Transcript Document

Vertex and Edge Covers with
Clustering Properties:
Complexity and Algorithms
Henning Fernau
Department of Computer Science
University of Trier
David Manlove
Department of Computing Science
University of Glasgow
Supported by EPSRC grant GR/R84597/01,
RSE / Scottish Exec Personal Research Fellowship
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What is a vertex cover?
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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
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Approximable within 2
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Karp, 1972
Result holds even for planar cubic graphs
 Garey and Johnson, 1979
Gavril, 1973
Not approximable within 105-21- for any >0 (1.36)
unless P=NP
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Dinur and Safra, 2002
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Connected vertex covers
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Given S V, let G[S] denote the subgraph of G induced by S
A connected vertex cover S is a vertex cover such that G[S]
is connected
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Connected vertex covers
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Given S V, let G[S] denote the subgraph of G induced 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
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Polynomial time-solvable for trees and for graphs of
maximum degree 3
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Garey and Johnson, 1977
Result holds even for planar graphs of maximum degree 4
Ueno et al, 1988
Approximable within 2
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Savage, 1982
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t-total vertex covers
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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
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t-total vertex covers
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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, … ?
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Blair, 2001 (2-total vertex cover = “total vertex cover”)
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Motivation: clustering
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Let S be 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]
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Motivation: clustering
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Let S be 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
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Dominating sets
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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
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Haynes, Hedetniemi and Slater, 1998
A minimum dominating set of size 3
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Total and connected
dominating sets
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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
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t-total vertex covers: our results
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For each t  2, the problem of finding a minimum
t-total vertex cover is:
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NP-hard
 even for planar bipartite graphs of maximum degree 3
Approximable within 2
Not approximable (asymptotically) within 10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
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FPT algorithm for 2-tvc
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FPT – “Fixed parameter tractable”
Idea – identify a “parameter” k – here k is the size of
the 2-tvc
Derive an O(ck f(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  k is solvable in O*(2.37k) time
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Connected vertex covers:
our results
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The problem of finding a minimum connected vertex
cover is:
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NP-hard
 even for planar bipartite graphs of maximum
degree 4
Not approximable (asymptotically) within 10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
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Previous algorithm: O*(6k) complexity
Guo et al, 2005
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What is an edge cover?
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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
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A minimum edge cover of size 3
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A minimum edge cover can be found in O(nm) time
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Norman and Rabin, 1959
Micali and Vazirani, 1980
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Connected and t-total edge covers
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Given S E, let G[S] denote the subgraph of G induced
by S (comprising S and all incident vertices)
A connected edge cover S is an edge cover such that
G[S] is connected
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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
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A minimum 2-total edge cover of size 4
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t-total edge covers : our results
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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:
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NP-hard
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Approximable within 2
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(For t = 2) Not approximable within  for some  > 0 unless P=NP
 even for bounded degree graphs
In FPT
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Open problems
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Polynomial-time algorithms for finding minimum t-total
vertex / edge covers in restricted classes of graphs
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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”
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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
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