Persistency of Linear Programming Relaxations for the Stable Set Problem

Abstract

The Nemhauser-Trotter theorem states that the standard linear programming (LP) formulation for the stable set problem has a remarkable property, also known as (weak) persistency: for every optimal LP solution that assigns integer values to some variables, there exists an optimal integer solution in which these variables retain the same values. While the standard LP is defined by only non-negativity and edge constraints, a variety of stronger LP formulations have been studied and one may wonder whether any of them has the this property as well. We show that any stronger LP formulation that satisfies mild conditions cannot have the persistency property on all graphs, unless it is always equal to the stable-set polytope.

A Deferred proofs

Lemma 1. Let $P,Q \subseteq \mathbb {R}^n$ be polytopes. If there exists a vector $c \in \mathbb {R}^n$ such that , then there exists a vector $c' \in \mathbb {R}^n$ such that is a vertex of Q, while is not a vertex of P.

Proof

Let $c' \in \mathbb {R}^n$ be such that holds, and among those, such that is minimum. Clearly, $c'$ is well-defined since satisfies the conditions.

Assume, for the sake of contradiction, that . Let and . Let $F_1, F_2, \dotsc , F_k$ be the facets of F. By $n(F,F_i)$ we denote the set of vectors $w \in \mathbb {R}^n$ such that . Since F is a polytope, $\bigcup _{i \in \{1,2,\dotsc ,k\}} n(F,F_i)$ contains a basis U of $\mathbb {R}^n$. Moreover, not all vectors $u \in U$ can lie in , the orthogonal complement of , since then would hold, contradicting . Let .

Now, for a sufficiently small $\varepsilon > 0$, for some $i \in \{1,2,\dotsc ,k\}$, and is a proper face of G. Thus, $c' + \varepsilon u$ satisfies the requirements at the beginning of the proof. However, contradicts the minimality assumption, which concludes the proof.    $\square$

Lemma 2. Let $P \subseteq \mathbb {R}^n$ be a non-empty polytope, let $c,a \in \mathbb {R}^n$ and let . The functions $h^=, h^{\le } : [\ell ,\infty ) \rightarrow \mathbb {R}$ defined via and are concave. Moreover, there exists a number $\beta ^\star \in [\ell ,\infty )$ such that $h^=$ and $h^\le$ are identical and strictly monotonically increasing on the interval $[\ell , \beta ^\star ]$, and $h^\le$ is constant on the interval $[\beta ^\star , \infty )$.

Proof

Let be the projection of P along a and c. By construction, holds. Considering that Q is a polytope of dimension at most 2, the claimed properties of $h^{\le }$ and $h^=$ are obvious (see Fig. 1).    $\square$

Fig. 1.

Illustration of Lemma 2. The graph of $h^{\le }$ is highlighted in red, while that of $h^=$ is highlighted in blue. (Color figure online)