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Homework Sheet 1: Minimum Dominating Set (MDS)

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Homework Sheet 1: Minimum Dominating Set (MDS)

Due Date: 30-04-2019

Sequential Algorithms

1. Recall the greedy sequential algorithm for Dominating Set , and let S be the solution output by it. Construct an example where |S |S|

∗

| ≥ log(∆), where S ∗ is an optimal solution and ∆ is the maximum degree.

2. Extend that the greedy sequential algorithm to the weighted case. Here, the input is a graph G(V, E) and a weight function w : V → N , and we want to compute a minimum weight dominating set of G. Show that this algorithm is a (ln(∆+1))+1-factor approximation to the optimal solution.

Distributed Algorithms

3. Consider the Dominating Set problem on trees. Design a distributed algorithm that outputs a constant factor approximation, and terminates in constantly many rounds.

4. (Bonous Exercise) Let G = (V, E) be a graph, G 2 = (V, E(G 2 )) is defined as a graph with vertex set V (G 2 ) = V (G) and the edge set

E(G 2 ) = {{u, v} | distance(u,v) in G is at most 2}.

The graph G 2 is called power of the graph G. Provide a local algorithm that runs in constant number of rounds and outputs a constant factor approximation for the MDS in power graph of trees (in T 2 ).

1

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