Facets, weak facets, and extreme functions of the Gomory–Johnson infinite group problem

Abstract

We investigate three competing notions that generalize the notion of a facet of finite-dimensional polyhedra to the infinite-dimensional Gomory–Johnson model. These notions were known to coincide for continuous piecewise linear functions with rational breakpoints. We show that two of the notions, extreme functions and facets, coincide for the case of continuous piecewise linear functions, removing the hypothesis regarding rational breakpoints. We prove an if-and-only-if version of the Gomory–Johnson Facet Theorem. Finally, we separate the three notions using discontinuous examples.

Appendices

Appendix A: Auxiliary result

In the proof of Theorem 3.1, we need the following elementary geometric estimate.

Lemma A.1

Let $F \subset [0,1]^2$ be a convex polygon with vertex set ${{\,\mathrm{vert}\,}}(F)$, and let $g:F \rightarrow {\mathbb {R}}$ be an affine linear function. Suppose that for each $v \in {{\,\mathrm{vert}\,}}(F)$, either $g(v) = 0$ or $g(v) \ge m$ for some $m > 0$. Let $S = \{\,x \in F\mid g(x) = 0\,\}$, and assume that S is nonempty. Then $g(x) \ge m\,d(x, S)/2$ for any $x \in F$, where d(x, S) denotes the Euclidean distance from x to S.

Proof

Let $x \in F$ be arbitrary. We may write

$$\begin{aligned} x = \sum _{v \in {{\,\mathrm{vert}\,}}(F)} \alpha _v v \end{aligned}$$

for some $\alpha _v \in [0,1]$ with $\sum _v \alpha _v = 1$.

Since S is a closed set, for each $v \in {{\,\mathrm{vert}\,}}(F)$, there exists $s_v \in S$ such that $d(v,S)=d(v, s_v)$. Let $s^* = \sum _{v \in {{\,\mathrm{vert}\,}}(F)} \alpha _v s_v$. We have that $s^* \in S$ since the set S is convex. Thus,

$$\begin{aligned} d(x, S)&\le d(x, s^*)&\text { (by definition)} \\&= d\left( \sum _{v} \alpha _v v, \sum _{v} \alpha _v s_v\right)&\\&\le \sum _{v} \alpha _v d(v, s_v) \quad \quad&\text { (by the triangle inequality)} \\&= \sum _{v} \alpha _v d(v, S).&\end{aligned}$$

For those $v \in {{\,\mathrm{vert}\,}}(F)$ with $g(v) = 0$, we have $v \in S$ by definition and thus $d(v,S) = 0$. Therefore,

$$\begin{aligned} d(x, S) \le \sum _{\begin{array}{c} v \in {{\,\mathrm{vert}\,}}(F) \\ g(v) \ge m \end{array}} \alpha _v d(v, S) \le 2 \sum _{\begin{array}{c} v \in {{\,\mathrm{vert}\,}}(F) \\ g(v) \ge m \end{array}} \alpha _v. \end{aligned}$$

Using the affine linearity of g, it thus follows that

$$\begin{aligned} g(x) = \sum _{v \in {{\,\mathrm{vert}\,}}(F)} \alpha _v g(v) = \sum _{\begin{array}{c} v \in {{\,\mathrm{vert}\,}}(F) \\ g(v) = 0 \end{array}} \alpha _v g(v) + \sum _{\begin{array}{c} v \in {{\,\mathrm{vert}\,}}(F) \\ g(v) \ge m \end{array}} \alpha _v g(v) \ge \frac{m\,d(x, S)}{2}. \end{aligned}$$

$\square$

Appendix B: Data of the function $\pi \,{=}\,$ kzh_minimal_has_only_crazy_perturbation_1()

The following pages provide tables with data of the piecewise linear function $\pi =\textsf {kzh\_minimal\_has\_only\_crazy\_perturbation\_1()}$ of Theorem 6.2.

Table 1 defines the function by listing the breakpoints $x_i$ and the values and the left and right limits at the breakpoints. (A version of this table has previously appeared in [19].)

Tables 2 and 3 list the faces $F = F(I, J, K)$ of the complex $\varDelta \mathcal {P}$ that we use for proving piecewise linearity of ${{\bar{\pi }}}$ outside of the special intervals, i.e., Claim (o) in the proof of Theorem 6.2. In all tables, the faces are listed by lexicographically increasing triples (I, J, K); and of the two equivalent faces F(I, J, K) and F(J, I, K), we only show the lexicographically smaller one.

Table 4 shows a list of faces F that satisfy $\varDelta {{\bar{\pi }}}_F(x,y)=0$ for all $(x, y) \in {{\,\mathrm{rel\,int}\,}}(F)$. This property can be verified by inspecting the provided list of vertices of each face. A selection of one vertex (u, v) for each listed face F, listed first in the table, suffices to form a full-rank homogeneous linear system of equations $\varDelta {{\bar{\pi }}}_F(u, v) = 0$. We obtained the selection of faces and their vertices by Gaussian elimination. The full-rank system, shown in Table 5, proves that ${{\bar{\pi }}}$ is 0 outside of the special intervals, Claim (i) in the proof of Theorem 6.2.

Finally, Tables 6 and 7 list the faces F whose projections $p_i({{\,\mathrm{rel\,int}\,}}(F))$, $i=1,2,3$, overlap with the special intervals ($n_F>0$). They are relevant for verifying Claims (ii), (iii), and (vi) in the proof of Theorem 6.2. For each face F, we list the values of the subadditivity slack $\varDelta \pi _F(u, v)$ for all vertices (u, v) of F in nondecreasing order from left to right. If there is an enclosing face $F'\supset F$ with $\varDelta \pi _{F'}(u, v) = \varDelta \pi _F(u,v)$ for all vertices (u, v) of F because of one-sided continuity, then we suppress F in the table.

All numbers have been rounded to 3 decimals for presentation. Claims (ii) and (iii) use faces with $\varDelta \pi _F(x,y) = 0$ for $(x, y)\in F$. To verify Claim (v), note that if $\varDelta \pi _F(u,v)=0$ for one vertex of F, then $\varDelta \pi _F(u,v)=0$ for all vertices of F. Next, note that for all other faces F with $n_F>0$, the inequality $\varDelta \pi _F(u,v) \ge n_F\cdot s$ (where $s \approx 0.001$) is satisfied and tight for at most one vertex (u, v) of each face. These vertices are marked by the word “(tight)” in the tables; we have $n_F=1$ for each of these faces. All remaining subadditivity slacks $\varDelta \pi _F(u,v)$ for vertices $(u, v)\in {{\,\mathrm{vert}\,}}(F)$ exceed $0.003 \ge 3\cdot s$.

Table 1 The piecewise linear function $\pi = \textsf {kzh\_minimal\_has\_only\_crazy\_perturbation\_1}()$, defined by its values and limits at the breakpoints

Table 2 Two-dimensional faces F with additivity on ${{\,\mathrm{int}\,}}(F)$ for proving piecewise linearity outside of the special intervals

Table 3 One-dimensional faces F with additivity on ${{\,\mathrm{rel\,int}\,}}(F)$ for proving piecewise linearity outside of the special intervals

Table 4 Faces F with additivity on ${{\,\mathrm{rel\,int}\,}}(F)$, one vertex of each providing an equation $\varDelta {{\bar{\pi }}}_F(u, v)=0$, to form a full-rank homogeneous linear system in the proof of Theorem 6.2

Table 5 Homogeneous linear system for determining the restriction of ${{\bar{\pi }}}$ to the non-special intervals in the proof of Theorem 6.2

Table 6 Subadditivity slacks $\varDelta \pi _F$ for $\dim F=1$ and $n_F>0$

Table 7 Subadditivity slacks $\varDelta \pi _F$ for $\dim F=2$ and $n_F>0$