The Dominating Set Problem: (a few) variants and results

Transcription

1 The Dominating Set Problem: (a few) variants and results Gerasimos G. Pollatos 23 November 2007

2 Definition and preliminaries Input: Graph G = (V, E), V = n and E = m. Output: Subset DS of V s.t. each vertex of G either belongs to DS or has a neighbor in DS. Cardinality of DS is minimum. Complexity: Status: NP-complete (the decision version). Bad news: not-approximable within (1 ǫ)c log n for any ǫ > 0. Good news: O(log n)-approximable (via Set Cover), admits a PTAS for planar and unit disk graphs.

3 The famous algorithm The classic algorithm for Set Cover (Lóvasz 1975, Johnson 1974). Why? Because DS problem is a special instance of set cover. Set Cover A universe U of elements and a collection C of subsets of U. Find the minimum number of members of C whose union is U. Dominating Set Set Cover For each u V create S u = u N(u). U is V and C = u V S u.

4 The famous algorithm (cont.) The algorithm: SOL While U is not empty Choose S v from C with maximum S v U. Add S v to solution SOL. U U S v Return SOL. Simple greedy idea and yet, approximation is O(log n) The best possible! Running time is also fast! (O(nm 2 ))

5 Connected Dominating Set Input: Connected graph G = (V, E), V = n and E = m. Output: Subset DS of V s.t. each vertex of G either belongs to DS or has a neighbor in DS. Cardinality of DS is minimum but also DS must be connected! Complexity: Status: NP-complete. Approximability: (ln + 3) in the unweighted case ( is the maximum degree) and (3 lnn) in the weighted case.

6 The (only) algorithm for CDS - Preliminaries Throughout execution nodes are marked white, black or gray. A piece is either a white node or a black connected component. The algorithm runs in two phases: Phase I: Mark nodes so as no white ones exist. (In other words every node is dominated.) Phase II: Connect the unconnected dominating set of Phase I. (In an elegant and efficient way.) Perform the above having the approximability in mind.

7 The (only) algorithm for CDS - Phase I Initially all nodes are marked white. At each step pick the node that decreases the most the pieces, paint it black and paint all its neighbors gray. By the end of Phase I, no white nodes are left. All nodes are dominated!

8 The (only) algorithm for CDS - Phase II Connect the black components to get a connected DS. By using a Steiner tree approximation algorithm. Iterativelly connect chains of two black nodes. Paint the path black. Solution is the set of black nodes.

9 Comments on the algorithm for CDS The 3 lnn-approximation for the weighted case is similar: In Phase I use a weighted set cover algorithm to dominate. In Phase II use weighted Steiner tree algorithm to connect. A 2(1 + ln )-approximate variation of the algorithm can be implemented in O(m)! A slightly different analysis reduces the approximation by a small constant. The algorithm is due to Guha and Khuller: Approximation Algorithms for Connected Dominating Sets. Algorithmica 1998.

10 Distributed Dominating Set O(k 2/k log )-approximation in O(k 2 ) rounds. k arbitrary parameter constant number of rounds! Each node sends O(k 2 ) messages, each of size O(log ). First an LP is solved. Then a distributed rounding approximate procedure is performed by each node. Due to: Kuhn and Wattenhofer Constant-Time Distributed Dominating Set Approximation, PODC 2003.

11 Distributed Connected Dominating Set O(log )-approximation, the maximum degree. O(log n) number of rounds. By using a distributed sparsification procedure that maintains connectivity in the graph. Due to: Dubhashi, Mei and Panconesi Fast Distributed Algorithms for (Weakly) Connected Dominating Sets and Linear-Size Skeletons, SODA There exist various results for certain classes of graphs!

12 Extended Dominating Set Input: Graph G = (V, E), V = n and E = m. Output: Subset DS of V s.t. each vertex of G either belongs to DS or has a neighbor in DS or has k 2-hop neighbors in the set. Cardinality of DS is minimum. Variation when the extended DS is connected. Both versions are NP-complete.

13 Extended Dominating Set (cont.) Extension of the algorithm of Guha and Khuller. No theoretical guarantees for general graphs. Constant ratio for special classes of graphs. Very good performance in practise. Due to: Wu, Cardei, Dai and Yang Extended Dominating Set and Its Applications in Ad Hoc Networks Using Cooperative Communication, IEEE TPDS 2006.

14 Other variants Variants meeting connectivity requirements. Weakly or strongly connected DS or EDS. Weighted variants. Edge or node weights. Seeking to minimize weight instead of cardinality. Neighboring variants.... Each node has a neighbor in DS (strong DS). Each dominated node has k-neighbors in DS.