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 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
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
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
Relations between measures Directed Treewidth Algorithms DAG-width Kelly-width Directed pathwidth Cycle rank Hardness results ISAAC 2008 – p. 5/18
Known results An O ( n k ) algorithm for Hamiltonian Cycle where k is the directed treewidth. [Johnson et al., 2001] An O ( n k ) algorithm for parity games where k is the DAG-width [Obdrzálek, 2006] A O ( n k ) algorithms for both where k is the kelly-width [Hunter and Kreutzer, 2007] No FPT algorithms are known! ISAAC 2008 – p. 6/18
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
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
The reduction Size Choice Satisfaction Construction has three parts. ISAAC 2008 – p. 9/18
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
The reduction Choice Satisfaction ... k vertices This is accomplished by using exactly k vertices. ISAAC 2008 – p. 9/18
The reduction Choice Satisfaction ... k vertices The second part represents a choice of dominating set. ISAAC 2008 – p. 9/18
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
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
The reduction ... G1 G3 G2 Gn ... ... k vertices n gadgets n-cycle This is accomplished by placing a gadget to check domination for each vertex of G . ISAAC 2008 – p. 9/18
Example 2 1 3 5 4 6 Suppose that we want to see if this graph has a dominating set of size 2. ISAAC 2008 – p. 10/18
Example G5 G1 G2 G3 G6 G4 2 1 3 2 vertices 5 6 gadgets 6-cycle 4 6 ISAAC 2008 – p. 11/18
The satisfaction gadget In Out 1 2 2 3 1 5 3 5 G1 4 6 Vertex 1 can be dominated in 4 ways: by picking 1,2,3 or 5. The gadget G 1 will have 4 inputs and 4 outputs. ISAAC 2008 – p. 12/18
The satisfaction gadget In Out 1 2 2 3 1 5 3 5 G1 4 6 For each input/output point use one vertex. ISAAC 2008 – p. 12/18
The satisfaction gadget In Out 1 2 2 3 1 5 3 5 G1 4 6 Connect them in a directed cycle. ISAAC 2008 – p. 12/18
The satisfaction gadget In Out 1 2 2 3 1 5 3 5 G1 4 6 This makes any Hamiltonian tour of the gadget exit from the same set of outputs it entered. ISAAC 2008 – p. 12/18
The satisfaction gadget In Out 1 2 2 3 1 5 3 5 G1 4 6 Example: Entering through input point 1. ISAAC 2008 – p. 12/18
The satisfaction gadget In Out 1 2 2 3 1 5 3 5 G1 4 6 Entering through input points 1 and 3. ISAAC 2008 – p. 12/18
The satisfaction gadget In Out 1 2 2 3 1 5 3 5 G1 4 6 Why this is important: The gadgets maintain the choices made in the second part of the graph. ISAAC 2008 – p. 12/18
Full example In Out 1 1 In Out In Out 1 1 2 2 2 2 3 3 3 In Out 3 4 5 In Out 5 4 5 4 In Out 5 5 6 6 6 Full construction. G ′ has Hamiltonian cycle for dominating set 2 , 5 . ISAAC 2008 – p. 13/18
Full example In Out 1 1 In Out In Out 1 1 2 2 2 2 3 3 3 In Out 3 4 5 In Out 5 4 5 4 In Out 5 5 6 6 6 Full construction. G ′ has Hamiltonian cycle for dominating set 2 , 5 . ISAAC 2008 – p. 13/18
Full example In Out 1 1 In Out In Out 1 1 2 2 2 2 3 3 3 In Out 3 4 5 In Out 5 4 5 4 In Out 5 5 6 6 6 Full construction. G ′ has Hamiltonian cycle for dominating set 2 , 5 . ISAAC 2008 – p. 13/18
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
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
Summary of results Hamiltonian Cycle MaxDiCut Treewidth FPT FPT Dir. Treewidth XP DAG-width XP Kelly-width XP Dir. Pathwidth XP XP W [2] -hard Cycle rank NP-complete ISAAC 2008 – p. 15/18
Summary of results Hamiltonian Cycle MaxDiCut Treewidth FPT FPT Dir. Treewidth XP W [2] -hard NP-complete XP W [2] -hard DAG-width NP-complete XP W [2] -hard Kelly-width NP-complete XP W [2] -hard Dir. Pathwidth NP-complete XP W [2] -hard Cycle rank NP-complete ISAAC 2008 – p. 15/18
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
Thank You! ISAAC 2008 – p. 17/18
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
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