5.112. dom_reachability

DESCRIPTIONLINKS
Origin

[QuesadaVanRoyDevilleCollet06]

Constraint

πšπš˜πš–_πš›πšŽπšŠπšŒπš‘πšŠπš‹πš’πš•πš’πšπš’πš‚π™Ύπš„πšπ™²π™΄,π™΅π™»π™Ύπš†_π™Άπšπ™°π™Ώπ™·,π™³π™Ύπ™Όπ™Έπ™½π™°πšƒπ™Ύπš_π™Άπšπ™°π™Ώπ™·,πšƒπšπ™°π™½πš‚π™Έπšƒπ™Έπš…π™΄_π™²π™»π™Ύπš‚πš„πšπ™΄_π™Άπšπ™°π™Ώπ™·

Arguments
πš‚π™Ύπš„πšπ™²π™΄πš’πš—πš
π™΅π™»π™Ύπš†_π™Άπšπ™°π™Ώπ™·πšŒπš˜πš•πš•πšŽπšŒπšπš’πš˜πš—(πš’πš—πšπšŽπš‘-πš’πš—πš,𝚜𝚞𝚌𝚌-πšœπšŸπšŠπš›)
π™³π™Ύπ™Όπ™Έπ™½π™°πšƒπ™Ύπš_π™Άπšπ™°π™Ώπ™·πšŒπš˜πš•πš•πšŽπšŒπšπš’πš˜πš—(πš’πš—πšπšŽπš‘-πš’πš—πš,𝚜𝚞𝚌𝚌-πšœπš’πš—πš)
πšƒπšπ™°π™½πš‚π™Έπšƒπ™Έπš…π™΄_π™²π™»π™Ύπš‚πš„πšπ™΄_π™Άπšπ™°π™Ώπ™·πšŒπš˜πš•πš•πšŽπšŒπšπš’πš˜πš—(πš’πš—πšπšŽπš‘-πš’πš—πš,𝚜𝚞𝚌𝚌-πšœπšŸπšŠπš›)
Restrictions
πš‚π™Ύπš„πšπ™²π™΄β‰₯1
πš‚π™Ύπš„πšπ™²π™΄β‰€|π™΅π™»π™Ύπš†_π™Άπšπ™°π™Ώπ™·|
πš›πšŽπššπšžπš’πš›πšŽπš(π™΅π™»π™Ύπš†_π™Άπšπ™°π™Ώπ™·,[πš’πš—πšπšŽπš‘,𝚜𝚞𝚌𝚌])
π™΅π™»π™Ύπš†_π™Άπšπ™°π™Ώπ™·.πš’πš—πšπšŽπš‘β‰₯1
π™΅π™»π™Ύπš†_π™Άπšπ™°π™Ώπ™·.πš’πš—πšπšŽπš‘β‰€|π™΅π™»π™Ύπš†_π™Άπšπ™°π™Ώπ™·|
π™΅π™»π™Ύπš†_π™Άπšπ™°π™Ώπ™·.𝚜𝚞𝚌𝚌β‰₯1
π™΅π™»π™Ύπš†_π™Άπšπ™°π™Ώπ™·.πšœπšžπšŒπšŒβ‰€|π™΅π™»π™Ύπš†_π™Άπšπ™°π™Ώπ™·|
πšπš’πšœπšπš’πš—πšŒπš(π™΅π™»π™Ύπš†_π™Άπšπ™°π™Ώπ™·,πš’πš—πšπšŽπš‘)
πš›πšŽπššπšžπš’πš›πšŽπš(π™³π™Ύπ™Όπ™Έπ™½π™°πšƒπ™Ύπš_π™Άπšπ™°π™Ώπ™·,[πš’πš—πšπšŽπš‘,𝚜𝚞𝚌𝚌])
|π™³π™Ύπ™Όπ™Έπ™½π™°πšƒπ™Ύπš_π™Άπšπ™°π™Ώπ™·|=|π™΅π™»π™Ύπš†_π™Άπšπ™°π™Ώπ™·|
π™³π™Ύπ™Όπ™Έπ™½π™°πšƒπ™Ύπš_π™Άπšπ™°π™Ώπ™·.πš’πš—πšπšŽπš‘β‰₯1
π™³π™Ύπ™Όπ™Έπ™½π™°πšƒπ™Ύπš_π™Άπšπ™°π™Ώπ™·.πš’πš—πšπšŽπš‘β‰€|π™³π™Ύπ™Όπ™Έπ™½π™°πšƒπ™Ύπš_π™Άπšπ™°π™Ώπ™·|
π™³π™Ύπ™Όπ™Έπ™½π™°πšƒπ™Ύπš_π™Άπšπ™°π™Ώπ™·.𝚜𝚞𝚌𝚌β‰₯1
π™³π™Ύπ™Όπ™Έπ™½π™°πšƒπ™Ύπš_π™Άπšπ™°π™Ώπ™·.πšœπšžπšŒπšŒβ‰€|π™³π™Ύπ™Όπ™Έπ™½π™°πšƒπ™Ύπš_π™Άπšπ™°π™Ώπ™·|
πšπš’πšœπšπš’πš—πšŒπš(π™³π™Ύπ™Όπ™Έπ™½π™°πšƒπ™Ύπš_π™Άπšπ™°π™Ώπ™·,πš’πš—πšπšŽπš‘)
πš›πšŽπššπšžπš’πš›πšŽπš(πšƒπšπ™°π™½πš‚π™Έπšƒπ™Έπš…π™΄_π™²π™»π™Ύπš‚πš„πšπ™΄_π™Άπšπ™°π™Ώπ™·,[πš’πš—πšπšŽπš‘,𝚜𝚞𝚌𝚌])
|πšƒπšπ™°π™½πš‚π™Έπšƒπ™Έπš…π™΄_π™²π™»π™Ύπš‚πš„πšπ™΄_π™Άπšπ™°π™Ώπ™·|=|π™΅π™»π™Ύπš†_π™Άπšπ™°π™Ώπ™·|
πšƒπšπ™°π™½πš‚π™Έπšƒπ™Έπš…π™΄_π™²π™»π™Ύπš‚πš„πšπ™΄_π™Άπšπ™°π™Ώπ™·.πš’πš—πšπšŽπš‘β‰₯1
πšƒπšπ™°π™½πš‚π™Έπšƒπ™Έπš…π™΄_π™²π™»π™Ύπš‚πš„πšπ™΄_π™Άπšπ™°π™Ώπ™·.πš’πš—πšπšŽπš‘β‰€|πšƒπšπ™°π™½πš‚π™Έπšƒπ™Έπš…π™΄_π™²π™»π™Ύπš‚πš„πšπ™΄_π™Άπšπ™°π™Ώπ™·|
πšƒπšπ™°π™½πš‚π™Έπšƒπ™Έπš…π™΄_π™²π™»π™Ύπš‚πš„πšπ™΄_π™Άπšπ™°π™Ώπ™·.𝚜𝚞𝚌𝚌β‰₯1
πšƒπšπ™°π™½πš‚π™Έπšƒπ™Έπš…π™΄_π™²π™»π™Ύπš‚πš„πšπ™΄_π™Άπšπ™°π™Ώπ™·.πšœπšžπšŒπšŒβ‰€|πšƒπšπ™°π™½πš‚π™Έπšƒπ™Έπš…π™΄_π™²π™»π™Ύπš‚πš„πšπ™΄_π™Άπšπ™°π™Ώπ™·|
πšπš’πšœπšπš’πš—πšŒπš(πšƒπšπ™°π™½πš‚π™Έπšƒπ™Έπš…π™΄_π™²π™»π™Ύπš‚πš„πšπ™΄_π™Άπšπ™°π™Ώπ™·,πš’πš—πšπšŽπš‘)
Purpose

Let π™΅π™»π™Ύπš†_π™Άπšπ™°π™Ώπ™·, π™³π™Ύπ™Όπ™Έπ™½π™°πšƒπ™Ύπš_π™Άπšπ™°π™Ώπ™· and πšƒπšπ™°π™½πš‚π™Έπšƒπ™Έπš…π™΄_π™²π™»π™Ύπš‚πš„πšπ™΄_π™Άπšπ™°π™Ώπ™· be three directed graphs respectively called the flow graph, the dominance graph and the transitive closure graph which all have the same vertices. In addition let πš‚π™Ύπš„πšπ™²π™΄ denote a vertex of the flow graph called the source node (not necessarily a vertex with no incoming arcs). The πšπš˜πš–_πš›πšŽπšŠπšŒπš‘πšŠπš‹πš’πš•πš’πšπš’ constraint holds if and only if the flow graph (and its source node) verifies:

  • The dominance relation expressed by the dominance graph (i.e.,Β if there is an arc (i,j) in the dominance graph then, within the flow graph, all the paths from the source node to j contain i; note that when there is no path from the source node to j then any node dominates j).

  • The transitive relation expressed by the transitive closure graph (i.e.,Β if there is an arc (i,j) in the transitive closure graph then there is also a path from i to j in the flow graph).

Example
1,πš’πš—πšπšŽπš‘-1𝚜𝚞𝚌𝚌-{2},πš’πš—πšπšŽπš‘-2𝚜𝚞𝚌𝚌-{3,4},πš’πš—πšπšŽπš‘-3𝚜𝚞𝚌𝚌-βˆ…,πš’πš—πšπšŽπš‘-4𝚜𝚞𝚌𝚌-βˆ…,πš’πš—πšπšŽπš‘-1𝚜𝚞𝚌𝚌-{2,3,4},πš’πš—πšπšŽπš‘-2𝚜𝚞𝚌𝚌-{3,4},πš’πš—πšπšŽπš‘-3𝚜𝚞𝚌𝚌-βˆ…,πš’πš—πšπšŽπš‘-4𝚜𝚞𝚌𝚌-βˆ…,πš’πš—πšπšŽπš‘-1𝚜𝚞𝚌𝚌-{1,2,3,4},πš’πš—πšπšŽπš‘-2𝚜𝚞𝚌𝚌-{2,3,4},πš’πš—πšπšŽπš‘-3𝚜𝚞𝚌𝚌-{3},πš’πš—πšπšŽπš‘-4𝚜𝚞𝚌𝚌-{4}

The flow graph, the dominance graph and the transitive closure graph corresponding to the second, third and fourth arguments of the πšπš˜πš–_πš›πšŽπšŠπšŒπš‘πšŠπš‹πš’πš•πš’πšπš’ constraint are respectively depicted by parts (A), (B) and (C) of FigureΒ 5.112.1. The πšπš˜πš–_πš›πšŽπšŠπšŒπš‘πšŠπš‹πš’πš•πš’πšπš’ holds since the following conditions hold.

  • The dominance relation expressed by the dominance graph is verified:

    • Since (1,2) belongs to the dominance graph all the paths from 1 to 2 in the flow graph pass through 1.

    • Since (1,3) belongs to the dominance graph all the paths from 1 to 3 in the flow graph pass through 1.

    • Since (1,4) belongs to the dominance graph all the paths from 1 to 4 in the flow graph pass through 1.

    • Since (2,3) belongs to the dominance graph all the paths from 1 to 3 in the flow graph pass through 2.

    • Since (2,4) belongs to the dominance graph all the paths from 1 to 4 in the flow graph pass through 2.

  • The graph depicted by the fourth argument of the πšπš˜πš–_πš›πšŽπšŠπšŒπš‘πšŠπš‹πš’πš•πš’πšπš’ constraint (i.e.,Β πšƒπšπ™°π™½πš‚π™Έπšƒπ™Έπš…π™΄_π™²π™»π™Ύπš‚πš„πšπ™΄_π™Άπšπ™°π™Ώπ™·) is the transitive closure of the graph depicted by the second argument (i.e.,Β π™΅π™»π™Ύπš†_π™Άπšπ™°π™Ώπ™·).

Figure 5.112.1. (A) Flow graph, (B) dominance graph and (C) transitive closure graph of the Example slot (taken fromΒ [Quesada06])
ctrs/dom_reachability1
Typical
|π™΅π™»π™Ύπš†_π™Άπšπ™°π™Ώπ™·|>2
Symmetries
  • Items of π™΅π™»π™Ύπš†_π™Άπšπ™°π™Ώπ™· are permutable.

  • Items of π™³π™Ύπ™Όπ™Έπ™½π™°πšƒπ™Ύπš_π™Άπšπ™°π™Ώπ™· are permutable.

  • Items of πšƒπšπ™°π™½πš‚π™Έπšƒπ™Έπš…π™΄_π™²π™»π™Ύπš‚πš„πšπ™΄_π™Άπšπ™°π™Ώπ™· are permutable.

Usage

The πšπš˜πš–_πš›πšŽπšŠπšŒπš‘πšŠπš‹πš’πš•πš’πšπš’ constraint was introduced in order to solve reachability problems (e.g., disjoint paths, simple path with mandatory nodes).

Remark

Within the name πšπš˜πš–_πš›πšŽπšŠπšŒπš‘πšŠπš‹πš’πš•πš’πšπš’, πšπš˜πš– stands for domination. In the context of path problems πš‚π™Ύπš„πšπ™²π™΄ refers to the start of the path we want to build.

Algorithm

It was shown inΒ [Quesada06] that, finding out wether a πšπš˜πš–_πš›πšŽπšŠπšŒπš‘πšŠπš‹πš’πš•πš’πšπš’ constraint has a solution or not is NP-hard. This was achieved by reduction to disjoint paths problemΒ [GareyJohnson79].

The first implementationΒ [QuesadaVanRoyDeville05] of the πšπš˜πš–_πš›πšŽπšŠπšŒπš‘πšŠπš‹πš’πš•πš’πšπš’ constraint was done in MozartΒ [Mozart06]. Later on, a second implementionΒ [Quesada06] was done in GecodeΒ [Gecode06]. Both implementations consist of the following two parts:

  • AlgorithmsΒ [Roditty03] for maintaining the lower bound of the transitive closure graph.

  • Algorithms for maintaining the upper bound of the transitive closure graph, while respecting the dominance constraintsΒ [Georgiadis05].

See also

common keyword: πš™πšŠπšπš‘, πš™πšŠπšπš‘_πšπš›πš˜πš–_𝚝𝚘 (path).

Keywords

combinatorial object: path.

constraint arguments: constraint involving set variables.

constraint type: predefined constraint, graph constraint.