Dominating Set of Fixed Size in Degenerated Graphs

2009-12-15

Abner Huang

Outline

• Materials• Terminologies• Algorithms in Degenerated Graphs– Domination– Connected Domination– Dominating Threshold Set

• Ref.: Alon, N. & Gutner, S., Linear Time Algorithms for Finding a Dominating Set of Fixed Size in Degenerated Graphs, ALGORITHMICA, 2009, 54, 544-556

Baumritter Professor of Mathematics and Computer Science in Tel Aviv University, Israel.

• Golovach, P. & Villanger, Y.Parameterized Complexity for Domination Problems on Degenerate GraphsGraph-Theoretic Concepts in Computer Science, Graph-Theoretic Concepts in Computer Science: 34th International Workshop, WG 2008, Durham, UK, June 30 -- July 2, 2008. Revised Papers, 2008, 5344, 195-205

What do we skip?

• minors,• Topological minors,• Induced cycles,• some computational complexity results.

Fixed-Parameter Tractable (FPT)

• A parameterized problem L is fixed-parameter tractable if the question “(x,k) in L” can be decided in running time f(k)nO(1), where f is an arbitrary function depending only on k. The corresponding complexity class is called FPT.

d-degenerated graphs

• A graph G is d-degenerated if every induced subgraph of G has a vertex of degree at most d.– E.g., Cactus graphs– A graph contains (d+2)-clique will not be d-

degenerated.

Black and White Graphs

• The input is a black and white graph, which simply means that the vertex set V of the graph G has been partitioned into two disjoint sets B and W of black and white vertices, respectively, i.e., V = B $ W, where $ denotes disjoint set union. Given a black and white graph G = (B $ W,E) and an integer k,

Properties

• A d-degenerated graph with n vertices has less than dn edges and therefore its average degree is less than 2d.

• Every k-degenerated graph G = (V ,E) admits an acyclic orientation such that the outdegree of each vertex is at most d.

• Many interesting families of graphs are degenerated (have bounded degeneracy). For example, degree-bounded graphs, graphs of bounded tree-width.

Outline

• Materials• Terminologies• Algorithms in Degenerated Graphs– Domination– Connected Domination– Dominating Threshold Set

Main Idea of The Algorithm

• Search Tree method.– Its algorithm shows the reason to prove this

theorem, i.e., they try to find a suitable threshold to traverse the search tree. If we have the suitable threshold, we can bound the time complexity.

Branch and Bound

Threshold: Linear Median Algorithm

Main Idea

If |B| < (4d +2)kDo Exhaustive

Search.Else

Branch the tree

Magic number

The most important combinatorial result

• Lemma 1 Let G = (B U W,E) be a d-degenerated black and white graph. If |B| is larger than (4d +2)k, then there are at most (4d +2)k vertices in G that dominate at least |B|/k vertices of B.

Proof

Outline

• Materials• Terminologies• Algorithms in Degenerated Graphs– Domination– Connected Domination– Dominating Threshold Set

• Connected k-Dominating Set

• k-Dominating threshold set

Connected k-Dominating Set

• The initial problem instance for a graph G will be (S = ,W = ,B = V, q = k).∅ ∅

• As long as |B| > (4d+2)q, add a leaf (S {u},W (N(u)∩B)\{u},B\N[u], q−1) to the ∪ ∪problem instance (S,W,B, q) for every vertex u

W B where |N[u] ∩ B| ≥ |B|/q.∈ ∪• Like the algorithm for dominating set on d-

degenerate graphs.

k-Dominating threshold set

• Similarly, While |B| ≥ (4d+2)q, we branch and create one new problem instance (S {u},W ∪

U,B\ U, q−1) for ∪ each vertex u (W B) \ S ∈ ∪that dominates at least |B|/q vertices in B, where U is the set of vertices in N[u] ∩ B that are dominated by r − 1 vertices in S.

Thank You