be fairly tractable in practice. Consequently, $\theta(G)$ achieves a hard-to-beat trade-off between computational effort and strength of the bound. Indeed, several attempts to improve the theta bound are documented, mainly based on playing around the application of the $N_+(\cdot)$ lifting operator of Lov\'asz and Schrijver to the classical formulation of the maximum stable set problem. Experience shows that solving such SDP-s often struggles against practical intractability and requires highly specialized methods. We investigate the application of such an operator to two different linear formulations based on clique and nodal inequalities, respectively. Fewer inequalities describe these two and yet guarantee that the resulting SDP bound is at least as strong as $\theta(G)$. Our computational experience, including larger graphs than those previously documented, shows that upper bounds stronger than $\theta(G)$ can be accessed by a reasonable additional effort using the clique-based formulation on sparse graphs and the nodal-based one on dense graphs.

• The paper introduces new SDP relaxations via the Lovász-Schrijver operator that significantly tighten upper bounds for the NP-hard Stable Set Problem.
• Computational experiments reveal that nodal-based relaxations outperform clique-based formulations on dense graphs by effectively improving bounds on α(G).
• The study establishes a theoretical hierarchy among the relaxations, balancing enhanced accuracy with manageable computational complexity.

Application of the Lovász-Schrijver Lift-and-Project Operator to Compact Stable Set Integer Programs

The paper investigates the use of the Lovász-Schrijver lift-and-project operator applied to linear programming relaxations tailored for the Stable Set Problem (SSP), which is an NP-hard combinatorial optimization problem concerned with finding the largest set of pairwise non-adjacent vertices in a graph. One primary focus is examining how this operator can improve upon the well-known Lovász theta function, denoted as $\theta(G)$, which provides a strong upper bound to the graph's stability number $\alpha(G)$ and can be efficiently computed using semidefinite programming (SDP).

Core Contributions

1. Introduction of New SDP Relaxations: The authors explore SDP relaxations resulting from the application of the Lovász-Schrijver lift-and-project operator to two different types of LP relaxations of the SSP: clique-based and nodal-based formulations. These formulations leverage the \N{} operator to tighten the bounds beyond $\theta(G)$.
2. Computational Experiments: Extensive computations illustrate that these new relaxations can significantly enhance the bounds on $\alpha(G)$, particularly for large and dense graphs, revealing the utility of these improved relaxations in practical settings.
3. Comparison and Hierarchies: The research establishes a theoretical hierarchy among the proposed relaxations, comparing them in terms of computational efficiency and the tightness of the resulting bounds. A central finding is that relaxations using nodal inequalities, when strengthened through the operator and suitably chosen coefficients like $\alpha(G[\Gamma(i)])$, deliver the most substantial improvements, making them particularly potent for dense graphs.
4. Theoretical Recovery of Weakness: Even when starting from weaker LP relaxations, the proposed methodology can recover a greater portion of the gap left by traditional SDP approaches through defining and refining the nodal inequalities efficiently.

Numerical Results and Implications

• Performance on Large Graphs:

Applying these enhanced relaxations to significantly large graphs documents notable progress over the traditional Lovász theta function. For instance, the clique-based relaxation, though theoretically cumbersome due to exponential inequality growth, is made operationally tractable and highly effective on sparse graphs.

• Efficiency in Dense Graphs:

In contrast, dense graphs see better performance with nodal-based relaxations facilitated through SDP, which harness the power of the \N{} operator to correct initial linear programming deficiencies effectively.

• Computational Trade-offs:

Although there is an increase in computational workload, the paper demonstrates that the workload scaling is smoother than anticipated, thus allowing experimentation on larger datasets than previously recorded.

Future Directions

The application of Lovász-Schrijver's operator presented in this paper sheds light on its dual role as both a mathematical tool and a practical technique. There is potential for further exploration to identify specific graph classes where the new SDP relaxations fully converge to the stable set polytope. Additionally, extending the methodological framework to other combinatorial optimization problems or exploring hybrid approaches combining these SDP relaxations with exact algorithms could unlock even more practical implications for complex and large-scale graph problems.