Vertex and Edge Covers with Clustering Properties: Complexity and Algorithms

1. Vertex and Edge Covers with Clustering Properties: Complexity and Algorithms Henning Fernau Department of Computer Science University of Trier David ManloveDepartment of Computing Science University of Glasgow Supported by EPSRC grant GR/R84597/01,RSE / Scottish Exec Personal Research Fellowship

2. What is a vertex cover? • Let G=(V,E) be a connected graph • A vertex cover S is a set of vertices SV, such that for each edge {u,v}E, either uS or vS • A minimum vertex cover of size 3 • Finding a minimum vertex cover is NP-hard • Karp, 1972 • Result holds even for planar cubic graphs • Garey and Johnson, 1979 • Approximable within 2 • Gavril, 1973 • Not approximable within 105-21- for any >0 (1.36) unless P=NP • Dinur and Safra, 2002

3. Connected vertex covers • Given S V, let G[S] denote the subgraph of Ginduced by S • A connected vertex cover S is a vertex cover such that G[S] is connected

4. Connected vertex covers • Given S V, let G[S] denote the subgraph of Ginduced by S • A connected vertex cover S is a vertex cover such that G[S] is connected • A minimum connected vertex cover of size 5 • Finding a minimum connected vertex cover is NP-hard • Garey and Johnson, 1977 • Result holds even for planar graphs of maximum degree4 • Polynomial time-solvable for trees and for graphs of maximum degree 3 • Ueno et al, 1988 • Approximable within 2 • Savage, 1982

5. t-total vertex covers • Let n =|V|, m =|E| 1 and 1 t  n • A t-total vertex cover S is a vertex cover such that each connected component of G[S] has at least t vertices

6. t-total vertex covers • Let n =|V|, m =|E| 1 and 1 t  n • A t-total vertex cover S is a vertex cover such that each connected component of G[S] has at least t vertices • A minimum 2-total vertex cover of size 4 • S is a 1-total vertex cover S is a vertex cover • S is a t-total vertex cover of size t  S is a connected vertex cover of size t • Other values of t ? t=2,t=3, … ? • Blair, 2001 (2-total vertex cover = “total vertex cover”)

7. Motivation: clustering • Let Sbe a set of vertices satisfying some property • Elements of S may be required to form clusters • We interpret cluster as connected component of G[S]

8. Motivation: clustering • Let Sbe a set of vertices satisfying some property • Elements of S may be required to form clusters • We interpret cluster as connected component of G[S] • We may wish to impose a lower bound t on the size of the connected component

9. Dominating sets • Let G=(V,E) be a connected graph • A dominating set S is a set of vertices in SV, such that each vertex uS is adjacent to some vS • Haynes, Hedetniemi and Slater, 1998 • A minimum dominating set of size 3

10. Total and connected dominating sets • A connected dominating set S is a dominating set such that G[S] is connected • A total dominating set S is a dominating set such that each connected component of G[S] has size 2

11. t-total vertex covers: our results • For each t  2, the problem of finding a minimum t-total vertex cover is: • NP-hard • even for planar bipartite graphs of maximum degree 3 • Approximable within 2 • Not approximable (asymptotically)within105-21- for any >0 (1.36) unless P=NP • Not approximable within t for some t >0 unless P=NP • even for bipartite graphs of maximum degree 3

12. FPT algorithm for 2-tvc • FPT – “Fixed parameter tractable” • Idea – identify a “parameter” k – here k is the size of the 2-tvc • Derive an O(ckf(n))=O*(ck) algorithm for finding a minimum 2-tvc, where n is the input size and f is some polynomial function in n • The problem of deciding whether there is a 2-tvc of size  kis solvable in O*(2.37k) time

13. Connected vertex covers: our results • The problem of finding a minimum connected vertex cover is: • NP-hard • even for planar bipartite graphs of maximum degree 4 • Not approximable (asymptotically)within105-21- for any >0 (1.36) unless P=NP • In FPT and solvable in O*(2.94k) time, where k is the size of the connected vertex cover • Previous algorithm: O*(6k) complexity • Guo et al, 2005

14. What is an edge cover? • Let G=(V,E) be a connected graph • Assume that n =|V| and m =|E| 1 • An edge cover S is a set of edges in SE, such that each vertex in V is incident to an edge in S • A minimum edge cover of size 3 • A minimum edge cover can be found in O(nm) time • Norman and Rabin, 1959 • Micali and Vazirani, 1980

15. Connected and t-total edge covers • Given S E, let G[S] denote the subgraph of Ginduced by S (comprising S and all incident vertices) • A connected edge cover S is an edge cover such that G[S] is connected • A minimum connected edge cover is a spanning tree and contains n-1 edges • For 1 t  m, a t-total edge cover S is an edge cover such that each connected component of G[S] has at least t edges • A minimum 2-total edge cover of size 4

16. t-total edge covers : our results • S is a 1-total edge cover S is an edge cover • S is an (n-1)-total edge cover of size n-1 S is a spanning tree • For each t  2, the problem of finding a minimum t-total edge cover is: • NP-hard • Approximable within 2 • (For t = 2) Not approximable within  for some >0 unless P=NP • even for bounded degree graphs • In FPT

17. Open problems • Polynomial-time algorithms for finding minimum t-total vertex / edge covers in restricted classes of graphs • chordal graphs, interval graphs, trees… • FPT algorithm for finding a minimum t-total vertex cover (for t>2) • “Clustering” variants of vertex and edge dominating sets • Interpolation from “opposite side” • Consider vertex / edge covers S where G[S] has t components H. Fernau and DFM, “Vertex and Edge Covers with Clustering Properties: Complexity and Algorithms”, Technical Report, University of Glasgow, Department of Computing Science, April 2006