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 other LP formulations have been studied and one may wonder whether any of them has this property as well. We show that any other 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

We repeat the statements of Lemmas 8 and 16 and provide their proofs.

Lemma 8 Let $P,Q \subseteq \mathbb {R}^n$ be polytopes. If there exists a vector $c \in \mathbb {R}^n$ such that $\dim ({\text {opt}}({Q},{c})) < \dim ({\text {opt}}({P},{c}))$, then there exists a vector $c' \in \mathbb {R}^n$ such that ${\text {opt}}({Q},{c'})$ is a vertex of Q, while ${\text {opt}}({P},{c'})$ is not a vertex of P.

Proof

Let $c' \in \mathbb {R}^n$ be such that $\dim ({\text {opt}}({Q},{c'})) < \dim ({\text {opt}}({P},{c'}))$ holds, and among those, such that $\dim ({\text {opt}}({Q},{c'}))$ is minimum. Clearly, $c'$ is well-defined since $c' {:}{=}c$ satisfies the conditions.

Assume, for the sake of contradiction, that $\dim ({\text {opt}}({Q},{c'})) > 0$. Let $F {:}{=}{\text {opt}}({P},{c'})$ and $G {:}{=}{\text {opt}}({Q},{c'})$. 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 ${\text {opt}}({F}{w}) \supseteq F_i$. 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 ${\text {aff}}(G)^{\perp }$, the orthogonal complement of ${\text {aff}}(G)$, since then ${\text {aff}}(G)^{\perp } = \mathbb {R}^n$ would hold, contradicting $\dim (G) > 0$. Let $u \in U {\setminus } {\text {aff}}(G)^{\perp }$.

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

Fig. 6

Illustration of Lemma 16. The graph of $h^{\le }$ is highlighted in red, while that of $h^=$ is highlighted in blue

Lemma 16 Let $P \subseteq \mathbb {R}^n$ be a non-empty polytope, let $c,a \in \mathbb {R}^n$ and let . The functions $h^=, h^{\le } : [\ell ,\infty ) \rightarrow \mathbb {R}$ with 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. 6). $\square$