On the Algorithmic Effectiveness of Digraph Decompositions and - - PowerPoint PPT Presentation

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On the Algorithmic Effectiveness of Digraph Decompositions and - - PowerPoint PPT Presentation

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On the Algorithmic Effectiveness of Digraph Decompositions and Complexity Measures Michael Lampis, Georgia Kaouri and Valia Mitsou ISAAC 2008 p. 1/18 Graph decompositions Treewidth (by Robertson and Seymour) is the most well-known and

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On the Algorithmic Effectiveness

  • f Digraph Decompositions and

Complexity Measures

Michael Lampis, Georgia Kaouri and Valia Mitsou

ISAAC 2008 – p. 1/18

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SLIDE 2

Graph decompositions

Treewidth (by Robertson and Seymour) is the most well-known and widely studied graph decomposition. Treewidth describes how much a graph looks like a tree. A large number of graph problems can be solved efficiently (in FPT time) for low

  • treewidth. (Courcelle’s theorem)

Many equivalent definitions (e.g. cops-and-robber games, minimum fill-in, elimination orderings).

ISAAC 2008 – p. 2/18

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Digraph decompositions

Treewidth is generally considered the right measure for undirected graphs. Treewidth can usually be employed for digraph problems as well: take the tree decomposition of the underlying undirected graph. This solution is not perfect. E.g. ignoring the direction of edges on a DAG may lead to a clique (large treewidth). But the problem may be trivial on DAGs (e.g. Hamiltonian Cycle).

ISAAC 2008 – p. 3/18

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Digraph decompositions

What is the right treewidth analogue for digraphs? Directed treewidth [Johnson et al., 2001] DAG-width [Obdrzálek, 2006] Kelly-width [Hunter and Kreutzer, 2007]

ISAAC 2008 – p. 4/18

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Relations between measures

Directed Treewidth DAG-width Kelly-width Directed pathwidth Cycle rank Hardness results Algorithms

ISAAC 2008 – p. 5/18

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Known results

An O(nk) algorithm for Hamiltonian Cycle where k is the directed treewidth. [Johnson et al., 2001] An O(nk) algorithm for parity games where k is the DAG-width [Obdrzálek, 2006] A O(nk) algorithms for both where k is the kelly-width [Hunter and Kreutzer, 2007] No FPT algorithms are known!

ISAAC 2008 – p. 6/18

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Our results

MaxDiCut is NP-complete when restricted to DAGs Hamiltonian Cycle is W[2]-hard when the parameter is the cycle rank of the input graph. Implication: Both problems are intractable for all considered complexity measures.

ISAAC 2008 – p. 7/18

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Hamiltonian cycle

Reduction from Dominating Set. We are given an undirected graph G and a number k. Does G have a dominating set of size k? Construct a digraph G′. G′ will be Hamiltonian iff G has a dominating set of size k. G′ will have small width (a function of k) under all definitions.

ISAAC 2008 – p. 8/18

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The reduction

Size Choice Satisfaction

Construction has three parts.

ISAAC 2008 – p. 9/18

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The reduction

Size Choice Satisfaction

Construction has three parts. The first part makes sure that G′ can only be Hamiltonian iff I pick a dominating set of size k.

ISAAC 2008 – p. 9/18

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The reduction

Choice Satisfaction ... k vertices

This is accomplished by using exactly k vertices.

ISAAC 2008 – p. 9/18

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The reduction

Choice Satisfaction ... k vertices

The second part represents a choice of dominating set.

ISAAC 2008 – p. 9/18

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The reduction

Satisfaction ... k vertices ... n-cycle

This is accomplished by using an n-cycle. The exit points from the cycle correspond to vertices in the dominating set.

ISAAC 2008 – p. 9/18

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The reduction

Satisfaction ... k vertices ... n-cycle

Finally, the third part makes sure that the choice is indeed a dominating set.

ISAAC 2008 – p. 9/18

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The reduction

... k vertices ... n-cycle

G1 G2 G3 Gn

... n gadgets

This is accomplished by placing a gadget to check domination for each vertex of G.

ISAAC 2008 – p. 9/18

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Example

1 4 5 6 2 3

Suppose that we want to see if this graph has a dominating set of size 2.

ISAAC 2008 – p. 10/18

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Example

1 4 5 6 2 3

2 vertices 6-cycle

G1 G2

6 gadgets

G3 G4 G5 G6

ISAAC 2008 – p. 11/18

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The satisfaction gadget

1 4 5 6 2 3

1 2 3 5 In Out G1

Vertex 1 can be dominated in 4 ways: by picking 1,2,3 or 5. The gadget G1 will have 4 inputs and 4

  • utputs.

ISAAC 2008 – p. 12/18

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The satisfaction gadget

1 4 5 6 2 3

1 2 3 5 In Out G1

For each input/output point use one vertex.

ISAAC 2008 – p. 12/18

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The satisfaction gadget

1 4 5 6 2 3

1 2 3 5 In Out G1

Connect them in a directed cycle.

ISAAC 2008 – p. 12/18

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The satisfaction gadget

1 4 5 6 2 3

1 2 3 5 In Out G1

This makes any Hamiltonian tour of the gadget exit from the same set of outputs it entered.

ISAAC 2008 – p. 12/18

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The satisfaction gadget

1 4 5 6 2 3

1 2 3 5 In Out G1

Example: Entering through input point 1.

ISAAC 2008 – p. 12/18

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The satisfaction gadget

1 4 5 6 2 3

1 2 3 5 In Out G1

Entering through input points 1 and 3.

ISAAC 2008 – p. 12/18

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The satisfaction gadget

1 4 5 6 2 3

1 2 3 5 In Out G1

Why this is important: The gadgets maintain the choices made in the second part of the graph.

ISAAC 2008 – p. 12/18

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Full example

1 2 3 4 5 6 1 1 2 2 3 3 4 5 5 6 In Out In Out In Out 1 2 3 5 4 5 6 In Out In Out In Out

Full construction. G′ has Hamiltonian cycle for dominating set 2, 5.

ISAAC 2008 – p. 13/18

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Full example

1 2 3 4 5 6 1 1 2 2 3 3 4 5 5 6 In Out In Out In Out 1 2 3 5 4 5 6 In Out In Out In Out

Full construction. G′ has Hamiltonian cycle for dominating set 2, 5.

ISAAC 2008 – p. 13/18

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Full example

1 2 3 4 5 6 1 1 2 2 3 3 4 5 5 6 In Out In Out In Out 1 2 3 5 4 5 6 In Out In Out In Out

Full construction. G′ has Hamiltonian cycle for dominating set 2, 5.

ISAAC 2008 – p. 13/18

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Completing the proof

What remains is to show that G′ has small width. If we remove the k vertices of the first part, we are left with an ordered set of n + 1 directed cycles. Each of these has small width.

ISAAC 2008 – p. 14/18

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Summary of results

Hamiltonian Cycle MaxDiCut Treewidth FPT FPT

  • Dir. Treewidth

XP DAG-width XP Kelly-width XP

  • Dir. Pathwidth

XP Cycle rank XP

ISAAC 2008 – p. 15/18

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Summary of results

Hamiltonian Cycle MaxDiCut Treewidth FPT FPT

  • Dir. Treewidth

XP DAG-width XP Kelly-width XP

  • Dir. Pathwidth

XP Cycle rank XP W[2]-hard NP-complete

ISAAC 2008 – p. 15/18

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Summary of results

Hamiltonian Cycle MaxDiCut Treewidth FPT FPT

  • Dir. Treewidth

XP W[2]-hard NP-complete DAG-width XP W[2]-hard NP-complete Kelly-width XP W[2]-hard NP-complete

  • Dir. Pathwidth

XP W[2]-hard NP-complete Cycle rank XP W[2]-hard NP-complete

ISAAC 2008 – p. 15/18

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Conclusion

Currently known digraph decompositions don’t work as well as treewidth. Why? Perhaps DAGs are not a good starting point. Perhaps different cops-and-robber games could reveal something interesting. What if we allow the robber to move backwards sometimes? Finding a good treewidth for digraphs is an interesting open problem.

ISAAC 2008 – p. 16/18

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Thank You!

ISAAC 2008 – p. 17/18

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References

[Hunter and Kreutzer, 2007] Hunter, P . and Kreutzer, S. (2007). Digraph measures: Kelly decompositions, games, and orderings. In Bansal, N., Pruhs, K., and Stein, C., editors, SODA, pages 637–644. SIAM. [Johnson et al., 2001] Johnson, T., Robertson, N., Seymour, P . D., and Thomas, R. (2001). Directed tree-width. J. Comb. Theory, Ser. B, 82(1):138–154. [Obdrzálek, 2006] Obdrzálek, J. (2006). Dag-width: connectivity measure for directed graphs. In SODA, pages 814–821. ACM Press.

ISAAC 2008 – p. 18/18