A Generic Construction for Crossovers of Graph-Like Structures

Abstract

In model-driven optimization (MDO), domain-specific models are used to define and solve optimization problems with evolutionary algorithms. Models are typically evolved using mutations, which can be formally specified as graph transformations. So far, only mutations have been used in MDO to generate new solutions from existing ones; a crossover mechanism has not yet been elaborated. In this paper, we present a generic crossover construction for graph-like structures that can be used to implement crossover operators in MDO. We prove basic properties of our construction and show how it can be used to implement a whole set of crossover operators that have been proposed for specific problems and situations on graphs.

A Proofs

The following lemma is the central ingredient for the proof of Proposition 1 and also used in the one of Theorem 3. For adhesive categories, it has already been stated in the extended version of [14]. Here, we present it in the more general context of $\mathcal {M}$-adhesive categories. Because of that, we need to additionally assume the existence of $\mathcal {M}$-effective unions.

Lemma 2 (Pullbacks as pushouts)

In an $\mathcal {M}$-adhesive category $(\mathcal {C},\mathcal {M})$ with $\mathcal {M}$-effective unions, let $(e_1,e_2): L_1, L_2 \hookrightarrow E$ be a pair of jointly epimorphic $\mathcal {M}$-morphisms. Then the pullback of $(e_1,e_2)$ is also a pushout.

Proof

Given the diagram below, where P arises as pullback of $(e_1,e_2)$, Q as pushout of $(p_1,p_2)$, and the morphism h from the universal property of Q, we show that h is an isomorphism.

First, since $e_1,e_2$ are $\mathcal {M}$-morphisms, the morphism h is an $\mathcal {M}$-morphism, assuming $\mathcal {M}$-effective unions. This means that h is a regular monomorphism (compare [25, Lemma 4.8], which is easily seen to also hold in $\mathcal {M}$-adhesive categories).

Secondly, given two morphisms $f,g: E \rightarrow X$ with $f \circ h = g \circ h$, it follows that $f \circ h \circ q_1 = g \circ h \circ q_1$ which implies $f \circ e_1 = g \circ e_1$; analogously, $f \circ e_2 = g \circ e_2$ holds. Since $e_1,e_2$ are jointly epimorphic, it follows that $f=g$, and h is an epimorphism. Thus, h is epi and regular mono and therefore an isomorphism.    $\square$

Proof

(of Proposition 1). Given a solution split as depicted in Fig. 5, it is straightforward to realize this split via the split construction. One just chooses the already given morphisms $s^1$ and $s^2$. As the bottom square in Fig. 5 is a pushout, $s^1$ and $s^2$ are jointly epimorphic. Moreover, in an $\mathcal {M}$-adhesive category that square is also a pullback because $E^I\hookrightarrow E^1$ (or, equally, $E^I\hookrightarrow E^2$) $\in \mathcal {M}$.

To show that the construction always computes a solution split, we have to show that it produces a commuting cube of $\mathcal {M}$-morphisms (with isomorphisms at the top) such that the bottom square is a pushout and the four vertical squares constitute ce-morphisms (i.e., are also pullbacks and are compatible with typing). It is well-known that, in every category, in a cube that is computed via pullbacks as stipulated by our construction, all squares are pullbacks; see, e.g., [3, 5.7 Exercises, 2. (b)]. By closedness of $\mathcal {M}$-morphism under pullbacks, this in turn implies that all morphisms are $\mathcal {M}$-morphisms (because e, $s^1$, and $s^2$ are). The two morphisms at the front of the top square are isomorphisms by assumption; the other two become isomorphisms by closedness of isomorphisms under pullback. Finally, in an $\mathcal {M}$-adhesive category with $\mathcal {M}$-effective unions, the pullback of jointly epimorphic $\mathcal {M}$-morphisms is always a pushout (see Lemma 2 above). Therefore, the bottom square (computed as pullback of the jointly epic $\mathcal {M}$-morphisms $s^1$ and $s^2$) is a pushout as desired. The typing of $\overline{E^1}$ and $\overline{E^2}$ is compatible with the typing of $\overline{E}$ by definition; moreover, the squares obtained from the typing morphisms are pullbacks by pullback composition.

For the last statement, it suffices to observe that $E^2$ can always be chosen as E, embedded via the identity morphism (which then leads to $E^I\cong E^1$).    $\square$

Proof

(of Lemma 1). To prove the statement, we have to show that there exists a ce-morphism $(a_P,a)$ from $\overline{ CP }:= (id: PI _P\hookrightarrow PI _P,t_{ PI _P}, tp \circ t_{ PI _P})$ to $\overline{E^I}$ such that $a_P$ is an isomorphism and $a \in \mathcal {M}$; the analogous statement for $\overline{F^I}$ is proved in exactly the same way.

Fig. 15.

Showing $\overline{ CP }$ to constitute a crossover point

We define such a ce-morphism using the isomorphism $a_P$ with $t_{E^I_P} \circ a_P = t_{ PI _P}$ that exists since $\overline{E^I}$ is an element of the search space of $\overline{ PI }$. Figure 15 depicts this. The square commutes and $a,e^I \circ a \in \mathcal {M}$ by closedness of $\mathcal {M}$ under isomorphisms and composition. Moreover, using the fact that $e^I$ is a monomorphism, it is also easy to check that the square constitutes a pullback. Finally, using $t_{E^I_P} \circ a_P = t_{ PI _P}$ we compute

$$\begin{aligned} t_{E^I} \circ e^I \circ a_P&= tp \circ t_{E^I_P} \circ a_P \\&= tp \circ t_{ PI _P} \end{aligned}$$

which shows $(a_P, e^I \circ a_P)$ to be type-compatible.    $\square$

Proof

(of Proposition 2). First, in an $\mathcal {M}$-adhesive category, pushouts along $\mathcal {M}$-morphisms exist. This means that, given two solution splits and a crossover point, crossover is always applicable. Since isomorphisms are closed under pushout, the top squares in the construction consist of isomorphisms only. In particular, $(E^1F^2)_P\cong PI _P\cong (E^2F^1)_P$ (because $E^1_P\cong PI _P\cong E^2_P$ by assumption).

By definition, $o_1$ is the unique morphism such that

$$\begin{aligned} o_1 \circ a_P = a \circ e^1 \text { and } o_1 \circ b_P = b \circ f^2, \end{aligned}$$

where $(a_P,a)$ and $(b_P,b)$ denote the ce-morphisms from $e^1$ and $f^2$ to $o_1$ (see Fig. 8). A standard diagram chase (using the facts that the top squares in Fig. 8 consist of isomorphisms only and that diagrams remain commutative if one replaces isomorphisms by their inverses) then shows that $a \circ e^1 \circ a_P^{-1}$ (or, equally, $b \circ f^2 \circ b_P^{-1}$) exhibits this universal property. Therefore, $o_1 = a \circ e^1 \circ a_P^{-1} \in \mathcal {M}$ as composition of $\mathcal {M}$-morphisms. Again, this uses the fact that $\mathcal {M}$ contains all isomorphisms.

Finally, that the typing morphisms of $\overline{O_1}$ induce even a pullback square over $tp$ (and not merely a commuting one) follows exactly as in the proof of Lemma 2.2 in [15], using the facts that the ambient category $\mathcal {C}$ is $\mathcal {M}$-adhesive and $tp \in \mathcal {M}$.    $\square$

Proof

(of Proposition 3). Let solution $\overline{O}$ be computed via a crossover from $\overline{E}$ and $\overline{F}$. It is immediately clear from the construction that there exist the two required ce-morphisms $\bar{i}$ and $\bar{j}$ such that i, j are jointly epic $\mathcal {M}$-morphisms because the projections of a pushout are jointly epi and $\mathcal {M}$-morphisms are closed under pushout.

For the converse direction, O is jointly covered by $E^1$ and $F^2$, which stem from subsolutions $\overline{E^1}$ and $\overline{F^1}$ of $\overline{E}$ and $\overline{F}$ by assumption. If the underlying category has $\mathcal {M}$-effective unions, pulling these morphisms back results in a pushout. Let $\overline{ CP }$ be the object resulting from that pullback (exactly as in the proof of Proposition 1). We merely have to show that there exist solution splits of $\overline{E}$ and $\overline{F}$ that split up $\overline{E}$ into $\overline{E^1}$ and some suitable subsolution $\overline{E^2}$ of $\overline{E}$ and $\overline{F}$ into $\overline{F^2}$ and some suitable subsolution $\overline{F^1}$ of $\overline{F}$ for which $\overline{ CP }$ can serve as a crossover point. As in (the proof of) the second part of Proposition 1, we can use $\overline{E}$ as $\overline{E^2}$ and, because of the symmetric nature of a solution split, $\overline{F}$ as $\overline{F^1}$ and obtain splits of $\overline{E}$ and $\overline{F}$ with $\overline{E^I}= \overline{E^1}$ and $\overline{F^I}= \overline{F^2}$. Hence, $\overline{ CP }$, together with the morphisms that stem from its computation as a pullback, can serve as a crossover point for these splits, and applying the crossover construction computes the given solution $\overline{O}$.    $\square$