## 5.401. tour

Origin
Constraint

$\mathrm{\pi \pi \pi \pi }\left(\mathrm{\pi ½\pi Ύ\pi ³\pi ΄\pi }\right)$

Synonyms

$\mathrm{\pi \pi \pi \pi \pi }$, $\mathrm{\pi \pi ’\pi \pi \pi }$.

Argument
 $\mathrm{\pi ½\pi Ύ\pi ³\pi ΄\pi }$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi \pi \pi ‘}-\mathrm{\pi \pi \pi },\mathrm{\pi \pi \pi \pi }-\mathrm{\pi \pi \pi \pi }\right)$
Restrictions
 $|\mathrm{\pi ½\pi Ύ\pi ³\pi ΄\pi }|\beta ₯3$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$$\left(\mathrm{\pi ½\pi Ύ\pi ³\pi ΄\pi },\left[\mathrm{\pi \pi \pi \pi \pi ‘},\mathrm{\pi \pi \pi \pi }\right]\right)$ $\mathrm{\pi ½\pi Ύ\pi ³\pi ΄\pi }.\mathrm{\pi \pi \pi \pi \pi ‘}\beta ₯1$ $\mathrm{\pi ½\pi Ύ\pi ³\pi ΄\pi }.\mathrm{\pi \pi \pi \pi \pi ‘}\beta €|\mathrm{\pi ½\pi Ύ\pi ³\pi ΄\pi }|$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$$\left(\mathrm{\pi ½\pi Ύ\pi ³\pi ΄\pi },\mathrm{\pi \pi \pi \pi \pi ‘}\right)$
Purpose

Enforce to cover an undirected graph $G$ described by the $\mathrm{\pi ½\pi Ύ\pi ³\pi ΄\pi }$ collection with a Hamiltonian cycle.

Example
$\left(\begin{array}{c}β©\begin{array}{cc}\mathrm{\pi \pi \pi \pi \pi ‘}-1\hfill & \mathrm{\pi \pi \pi \pi }-\left\{2,4\right\},\hfill \\ \mathrm{\pi \pi \pi \pi \pi ‘}-2\hfill & \mathrm{\pi \pi \pi \pi }-\left\{1,3\right\},\hfill \\ \mathrm{\pi \pi \pi \pi \pi ‘}-3\hfill & \mathrm{\pi \pi \pi \pi }-\left\{2,4\right\},\hfill \\ \mathrm{\pi \pi \pi \pi \pi ‘}-4\hfill & \mathrm{\pi \pi \pi \pi }-\left\{1,3\right\}\hfill \end{array}βͺ\hfill \end{array}\right)$

The $\mathrm{\pi \pi \pi \pi }$ constraint holds since its $\mathrm{\pi ½\pi Ύ\pi ³\pi ΄\pi }$ argument depicts the following Hamiltonian cycle visiting successively the vertices 1, 2, 3 and 4.

Symmetry

Items of $\mathrm{\pi ½\pi Ύ\pi ³\pi ΄\pi }$ are permutable.

Algorithm

When the number of vertices is odd (i.e.,Β $|\mathrm{\pi ½\pi Ύ\pi ³\pi ΄\pi }|$ is odd) a necessary condition is that the graph is not bipartite. Other necessary conditions for filtering the $\mathrm{\pi \pi \pi \pi }$ constraint are given inΒ [Cymer13], [CymerPhD13].

Keywords
Arc input(s)

$\mathrm{\pi ½\pi Ύ\pi ³\pi ΄\pi }$

Arc generator
$\mathrm{\pi Ά\pi Ώ\pi Ό\pi \pi \pi Έ}$

Arc arity
Arc constraint(s)
$\begin{array}{c}\mathrm{\pi \pi }_\mathrm{\pi \pi \pi }\left(\mathrm{\pi \pi \pi \pi \pi }\mathtt{2}.\mathrm{\pi \pi \pi \pi \pi ‘},\mathrm{\pi \pi \pi \pi \pi }\mathtt{1}.\mathrm{\pi \pi \pi \pi }\right)\beta \hfill \\ \mathrm{\pi \pi }_\mathrm{\pi \pi \pi }\left(\mathrm{\pi \pi \pi \pi \pi }\mathtt{1}.\mathrm{\pi \pi \pi \pi \pi ‘},\mathrm{\pi \pi \pi \pi \pi }\mathtt{2}.\mathrm{\pi \pi \pi \pi }\right)\hfill \end{array}$
Graph property(ies)
$\mathrm{\pi \pi \pi \pi }$$=|\mathrm{\pi ½\pi Ύ\pi ³\pi ΄\pi }|*|\mathrm{\pi ½\pi Ύ\pi ³\pi ΄\pi }|-|\mathrm{\pi ½\pi Ύ\pi ³\pi ΄\pi }|$

Arc input(s)

$\mathrm{\pi ½\pi Ύ\pi ³\pi ΄\pi }$

Arc generator
$\mathrm{\pi Ά\pi Ώ\pi Ό\pi \pi \pi Έ}$

Arc arity
Arc constraint(s)
$\mathrm{\pi \pi }_\mathrm{\pi \pi \pi }$$\left(\mathrm{\pi \pi \pi \pi \pi }\mathtt{2}.\mathrm{\pi \pi \pi \pi \pi ‘},\mathrm{\pi \pi \pi \pi \pi }\mathtt{1}.\mathrm{\pi \pi \pi \pi }\right)$
Graph property(ies)
 $\beta ’$$\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi }$$=|\mathrm{\pi ½\pi Ύ\pi ³\pi ΄\pi }|$ $\beta ’$$\mathrm{\pi \pi \pi }_\mathrm{\pi \pi }$$=2$ $\beta ’$$\mathrm{\pi \pi \pi }_\mathrm{\pi \pi }$$=2$ $\beta ’$$\mathrm{\pi \pi \pi }_\mathrm{\pi \pi }$$=2$ $\beta ’$$\mathrm{\pi \pi \pi }_\mathrm{\pi \pi }$$=2$

Graph model

The first graph property enforces the subsequent condition: If we have an arc from the ${i}^{th}$ vertex to the ${j}^{th}$ vertex then we have also an arc from the ${j}^{th}$ vertex to the ${i}^{th}$ vertex. The second graph property enforces the following constraints:

• We have one strongly connected component containing $|\mathrm{\pi ½\pi Ύ\pi ³\pi ΄\pi }|$ vertices,

• Each vertex has exactly two predecessors and two successors.

PartΒ (A) of FigureΒ 5.401.1 shows the initial graph from which we start. It is derived from the set associated with each vertex. Each set describes the potential values of the $\mathrm{\pi \pi \pi \pi }$ attribute of a given vertex. PartΒ (B) of FigureΒ 5.401.1 gives the final graph associated with the Example slot. The $\mathrm{\pi \pi \pi \pi }$ constraint holds since the final graph corresponds to a Hamiltonian cycle.

Signature

Since the maximum number of vertices of the final graph is equal to $|\mathrm{\pi ½\pi Ύ\pi ³\pi ΄\pi }|$, we can rewrite the graph property $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi }=|\mathrm{\pi ½\pi Ύ\pi ³\pi ΄\pi }|$ to $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi }\beta ₯|\mathrm{\pi ½\pi Ύ\pi ³\pi ΄\pi }|$ and simplify $\underset{Μ²}{\stackrel{Β―}{\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi }}}$ to $\stackrel{Β―}{\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi }}$.