Necessary Condition for Path Partitioning Constraints Nicolas - - PowerPoint PPT Presentation

Necessary Condition for Path Partitioning Constraints

Nicolas Beldiceanu and Xavier Lorca

Ecole des Mines de Nantes, LINA FRE CNRS 2729, FR-44307 Cedex 3 {Nicolas.Beldiceanu,Xavier.Lorca}@emn.fr

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Context and Contribution

Graph partitioning constraints:

with paths [Quesada06] [Sellmann03]. with cycles [Hooker06]. with trees [Beldiceanu05].

Graph theory: K node-disjoint paths problem [Steiner03] [Vygen95]. Contribution:

structural conditions for partitioning a digraph by one or several paths.

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Applications

Vehicle routing Network robustness Mission planning

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Application: routing (1)

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Covering a set of tasks (e.g., deliveries, flights) by a minimum of resources (e.g., lorries, planes).

time

a b b a a b b c b b c d a a b d d d a c c d d c 3 9 1 2 4 8 7 10 5 6 11

Application: routing (2)

Graph modelling:

5 1 2 3 4 5 6 7 8 9 10 11 [a,b] [c,b] [d,c] [c,a] [b,a] [a,d] [d,a] [d,b] [d,c] [b,d] [b,c] [a,b]

A path partition with 6 paths.

Outline

• A flow based approach evaluating the minimum number of paths
• In the case of DAGs
• In general
• Improving the flow based approach
• According to SCCs of the digraph
• According to the way the nodes of 2 SCCs are inter-connected
• A path partitioning constraint
• Modelling the constraint
• Checking feasibility
• Filtering
• Conclusion and Open questions

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Evaluating the minimum number of paths in DAGs

The digraph G

1 3 4 5 6 7 2

Network associated to G

s 0' 0'' 1'' 1' 2' 2'' 3' 3'' 4'' 4' 5' 5'' 6'' 6' 7' 7'' t

• Each node of the DAG

is splitted: fi = [1,1]

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• A backward arc is

added: fts = [1,K]

Evaluating the minimum number of paths in DAGs

The digraph G

1 3 4 5 6 7 2

Network associated to G

s 0' 0'' 1'' 1' 2' 2'' 3' 3'' 4'' 4' 5' 5'' 6'' 6' 7' 7'' t

• The path extremities

arcs are added: fsi = [0,K], fit = [0,K]

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Evaluating the minimum number of paths in DAGs

The digraph G

1 3 4 5 6 7 2

Network associated to G

s 0' 0'' 1'' 1' 2' 2'' 3' 3'' 4'' 4' 5' 5'' 6'' 6' 7' 7'' t

• The DAGs arcs are

added: fij = [0,K]

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Evaluating the minimum number of paths in the case of DAGs

A digraph G can be partitioned in K node-disjoint paths iff there exist a feasible flow fts= K in the network associated to G.

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Outline

• A flow based approach evaluating the minimum number of paths
• In the case of DAGs
• In general
• Improving the flow based approach
• According to SCCs of the digraph
• According to the way the nodes of 2 SCCs are inter-connected
• A path partitioning constraint
• Modelling the constraint
• Checking feasibility
• Filtering
• Conclusion and Open questions

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A flow based approach for the K node-disjoint paths problem

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The reduced digraph Gr

1 3 4 5 6 7 2 1 3 2 4 5 6 7 9 8 10 12 11 13 14 15

The digraph G

A flow based approach for the K node-disjoint paths problem

• How to take into account the SCCs of G in the

flow?

• How to take into account the ways two nodes of

distinct SCCs are inter-connected?

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A flow based approach for the K node-disjoint paths problem

1 3 4 5 6 7 2

Network associated with Gr

s 0' 0'' 1'' 1' 2' 2'' 3' 3'' 4'' 4' 5' 5'' 6'' 6' 7' 7'' t

Reduced digraph Gr associated with G

• Splitted nodes: fi = [1,wi]

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A flow based approach for the K node-disjoint paths problem

1 3 4 5 6 7 2

Network associated with Gr

s 0' 0'' 1'' 1' 2' 2'' 3' 3'' 4'' 4' 5' 5'' 6'' 6' 7' 7'' t

Reduced digraph Gr associated with G

• inter-scc arcs: fij = [0,cij]

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A flow based approach for the K node-disjoint paths problem

Thus, the two key points of the flow based approach: Estimating the minimum number of paths partitioning each SCC of the digraph G. Estimating the way the nodes of two SCCs are inter-connected (i.e., path bottleneck).

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Outline

• A flow based approach evaluating the minimum number of paths
• In the case of DAGs
• In general
• Improving the flow based approach
• According to SCCs of the digraph
• According to the way the nodes of 2 SCCs are inter-connected
• A path partitioning constraint
• Modelling the constraint
• Checking feasibility
• Filtering
• Conclusion and Open questions

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Estimating the number of paths partitioning a SCC

18 1 2 3 5 6 4 13 7 12 11 9 10 8

A scc S of a digraph G

Estimating the number of paths partitioning a SCC

In the scc S of G... ...Node 13 dominates nodes {7,8,10,12} according to nodes {0,1,2,3,4,5,6}... ... The removal of node 13 creates 5 new SCCs.

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Estimating the number of paths partitioning a SCC

1 2 3 5 6 4 13 7 12 11 9 10 8 20

Estimating the number of paths partitioning a SCC

Among the new SCCs, two kinds are distinguished: Those from which final nodes can be reached (Δout). The others (Δin).

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Estimating the number of paths partitioning a SCC

1 2 3 5 6 4 13 7 12 11 9 10 8 22

Δin Δout

Estimating the number of paths partitioning a SCC

If there exists a path partition of G then, there exists a Hamiltonian path in Δin, finishing on the dominator node 13.

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Thus, the minimum number of paths partitioning the scc S is provided by the minimum number of paths partitioning Δout (i.e., 3 paths)... ... in fact, not exactly, assume the following partition...

Estimating the number of paths partitioning a SCC

1 2 3 5 6 4 13 7 12 11 9 10 8 24

Δin Δout

scc7 scc12

Estimating the number of paths partitioning a SCC

Let Pham be the Hamiltonian path covering Δin, there exists a path P, from scc7 to scc12, in S such that: P = scc7 U Pham U {13} U scc12 Thus, at least 2 paths are necessary to partition scc S of the digraph G.

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Estimating the number of paths partitioning a SCC

Let i be a node of the reduced digraph Gr associated to the digraph G. Let li be the minimum number of node-disjoint paths partitioning the SCC scci of G. Let Di be the set of dominators nodes of scci. Let li

d be the lower bound on the minimum

number of paths partitioning scci according to the dominator node d of Di.

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Estimating the number of paths partitioning a SCC

A lower bound of the flow going through a node splitted arc of the network N associated with G: 1 ≤ max({li

d | d in Di}) ≤ li 27

Outline

• A flow based approach evaluating the minimum number of paths
• In the case of DAGs
• In general
• Improving the flow based approach
• According to SCCs of the digraph
• According to the way the nodes of 2 SCCs are

inter-connected

• A path partitioning constraint
• Modelling the constraint
• Checking feasibility
• Filtering
• Conclusion and Open questions

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Estimating the number of paths bewteen two SCCs

1 6 5 4 3 2 10 9 8 7 11 13 12

scc1 scc2

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Estimating the number of paths bewteen two SCCs

The maximum number of paths going from scc1 to scc2 can be computed by a maximum matching. The bipartite graph is provided by the nodes incidents to the arcs emanating from scc1 and entering scc2.

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Estimating the number of paths bewteen two SCCs

scc1 scc2

6 5 4 3 10 9 8 7 31

Estimating the number of paths bewteen two SCCs

At most 3 distinct paths can connect scc1

to scc2. scc1 and scc2 can be partitionned in respectively at least 4 distinct paths. Globally, the digraph can be partitionned in at

most 5 paths.

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Estimating the number of paths bewteen two SCCs

1 6 5 4 3 2 10 9 8 7 11 13 12

scc1 scc2

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Estimating the number of paths bewteen two SCCs

The capacity cij going through an inter-scc arc (i'',j') of the network N associated with a digraph G is bounded by the size of maximum matching between scci and sccj.

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Outline

• A flow based approach evaluating the minimum number of paths
• In the case of DAGs
• In general
• Improving the flow based approach
• According to SCCs of the digraph
• According to the way the nodes of 2 SCCs are inter-connected
• A path partitioning constraint
• Modelling the constraint
• Checking feasibility
• Filtering
• Conclusion and Open questions

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Modelling a graph constraint

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path(NPATH, VERTICES)

An integer variable: number of trees to cover the graph. A collection of n vertices, with the following attributes:

• index: integer between 1 and n.
• father: integer variable whose

domain is subset of the values of [1,n].

• degree: integer variable whose

domain is [0,1].

Checking feasibility

If there is a solution to a path constraint then: there exists a feasible flow in the network N associated with G. for each SCC scci of G, for each dominator node d of scci there exist at most one path partitioning Δin.

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Filtering a path constraint

Adjust the minimum value of the number of paths allowed to partition G with the value of a minimum feasible flow in the network N associated with G. Filter each SCC of G according to dominators nodes detected.

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Filtering a path constraint

1 2 3 5 6 4 13 7 12 11 9 10 8 39

Δin Δout

Open questions

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In this approach, each dominator node is considered independently from the other... ... Do there exist an interaction between them? ... How to take into account of this interaction? How to efficiently detect the arcs of the network N associated with G that do not belong to any feasible flow?

Conclusion

Structural conditions for path partitioning constraints which are independent from the number of paths. Combine structure of each SCC and the way they are inter-connected. Implementation on its way (integrated in the tree constraint). The dominator nodes [Quesada06] provide a lot of useful structural information also in the context of multiple paths.

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