On the complexity of fixed parameter clique and dominating set - - PowerPoint PPT Presentation
On the complexity of fixed parameter clique and dominating set Friedrich Eisenbrand, Fabrizio Grandoni(2004) Present by Xiaoyan Zhao Paper Outline Fixed parameter clique problem clique in dense graphs(*) clique in sparse graphs
Present by Xiaoyan Zhao
Fixed parameter clique problem
clique in dense graphs(*) clique in sparse graphs
Fixed parameter dominating sets(*) Detection of the induced diamonds The first two are believed not to be fixed
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Clique
A graph such that each pair of distinct nodes is
adjacent, e.g., triangle is a 3 nodes clique.
Dominating set
A subset V' dominates a node u in V if either u belongs
to V' or u is adjacent to at least one node in V'. The set V' is called a dominating set of G if all the nodes of G are dominated by V '.
Induced subgraph G' =(V ', E ' ) of G
Two nodes are adjacent in G' if and only if they are
adjacent in G. G' is denoted by G[V' ].
W(r,s,t) denotes the running time of the
multiplication of an matrix by an matrix.
n n
r s
×
n n
s t
×
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Definition: determine whether a graph G of n
nodes contains a clique of l nodes, where l is the parameter.
Two facts about the clique problem:
detection of a triangle: , where is the
exponent of fast square matrix multiplication.
a node v is contained in an l clique if and only if the graph
G[N(v)] induced on G contains an l-1 clique.
Major improvement on this paper:
previous result: new result:
O n ( )
ω
ω < 2 3 7 6
.
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Previous best algorithm outline Clique(G. l):
if l == 3h, then creates an auxiliary graph G' in the
following way:
creates a node for each h-node clique Creates an edge between two nodes if and only if
them form a 2h-node clique in G.
G contains a 3h-node clique if and only if G'
contains a triangle, and the time bound to find a triangle in G' is .
else for each node v in V apply the above algorithm on
the induced graph G[N(v)] to detect l-1 node clique. The time bound is where .
O n
h
( )
ω
O n
l
( ) ,
( )
α
α ω
( ) / ( m o d ) l l l
= +
3 3
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New algorithm outline Clique(G,l):
Let , , . Create a 3-partite
auxiliary graph G' in the following way:
Partition the nodes into sets , where the nodes in are the
cliques of order of G, for i ={1,2,3}.
Create an edge between a node u in and a node v in , if
and only if these two nodes induce an clique in G.
Detect a triangle of G' in the following way:
for each pair of nodes {u, v} where u in and v in , compute
the number P(u,v) of 2-length paths between u and v through a node of . The graph G' contains a triangle if and only if there is a pair of adjacent nodes {u, v} where u in and v in such that P(u,v)>0.
The time is bounded by adjacency matrices multiplication.
l l
1
3
=
/
l l
2
1 3
= −
( ) /
l l
3
3
=
/ V
i
V
i
l i
V
i
V
j
l l
i j
+
V
1
V
3
V
2
V
1
V
3
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Definition: determine whether an undirected
A previous fastest known algorithm:
A new improved algorithm based on fast
O n
l
( )
+ 1
O n
l l
( )
( , , )
ω
1 2
1
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The new algorithm outline:
: the set of subsets of V of cardinality h : a 0-1 matrix whose rows are indexed by the
elements of and whose columns are indexed by the elements of V. Given w in and v in V, if and only if w dominates v.
Compute the matrix , where , .
Clearly that D' contains a zero entry if and only if G admits a dominating set of size l.
The time is bounded by the matrix multiplication
cost.
V
h
D
h
V
h
V
h
[ ]
D w v
h
,
= 0
D D D
l l T
' =
⋅
1 2
l l
1
2
=
/
l l
2
2
=
/
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Present by Xiaoyan Zhao