## 5.48. balance_path

Origin
Constraint

$\mathrm{\pi \pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi }\left(\mathrm{\pi ±\pi °\pi »\pi °\pi ½\pi ²\pi ΄},\mathrm{\pi ½\pi Ύ\pi ³\pi ΄\pi }\right)$

Arguments
 $\mathrm{\pi ±\pi °\pi »\pi °\pi ½\pi ²\pi ΄}$ $\mathrm{\pi \pi \pi \pi }$ $\mathrm{\pi ½\pi Ύ\pi ³\pi ΄\pi }$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi \pi \pi ‘}-\mathrm{\pi \pi \pi },\mathrm{\pi \pi \pi \pi }-\mathrm{\pi \pi \pi \pi }\right)$
Restrictions
 $\mathrm{\pi ±\pi °\pi »\pi °\pi ½\pi ²\pi ΄}\beta ₯0$ $\mathrm{\pi ±\pi °\pi »\pi °\pi ½\pi ²\pi ΄}\beta €\mathrm{\pi \pi \pi ‘}\left(0,|\mathrm{\pi ½\pi Ύ\pi ³\pi ΄\pi }|-2\right)$ $\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)$ $\mathrm{\pi ½\pi Ύ\pi ³\pi ΄\pi }.\mathrm{\pi \pi \pi \pi }\beta ₯1$ $\mathrm{\pi ½\pi Ύ\pi ³\pi ΄\pi }.\mathrm{\pi \pi \pi \pi }\beta €|\mathrm{\pi ½\pi Ύ\pi ³\pi ΄\pi }|$
Purpose

Consider a digraph $G$ described by the $\mathrm{\pi ½\pi Ύ\pi ³\pi ΄\pi }$ collection. Partition $G$ into a set of vertex disjoint paths in such a way that each vertex of $G$ belongs to a single path. $\mathrm{\pi ±\pi °\pi »\pi °\pi ½\pi ²\pi ΄}$ is equal to the difference between the number of vertices of the largest path and the number of vertices of the smallest path.

Example
 $\left(\begin{array}{c}3,β©\begin{array}{cc}\mathrm{\pi \pi \pi \pi \pi ‘}-1\hfill & \mathrm{\pi \pi \pi \pi }-1,\hfill \\ \mathrm{\pi \pi \pi \pi \pi ‘}-2\hfill & \mathrm{\pi \pi \pi \pi }-3,\hfill \\ \mathrm{\pi \pi \pi \pi \pi ‘}-3\hfill & \mathrm{\pi \pi \pi \pi }-5,\hfill \\ \mathrm{\pi \pi \pi \pi \pi ‘}-4\hfill & \mathrm{\pi \pi \pi \pi }-4,\hfill \\ \mathrm{\pi \pi \pi \pi \pi ‘}-5\hfill & \mathrm{\pi \pi \pi \pi }-1,\hfill \\ \mathrm{\pi \pi \pi \pi \pi ‘}-6\hfill & \mathrm{\pi \pi \pi \pi }-6,\hfill \\ \mathrm{\pi \pi \pi \pi \pi ‘}-7\hfill & \mathrm{\pi \pi \pi \pi }-7,\hfill \\ \mathrm{\pi \pi \pi \pi \pi ‘}-8\hfill & \mathrm{\pi \pi \pi \pi }-6\hfill \end{array}βͺ\hfill \end{array}\right)$ $\left(\begin{array}{c}0,β©\begin{array}{cc}\mathrm{\pi \pi \pi \pi \pi ‘}-1\hfill & \mathrm{\pi \pi \pi \pi }-2,\hfill \\ \mathrm{\pi \pi \pi \pi \pi ‘}-2\hfill & \mathrm{\pi \pi \pi \pi }-3,\hfill \\ \mathrm{\pi \pi \pi \pi \pi ‘}-3\hfill & \mathrm{\pi \pi \pi \pi }-4,\hfill \\ \mathrm{\pi \pi \pi \pi \pi ‘}-4\hfill & \mathrm{\pi \pi \pi \pi }-4,\hfill \\ \mathrm{\pi \pi \pi \pi \pi ‘}-5\hfill & \mathrm{\pi \pi \pi \pi }-6,\hfill \\ \mathrm{\pi \pi \pi \pi \pi ‘}-6\hfill & \mathrm{\pi \pi \pi \pi }-7,\hfill \\ \mathrm{\pi \pi \pi \pi \pi ‘}-7\hfill & \mathrm{\pi \pi \pi \pi }-8,\hfill \\ \mathrm{\pi \pi \pi \pi \pi ‘}-8\hfill & \mathrm{\pi \pi \pi \pi }-8\hfill \end{array}βͺ\hfill \end{array}\right)$ $\left(\begin{array}{c}6,β©\begin{array}{cc}\mathrm{\pi \pi \pi \pi \pi ‘}-1\hfill & \mathrm{\pi \pi \pi \pi }-2,\hfill \\ \mathrm{\pi \pi \pi \pi \pi ‘}-2\hfill & \mathrm{\pi \pi \pi \pi }-3,\hfill \\ \mathrm{\pi \pi \pi \pi \pi ‘}-3\hfill & \mathrm{\pi \pi \pi \pi }-4,\hfill \\ \mathrm{\pi \pi \pi \pi \pi ‘}-4\hfill & \mathrm{\pi \pi \pi \pi }-5,\hfill \\ \mathrm{\pi \pi \pi \pi \pi ‘}-5\hfill & \mathrm{\pi \pi \pi \pi }-6,\hfill \\ \mathrm{\pi \pi \pi \pi \pi ‘}-6\hfill & \mathrm{\pi \pi \pi \pi }-7,\hfill \\ \mathrm{\pi \pi \pi \pi \pi ‘}-7\hfill & \mathrm{\pi \pi \pi \pi }-7,\hfill \\ \mathrm{\pi \pi \pi \pi \pi ‘}-8\hfill & \mathrm{\pi \pi \pi \pi }-8\hfill \end{array}βͺ\hfill \end{array}\right)$

In the first example we have the following four paths: $2\beta 3\beta 5\beta 1$, $8\beta 6$, 4, and 7. Since $\mathrm{\pi ±\pi °\pi »\pi °\pi ½\pi ²\pi ΄}=3$ is the difference between the number of vertices of the largest path (i.e.,Β 4) and the number of vertices of the smallest path (i.e.,Β 1) the corresponding $\mathrm{\pi \pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi }$ constraint holds.

All solutions

FigureΒ 5.48.1 gives all solutions to the following non ground instance of the $\mathrm{\pi \pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi }$ constraint: $\mathrm{\pi ±\pi °\pi »\pi °\pi ½\pi ²\pi ΄}=0$, ${S}_{1}\beta \left[1,2\right]$, ${S}_{2}\beta \left[1,3\right]$, ${S}_{3}\beta \left[3,5\right]$, ${S}_{4}\beta \left[3,4\right]$, ${S}_{5}\beta \left[2,5\right]$, ${S}_{6}\beta \left[5,6\right]$, $\mathrm{\pi \pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi }$$\left(\mathrm{\pi ±\pi °\pi »\pi °\pi ½\pi ²\pi ΄},\beta ©1{S}_{1},2{S}_{2},3{S}_{3},4{S}_{4},5{S}_{5},6{S}_{6}\beta ͺ\right)$.

Typical
$|\mathrm{\pi ½\pi Ύ\pi ³\pi ΄\pi }|>2$
Symmetry

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

Arg. properties

Functional dependency: $\mathrm{\pi ±\pi °\pi »\pi °\pi ½\pi ²\pi ΄}$ determined by $\mathrm{\pi ½\pi Ύ\pi ³\pi ΄\pi }$.

Counting
 Length ($n$) 2 3 4 5 6 7 8 Solutions 3 13 73 501 4051 37633 394353

Number of solutions for $\mathrm{\pi \pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi }$: domains $0..n$

Length ($n$)2345678
Total31373501405137633394353
 Parameter value

037371211201504162161
1-612200210886224416
2--24601560525097776
3---1203601092062160
4----720252087360
5-----504020160
6------40320

Solution count for $\mathrm{\pi \pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi }$: domains $0..n$

related: $\mathrm{\pi \pi \pi \pi \pi \pi \pi }$Β (equivalence classes correspond to vertices in same path rather than variables assigned to the same value), $\mathrm{\pi \pi \pi \pi }$Β (do not care how many paths but how balanced the paths are).

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}$$\beta ¦\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi \pi \pi }\mathtt{1},\mathrm{\pi \pi \pi \pi \pi }\mathtt{2}\right)$

Arc arity
Arc constraint(s)
$\mathrm{\pi \pi \pi \pi \pi }\mathtt{1}.\mathrm{\pi \pi \pi \pi }=\mathrm{\pi \pi \pi \pi \pi }\mathtt{2}.\mathrm{\pi \pi \pi \pi \pi ‘}$
Graph property(ies)
 $\beta ’$$\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi }$$\beta €1$ $\beta ’$$\mathrm{\pi \pi \pi }_\mathrm{\pi \pi }$$\beta €1$ $\beta ’$$\mathrm{\pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi }$$=\mathrm{\pi ±\pi °\pi »\pi °\pi ½\pi ²\pi ΄}$

Graph class
$\mathrm{\pi Ύ\pi ½\pi ΄}_\mathrm{\pi \pi \pi ²\pi ²}$

Graph model

In order to express the binary constraint that links two vertices one has to make explicit the identifier of the vertices. This is why the $\mathrm{\pi \pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi }$ constraint considers objects that have two attributes:

• One fixed attribute $\mathrm{\pi \pi \pi \pi \pi ‘}$ that is the identifier of the vertex,

• One variable attribute $\mathrm{\pi \pi \pi \pi }$ that is the successor of the vertex.

We use the graph property $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi }$$\beta €1$ in order to specify the fact that the size of the largest strongly connected component should not exceed one. In fact each root of a tree is a strongly connected component with a single vertex. The graph property $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi }$$\beta €1$ constraints the maximum in-degree of the final graph to not exceed 1. $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi }$ does not consider loops: This is why we do not have any problem with the final node of each path.

PartsΒ (A) andΒ (B) of FigureΒ 5.48.2 respectively show the initial and final graph associated with the first example of the Example slot. Since we use the $\mathrm{\pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi }$ graph property, we show the connected components of the final graph. The constraint holds since all the vertices belong to a path and since $\mathrm{\pi ±\pi °\pi »\pi °\pi ½\pi ²\pi ΄}$ $=$ $\mathrm{\pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi }$ $=$ 3.