Bounds of MIN_NCC and MAX_NCC and filtering scheme for graph domain variables

We use the notations introduced in [1].

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  $G$: The graph domain variable.

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  $V_{T}$: The set of mandatory vertices.

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  $E_{T}$: The set of mandatory arcs.

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  $V_{U}$: The set of potential vertices.

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  $E_{U}$: The set of potential arcs.

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  $V_{F}$: The set of removed vertices.

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  $E_{F}$: The set of removed arcs.

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  $G(V_{T},E_{T})$: The graph kernel.

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  $G(V_{TU},E_{TU})$: The graph envelope.

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  $cc(G)$: The set of connected components of $G$, $cc_{[cond]}(G)$ denotes the set of connected components of $G$ verifying $cond$.

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  If $i$ is a connected component, $l_{i}$ is the length of any elementary sequence made from all the vertices of $i$, or more simple, $l_{i}$ is the size of $i$.

Given a graph domain variable $G$ and an integer domain variable $P$, the constraint $\text{MIN\_NCC}(G,P)$ holds if $P$ is the size of the smallest connected component of $G$. We denote by $\underline{\text{MIN\_NCC}}(G)$ and $\overline{\text{MIN\_NCC}}(G)$ the lower and upper bounds of the MIN_NCC graph property, computed from $G$. Sharp bounds of MIN_NCC had been defined in [1] for graph-based representation of global constraints, in the particular case of the path_with_loops graph class. We adjusted them for the context of graph domain variables, and for any graph class.

We now suggest a filtering scheme for the $\text{MIN\_NCC}(G,P)$ constraint that is adapted from the filtering scheme introduced in [1] for the path_with_loops graph class in the context of graph-based representation of global constraints.

1. 1.

   Trivial case: if $\underline{P}>|V_{TU}|$ then fail.

2. 2.

   Trivial case: if $\underline{P}>0$ and $|E_{TU}|<\max(1,\underline{P}-1)$ then fail.

3. 3.

   If $\overline{\text{MIN\_NCC}}(G)<\underline{P}$ then fail. Note that a sufficient condition is $|\textit{cc}_{[|V_{TU}|<\underline{P}\land|V_{T}|\geq 1]}(G(V_{TU},E_{TU}))|\geq 1$, that is the existence of a mandatory connected component in the envelope strictly smaller than $\underline{P}$.

4. 4.

   If $\underline{\text{MIN\_NCC}}(G)>\overline{P}$ then fail.

5. 5.

   If $\underline{P}<\underline{\text{MIN\_NCC}}(G)$ then set $\underline{P}$ to $\underline{\text{MIN\_NCC}}(G)$.

6. 6.

   If $\overline{P}>\overline{\text{MIN\_NCC}}(G)$ then set $\overline{P}$ to $\overline{\text{MIN\_NCC}}(G)$.

7. 7.

   Every $U$-vertex of any optional connected component smaller than $\underline{P}$ (that is, in $\textit{cc}_{[|V_{TU}|<\underline{P}\land|V_{T}|=0]}\allowbreak(G(V_{TU},E_{TU}))$) is turned into a $F$-vertex.

8. 8.

   If $\underline{\text{MIN\_NCC}}(G)<\underline{P}$ then:

   1. (a)

      If $|V_{T}|\geq 1$, every $U$-vertex of any mandatory connected component of size $\underline{P}$ (that is, in $\textit{cc}_{[|V_{TU}|=\underline{P}\land|V_{T}|\geq 1]}(G(V_{TU},E_{TU}))$), is turned into a $T$-vertex. $P$ can then be instantiated to $\underline{P}$.

   2. (b)

      Consider each connected component $i\in\textit{cc}_{[|V_{T}|<\underline{P}]}(G(V_{T},E_{T}))$, if there is exactly one $U$-arc ($x$, $y$) outgoing from $i$ ($x\in i\land y\notin i$), it is turned into a $T$-arc. If $y$ is a $U$-vertex, it is turned into a $T$-vertex. Similarly, if there are several $U$-arcs outgoing from $i$ but all of them are connected to a single $U$-vertex, then it is turned into a $T$-vertex.

9. 9.

   If $G$ had been modified in step 8b, recompute $\underline{\text{MIN\_NCC}}(G)$ and repeat steps 4 and 5.

10. 10.

    If $\overline{\text{MIN\_NCC}}(G)>\overline{P}$ then:

    1. (a)

       If $\overline{P}=0$ every $U$-vertex is turned into a $F$-vertex.

    2. (b)

       If $\overline{P}=1\land|V_{U}|=1\land\min_{i\in\textit{cc}(G(V_{T},E_{T}))}l_{i}>1$ then the only $U$-vertex is turned into a $T$-vertex and every $U$-arc outgoing from it and which is not a loop is turned into a $F$-arc.

Given a graph domain variable $G$ and an integer domain variable $P$, the constraint $\text{MAX\_NCC}(G,P)$ holds if $P$ is the size of the largest connected component of $G$. We denote by $\underline{\text{MAX\_NCC}}(G)$ and $\overline{\text{MAX\_NCC}}(G)$ the lower and upper bounds of the MAX_NCC graph property, computed from $G$. Sharp bounds of MAX_NCC had been defined in [1] for graph-based representation of global constraints, in the particular case of the path_with_loops graph class. Those bounds remain unchanged in the context of graph domain variables, for any graph class.

We now suggest a filtering scheme for the $\text{MAX\_NCC}(G,P)$ constraint that is adapted from the filtering scheme introduced in [1] for the path_with_loops graph class in the context of graph-based representation of global constraints.

1. 1.

   Trivial case: if $\underline{P}>|X_{TU}|$ then fail.

2. 2.

   Trivial case: if $\underline{P}>0$ and $|E_{TU}|<\max(1,\underline{P}-1)$ then fail.

3. 3.

   If $\overline{\textbf{MAX\_NCC}}(G)<\underline{P}$ then fail.

4. 4.

   If $\underline{\textbf{MAX\_NCC}}(G)>\overline{P}$ then fail. Note that a sufficient condition is $|\textit{cc}_{[|X_{T}|>\overline{P}]}(G(X_{T},E_{T}))|\geq 1$, that is the existence of connected component in the kernel strictly greater than $\overline{P}$.

5. 5.

   If $\underline{P}<\underline{\text{MAX\_NCC}}(G)$ then set $\underline{P}$ to $\underline{\text{MAX\_NCC}}(G)$.

6. 6.

   If $\overline{P}>\overline{\text{MAX\_NCC}}(G)$ then set $\overline{P}$ to $\overline{\text{MAX\_NCC}}(G)$.

7. 7.

   If $\overline{\text{MAX\_NCC}}(G)>\overline{P}$ then:

   1. (a)

      If $\overline{P}=1$ then every $U$-arc $(x,y)$ such that $x\neq y$ is turned into a $F$-arc.

   2. (b)

      If $\overline{P}=0$ then every $U$-vertex is turned into a $F$-vertex.

   3. (c)

      Every $U$-arc linked to a maximal size connected component in the kernel, that is in $\textit{cc}_{[|X_{T}|=\overline{P}]}(G(X_{T},E_{T}))$ is turned into a $F$-arc.

   4. (d)

      Any $U$-arc linking two connected components of $G(X_{T},E_{T})$ of size $s_{1}$ and $s_{2}$ such that $s_{1}+s_{2}>\overline{P}$ is turned into a $F$-arc.

   5. (e)

      If $G$ had been modified in step 7d, recompute $\overline{\text{MAX\_NCC}}(G)$ and repeat steps 3 and 6.

8. 8.

   If $|\textit{cc}_{[|V_{TU}|\geq\underline{P}]}(G(V_{TU},E_{TU}))|=1$ and the size of this single candidate component is exactly $\underline{P}$ then any $U$-vertex of it is turned into a $T$-vertex. $P$ can then be instantiated to $\underline{P}$.