On Perturbation Spaces of Minimal Valid Functions: Inverse Semigroup Theory and Equivariant Decomposition Theorem

Abstract

The non-extreme minimal valid functions for the Gomory–Johnson infinite group problem are those that admit effective perturbations. For a class of piecewise linear functions for the 1-row problem we give a precise description of the space of these perturbations as a direct sum of certain finite- and infinite-dimensional subspaces. The infinite-dimensional subspaces have partial symmetries; to describe them, we develop a theory of inverse semigroups of partial bijections, interacting with the functional equations satisfied by the perturbations. Our paper provides the foundation for grid-free algorithms for testing extremality and for computing liftings of non-extreme functions. The grid-freeness makes the algorithms suitable for piecewise linear functions whose breakpoints are rational numbers with huge denominators.

The authors wish to thank C.Y. Hong, who worked on a first grid-free implementation in 2013, and Q. Louveaux and R. La Haye for valuable discussions during 2013/14. A preliminary version of the development in this paper appeared in Y.Z.’s 2017 Ph.D. thesis  [14]. The authors gratefully acknowledge partial support from the National Science Foundation through grants DMS-0914873 (R.H., M.K.) and DMS-1320051 (M.K., Y.Z.) Part of this work was done while R.H. and M.K. were visiting the Simons Institute for the Theory of Computing in Fall 2017. It was partially supported by the DIMACS/Simons Collaboration on Bridging Continuous and Discrete Optimization through NSF grant CCF-1740425.

A Some Omitted Proofs and Results

We summarize some results on the interactions of the closure axioms. While the inverse semigroup generated by a join-closed ensemble is not automatically join-closed, the following holds: Applying (join) or the stronger (extend) to an inverse semigroup preserves the inverse semigroup properties,

$$\begin{aligned} \mathop {\mathrm {join}}(\mathop {\mathrm {i semi}}({\varOmega }))&= \mathop {\mathrm {i semi}}(\mathop {\mathrm {join}}(\mathop {\mathrm {i semi}}({\varOmega }))), \end{aligned}$$

(12a)

$$\begin{aligned} \mathop {\mathrm {extend}_{}}(\mathop {\mathrm {i semi}}({\varOmega }))&= \mathop {\mathrm {i semi}}(\mathop {\mathrm {extend}_{}}(\mathop {\mathrm {i semi}}({\varOmega }))). \end{aligned}$$

(12b)

Also applying limit and then join to a semigroup retains the semigroup properties , i.e., $\mathop {\mathrm {join}}(\mathop {\mathrm {lim}}(\mathop {\mathrm {join}}(\mathop {\mathrm {i semi}}({\varOmega }))))$ is a semigroup.

For some of the individual closure properties, generation theorems are available. Let ${{\varOmega }}^{\mathrm {inv}}$ be the union of the move ensemble ${\varOmega }$ and the inverses of all of its moves; this is clearly the smallest ensemble containing ${\varOmega }$ that satisfies (inv). Also $\mathop {\mathrm {i semi}}({\varOmega })$, defined as the smallest inverse semigroup containing ${\varOmega }$, is, of course, the set of all finite compositions $\gamma ^k|_{D_k} \circ \dots \circ \gamma ^1|_{D_1}$ of moves $\gamma ^i|_{D_i} \in {{\varOmega }}^{\mathrm {inv}}$. The joined ensemble $\mathop {\mathrm {join}}({\varOmega })$ and the extended ensemble $\mathop {\mathrm {extend}_{}}({\varOmega })$ consist of the following moves:

$$\begin{aligned} \mathop {\mathrm {join}}({\varOmega })&= \bigl \{ \, \gamma |_D \mathrel {\big |}\, D\subseteq C_\gamma \,\bigr \}, \end{aligned}$$

(13a)

$$\begin{aligned} \mathop {\mathrm {extend}_{}}({\varOmega })&= \bigl \{ \, \gamma |_D \mathrel {\big |}\, D\subseteq {{\,\mathrm{cl}\,}}(C_\gamma ) \,\bigr \}, \quad \text {where } C_\gamma := \bigcup \{\, I \mid \gamma |_I \in {\varOmega } \,\}. \end{aligned}$$

(13b)

In particular, joined ensembles and extended ensembles have maximal elements in the restriction partial order, which is not true for arbitrary ensembles.

For closures with respect to more complex combinations of our axioms, no generation theorem is available. In particular, we have no generation theorem for the moves closure. It is unknown if $\mathop {\mathrm {clsemi}_{}}({\varOmega })$ can be obtained by a finite number of applications of closures with respect to the individual axioms similar to (12). Instead we reason about the moves closure in an indirect way, as follows: Let $\mathbb {L}$ be the family of closed move semigroups containing ${\varOmega }$. Then $\mathop {\mathrm {clsemi}_{}}({\varOmega }) = \bigcap \mathbb {L}= \bigcap _{{\varOmega }' \in \mathbb {L}} {\varOmega }'$; this holds because of the structure of our axioms.

In addition to the various closure properties, we use the following lemma in the proof of the structure and generation theorem for finitely presented moves closures (Theorem 4.8). Recall that the initial moves ensemble ${\varOmega }^0$ is join-closed.

Lemma A.1

(Filtration of $\mathop {\mathrm {i semi}}({\varOmega }^0|_U)$ by word length; maximal moves). Under Assumptions 4.1/4.2/4.6, for $k\in \mathbb N$, let

$$\begin{aligned} {\varOmega }^0|_U{}^k = \bigl \{\, \gamma ^k|_{D^k} \circ \dots \circ \gamma ^1|_{D^1} \mathrel {\big |}\,&\gamma ^i|_{D^i} \in {\varOmega }^0|_U \text { for }1\le i \le k \,\bigr \}. \end{aligned}$$

Then ${\varOmega }^0|_U{}^1 \subseteq {\varOmega }^0|_U{}^2 \subseteq \dots$ and $\mathop {\mathrm {i semi}}({\varOmega }^0|_U) = \bigcup _{k\in \mathbb N} {\varOmega }^0|_U{}^k$. For each $k\in \mathbb N$, the ensemble ${\varOmega }^0|_U{}^k$ is equal to the set of restrictions of the ensemble ${{\,\mathrm{Max}\,}}({\varOmega }^0|_U{}^k)$ of its maximal elements in the restriction partial order, which is a finite set. For $\gamma |_{(a,b)} \in {{\,\mathrm{Max}\,}}( {\varOmega }^0|_U{}^k)$, we have $a, b, \gamma (a), \gamma (b) \in X \cup Y$.

1.1 Proof of Theorem 4.8 (Structure and generation theorem)

Part (a). Let ${\varOmega }'$ denote the right hand side of the equation in part (a). Clearly, ${\varOmega }^0 \subseteq {\varOmega }' \subseteq \mathop {\mathrm {clsemi}_{}}({\varOmega }^0)$. We now show that ${\varOmega }'$ is a closed move semigroup. By Remark 4.7-(a), we have that $\mathop {\mathrm {clsemi}_{}}({\varOmega }^0|_U) \subseteq \mathop {\mathrm {restrict}}({\varOmega }|_U) \subseteq \mathop {\mathrm {moves}}\nolimits (U \times U),$ where $\mathop {\mathrm {restrict}}({\varOmega }|_U)$ is the set of restrictions of moves $\gamma |_D \in {\varOmega }$ to domains that are open subintervals of $D\cap U$ (or $\emptyset$); and $\mathop {\mathrm {clsemi}_{}}({\varOmega }^0|_C) = \bigcup _{i=1}^k \mathop {\mathrm {moves}}\nolimits (C_{i} \times C_{i}) \subseteq \mathop {\mathrm {moves}}\nolimits (C \times C),$ where the open sets U and C are disjoint. Thus, we have that $\mathop {\mathrm {clsemi}_{}}({\varOmega }^0|_U) \cup \mathop {\mathrm {clsemi}_{}}({\varOmega }^0|_C)$ is a move semigroup, under Assumption 4.2. It follows from (12b) that ${\varOmega }'$ is a move semigroup that satisfies (extend). Note that for any open intervals D and I such that $\mathop {\mathrm {moves}}\nolimits (D\times I) \subseteq \mathop {\mathrm {clsemi}_{}}({\varOmega }^0)$, we have $\mathop {\mathrm {moves}}\nolimits (D\times I) \subseteq \mathop {\mathrm {clsemi}_{}}({\varOmega }^0|_C)$. Therefore, ${\varOmega }'$ also satisfies (kaleido) and (lim) by Theorem 4.5. We conclude that ${\varOmega }'$ is a closed move semigroup. Hence, part (a) holds.

Part (b). By restricting the moves ensembles on both sides of the equation in part (a) to domain U, we obtain that

$$\begin{aligned} \mathop {\mathrm {restrict}}({\varOmega }|_U) = \mathop {\mathrm {clsemi}_{}}({\varOmega }^0)|_U = \mathop {\mathrm {clsemi}_{}}({\varOmega }^0|_U) \end{aligned}$$

(14)

Next, we show that

$$\begin{aligned} \mathop {\mathrm {clsemi}_{}}({\varOmega }^0|_U) = \mathop {\mathrm {extend}_{}}(\mathop {\mathrm {i semi}}({\varOmega }^0|_U)). \end{aligned}$$

(15)

It follows from (12b) that $\mathop {\mathrm {extend}_{}}(\mathop {\mathrm {i semi}}({\varOmega }^0|_U))$ is a move semigroup that satisfies (extend) (and also (join)). Since

$$\begin{aligned} \mathop {\mathrm {extend}_{}}(\mathop {\mathrm {i semi}}({\varOmega }^0|_U)) \subseteq \mathop {\mathrm {clsemi}_{}}({\varOmega }^0 |_{U}) = \mathop {\mathrm {restrict}}({\varOmega }|_U), \end{aligned}$$

(16)

where the equality follows from (14), and ${\varOmega } |_{U}$ is a finite move ensemble by Remark 4.7-(b), we obtain that the move semigroup $\mathop {\mathrm {extend}_{}}(\mathop {\mathrm {i semi}}({\varOmega }^0|_U))$ also satisfies (kaleido) and (lim). Therefore, $\mathop {\mathrm {extend}_{}}(\mathop {\mathrm {i semi}}({\varOmega }^0|_U))$ is a closed move semigroup which contains ${\varOmega }^0|_U$. Since $\mathop {\mathrm {clsemi}_{}}({\varOmega }^0 |_{U})$ is the smallest closed move semigroup containing ${\varOmega }^0|_U$, we have $\mathop {\mathrm {clsemi}_{}}({\varOmega }^0 |_{U}) \subseteq \mathop {\mathrm {extend}_{}}(\mathop {\mathrm {i semi}}({\varOmega }^0|_U))$. Together with (16), we conclude that (15) holds. Since ${\varOmega }$ has only maximal moves, (14) and (15) imply the equation in part (b).

Part (c). Let $\gamma |_{(a,b)} \in {\varOmega } |_{U}$. By symmetry, it suffices to show that $a, b \in X \cup Y$. Consider $x=a$ or $x=b$. Part (b) implies that

$${\varOmega }|_U = {{\,\mathrm{Max}\,}}(\mathop {\mathrm {extend}_{}}(\mathop {\mathrm {join}}(\mathop {\mathrm {i semi}}({\varOmega }^0|_U)))).$$

Together with (13b), we know that x is the limit of a sequence $\{x^j\}_{j \in \mathbb N}$, where $x^j$ is an endpoint of the domain $D^j$ of a move $\gamma |_{D^j} \in {{\,\mathrm{Max}\,}}(\mathop {\mathrm {join}}(\mathop {\mathrm {i semi}}({\varOmega }^0|_U)))$. By Lemma A.1 and (13a), for any $j \in \mathbb N$, we have that $D^j$ is a maximal subinterval of $\bigcup \{\,D \mid \gamma |_D \in \bigcup _{k\in \mathbb N} {{\,\mathrm{Max}\,}}({\varOmega }^0|_U{}^{k})\,\}$. Thus for every $j\in \mathbb N$, there exists a sequence $\{x^j_k\}_{k \in \mathbb N}$ such that each $x^j_k$ is an endpoint of the domain of a move $\gamma |_{D^j_k} \in {{\,\mathrm{Max}\,}}({\varOmega }^0|_U{}^{k})$, and $x^j_k \rightarrow x^j$ as $k \rightarrow \infty$. We obtain that $x^k_k \rightarrow x$ as $k \rightarrow \infty$, where each $x^k_k \in X\cup Y$ by Lemma A.1. Since $X \cup Y$ is a finite discrete set under Assumption 4.2, we obtain that $x \in X \cup Y$.    $\square$