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Linear Time Algorithms for Finding a Dominating Set of
Fixed Size in Degenerated Graphs
Noga Alon Shai Gutner
Lecturer : Daniel Motil
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22.12.2008Linear Time Algorithms for Finding a Dominating Set of Fixed Size in Degenerated Graphs
Motivation
This is the most general class of graphs for which fixed-parameter tractability for this problem has been established
Many interesting families of graphs are degenerated
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22.12.2008Linear Time Algorithms for Finding a Dominating Set of Fixed Size in Degenerated Graphs
It is hard…
The dominating set problem on general graphs is known to be W [2]-complete
This means that most likely there is no f(k)·nc algorithm for finding a dominating set of size at most k in a graph of size n for any computable function f : N → N and constant c
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22.12.2008Linear Time Algorithms for Finding a Dominating Set of Fixed Size in Degenerated Graphs
Previous results
O(227√kn) time algorithm for the dominating set problem in planar graphs
Fixed-parameter algorithms are now known also for map graphs and for constant powers of H-minor-free graphs
The running time given in for finding a dominating set of size k in an H-minor free graph G with n vertices is 2O(√k)nc , where c is a constant depending only on H
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22.12.2008Linear Time Algorithms for Finding a Dominating Set of Fixed Size in Degenerated Graphs
To summarize
fixed-parameter tractable algorithms for the dominating set problem were known for fixed powers of H-minor-free graphs and for map graphs
Linear time algorithms were established only for planar graphs
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22.12.2008Linear Time Algorithms for Finding a Dominating Set of Fixed Size in Degenerated Graphs
Our main result
kO(dk)n time algorithm for finding a dominating set of size k in a d-degenerated graph with n vertices
The algorithm is linear in the number of vertices of the graph
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22.12.2008Linear Time Algorithms for Finding a Dominating Set of Fixed Size in Degenerated Graphs
More results
We improve the dependence on k for the following specific families of degenerated graphs
(O(h))hk∙n time algorithm for graphs that do not contain Kh as a topological minor
O(logh)hk/2 ∙n time algorithm for graphs which are Kh - minor-free
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22.12.2008Linear Time Algorithms for Finding a Dominating Set of Fixed Size in Degenerated Graphs
Induced Cycles
The article is also address the problem of finding an induced cycle of a given length for the families of graphs discussed above
O(n) expected time algorithm for finding an induced k-cycle in graphs with an excluded minor
Deterministic O(n∙logn) time algorithm for the problem
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22.12.2008Linear Time Algorithms for Finding a Dominating Set of Fixed Size in Degenerated Graphs
Some notation
For an undirected graph G = (V ,E) and a vertex v ∈ V we denote:
N(v) – the set the set of all vertices adjacent to v (not including v itself)
We say that v dominates the vertices of N(v) ∪ {v}
The subgraph of G induced by some set V’ ⊆ V is denoted by G[V’]
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22.12.2008Linear Time Algorithms for Finding a Dominating Set of Fixed Size in Degenerated Graphs
Degenerated Graphs
A graph G is d-degenerated if every induced subgraph of G has a vertex v ∈ V such that d(v) ≤ d
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22.12.2008Linear Time Algorithms for Finding a Dominating Set of Fixed Size in Degenerated Graphs
Lemma
A d-degenerated graph with n vertices has less than dn edges
Proof – by induction on n– For n = 1 this is obviously true– In the general case, let v be a vertex of degree at most d. after
deleting v we get a d-degenerated graph with n-1 vertices. By induction it has less than d(n-1) edges. So original graph has less than dn edges
conclusion - average degree is less than 2d
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22.12.2008Linear Time Algorithms for Finding a Dominating Set of Fixed Size in Degenerated Graphs
Generalization of the problem
parameterized dominating set problem Input: undirected graph G = (V ,E), a
parameter k Problem: finding a set of at most k vertices
that dominate all the other vertices Let us generalize the problem for black and
white graphs
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22.12.2008Linear Time Algorithms for Finding a Dominating Set of Fixed Size in Degenerated Graphs
black and white graph
A black and white graph is a graph G = (V, E) such that the vertex set V of the graph G has been partitioned into two disjoint sets B and W of black and white vertices i.e., V = B ⊎ W
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22.12.2008Linear Time Algorithms for Finding a Dominating Set of Fixed Size in Degenerated Graphs
Dominating set problem for black and white graphs
Input: black and white graph G = (B ⊎ W,E) and an integer k
Problem: finding a set of at most k vertices that dominate that dominate the black vertices
i.e., finding subset U ⊆ B ⊎ W, such that |U| ≤ k andevery vertex v ∈ B − U satisfies N(v) ∩ U = ∅
Clearly this is a generalization of the problem as we can take W = ∅
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22.12.2008Linear Time Algorithms for Finding a Dominating Set of Fixed Size in Degenerated Graphs
Lemma 1
Let G = (B ⊎ W,E) be a d-degenerated black and white graph.If |B| > (4d +2)k, then there are at most (4d +2)k vertices in G that dominate at least |B|/k vertices of B
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22.12.2008Linear Time Algorithms for Finding a Dominating Set of Fixed Size in Degenerated Graphs
Proof
R = {v ∈ B ⊎ W | |(N(v) ∩ {v}) ∩ B| ≥ |B|/k} By contradiction, assume that |R| > (4d +2)k Let G’ be the induced graph G[R ∪ B] Clearly G’ has less than |R| + |B| vertices For every r ∈ R – d(r) > |B|/k – 1 ∑ d(v) ≥ |R| ∙ (|B|/k – 1) v ∈ R ∪ B
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22.12.2008Linear Time Algorithms for Finding a Dominating Set of Fixed Size in Degenerated Graphs
|B| |R|1)– /k |B|(|R| - |)B| |Rk(|
BR
|)B| |Rk(|k|R|
1 - |)B| |Rk(||B||R|
1 - )k |B| , |R| 2max(|B||R|
1 - 2k|)B|, |Rmin(| 1 - 2k
2)k (4d 2d
≤
≤
≤ ≤ >
So the average degree of G’ is at least
This contradicts the fact that G[R ∪ B] is d-degenerated
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22.12.2008Linear Time Algorithms for Finding a Dominating Set of Fixed Size in Degenerated Graphs
Theorem 1
There is a kO(dk)n time algorithm for finding a dominating set of size atmost k in a d-degenerated black and white graph with n verticesthat contains such a set
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22.12.2008Linear Time Algorithms for Finding a Dominating Set of Fixed Size in Degenerated Graphs
Algorithm 1 DominatingSetDegenerated(G, k)
Input: Black and white d-degenerated graph G = (B ⊎ W,E),integers k, d
Output: A set dominating all vertices of B of size at most k or NONE if no such set exists if B = ∅ then
– return ∅ else if k = 0 then
– return NONE else if |B| ≤ (4d +2)k then
– forall possible ways of splitting B into k (possibly empty) disjoint pieces B 1 , … , Bk do if each piece Bi has a vertex vi that dominates it then
– return {v1, . . . , vk}– return NONE
else– R ←{v ∈ B ⊎ W |(N(v) ∩ {v}) ∩ B| ≥ |B| / k} – forall v ∈ R do
Create a new graph G’ from G by marking all the elements of N(v) as white and removing v from the graph D←DominatingSetDegenerated(G, k −1) if D ≠ NONE then
– return D ∪ {v}– return NONE
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22.12.2008Linear Time Algorithms for Finding a Dominating Set of Fixed Size in Degenerated Graphs
Algorithm explanation
If |B| ≤ (4d + 2)k If there is a dominating set of size at most k,
we can split B into k disjoint pieces, so that each piece has a vertex that dominates it
At most k)4d+2)k ways for those splits check in O(kdn) time whether every piece is
dominated by a vertex Running time - k)4d+2)k ∙ O(kdn) = kO(dk)∙n
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22.12.2008Linear Time Algorithms for Finding a Dominating Set of Fixed Size in Degenerated Graphs
If |B| > (4d + 2)k By lemma 1 we get |R| < (4d + 2)k A dominating set of size at most k must
contain a vertex r ∈ R Check for every r if G’ has a dominating set of
size at most k-1 Number of recursive calls - (4d + 2)kk! Algorithm Running time
(4d + 2)kk! ∙ kO(dk)∙n = kO(dk)∙n
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22.12.2008Linear Time Algorithms for Finding a Dominating Set of Fixed Size in Degenerated Graphs
Topological minor
An edge is said to be subdivided when it is deleted and replaced by a path of length two
A subdivision of a graph G is a graph that can be obtained from G by a sequence of edge subdivisions
Graph H is a topological minor of a graph G if H is a subdivision of a subgraph of G
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22.12.2008Linear Time Algorithms for Finding a Dominating Set of Fixed Size in Degenerated Graphs
Minor of a graph
A graph H is called a minor of a graph G if it can be obtained from a subgraph of G by a series of edge contractions
edge contractions – removing the edge and combining its two endpoints
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22.12.2008Linear Time Algorithms for Finding a Dominating Set of Fixed Size in Degenerated Graphs
Graphs with an Excluded Minor
Graphs with either an excluded minor or with no topological minor are known to be degenerated
Proposition 1 There exists a constant c such that, for every h, every graph that does not contain Kh as a topological minor is ch2-degenerated
Proposition 2 There exists a constant c such that, for every h, every graph with no Kh minor is ch√logh-degenerated
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22.12.2008Linear Time Algorithms for Finding a Dominating Set of Fixed Size in Degenerated Graphs
Lemma 2
If a graph G with n vertices isd-degenerated, then for every h ≥ 1, G contains at most - n copies of Kh
dh
1
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22.12.2008Linear Time Algorithms for Finding a Dominating Set of Fixed Size in Degenerated Graphs
Proof
By induction on n For n = 1 this is obviously true In the general case, let v ∈ V with d(v) ≤ d The number of Kh containing v ≤ By induction hypothesis, the number of Kh
in G – v is at most (n – 1) The number of Kh in G is at most n d
h
1
dh
1
dh
1
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22.12.2008Linear Time Algorithms for Finding a Dominating Set of Fixed Size in Degenerated Graphs
Definition 1
A black and white graph G = (B ⊎ W,E) is called reduced if it satisfies the following conditions:– W is an independent set– All the vertices of W have degree at least 2– N(w1) ≠ N(w2) for every two distinct vertices
w1,w2 ∈W.
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22.12.2008Linear Time Algorithms for Finding a Dominating Set of Fixed Size in Degenerated Graphs
Theorem 2
There exists a constant c > 0, such that for every reduced black and white graph G = (B ⊎ W,E), if G does not contain Kh as a topological minor, then there exists a vertex b ∈ B of degree at most (ch)h
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22.12.2008Linear Time Algorithms for Finding a Dominating Set of Fixed Size in Degenerated Graphs
Proof
Denote |B| = n > 0 and d = sh2 where s is the constant from Proposition 1
For each w ∈ W check if there exist two vertices b1, b2 ∈ N(w), such that b1 and b2 are not connected
If exist - add the edge {b1, b2} and remove the vertex w from the graph
b1 b1
w b2 b2
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22.12.2008Linear Time Algorithms for Finding a Dominating Set of Fixed Size in Degenerated Graphs
Denote the resulting graph G’ = (B ⊎ W,E) W’ – all w ∈ W such as each b1,b2 ∈ N(w) are
connected G’[B] does not contain Kh as a topological minor
because every edge in G’[B] can be subdivided to an edge in G
G’[B] is d-degenerated and therefore has at most dn edges
We deleted less then dn vertices from G
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22.12.2008Linear Time Algorithms for Finding a Dominating Set of Fixed Size in Degenerated Graphs
Let’s try to bound |W’| Every w ∈ W’ has degree ≥ 2 Every w ∈ W’ has degree < h-1
otherwise w ∪ N(w) is a Kh So, for each w ∈ W’ with d(w) = k, 2 ≤ k ≤ h-2 N(w) – a clique of size k For each w1,w2 ∈ W’ we get a different clique
because N(w1) ≠ N(w2) by lemma 2, the number of d(w) = k is bounded
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22.12.2008Linear Time Algorithms for Finding a Dominating Set of Fixed Size in Degenerated Graphs
|W’| ≤ [ + + … + ]n|W| ≤ |W’| + dn|E| ≤ d(|B| + |W|) ≤ d(n + dn + [ + … + ]n) ≤ d[3d + + … + ]n d1
d 1 dh
3
dh
3
dh
3
d2
d1
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22.12.2008Linear Time Algorithms for Finding a Dominating Set of Fixed Size in Degenerated Graphs
Obviously, there exists a black vertex of degree at most 2|E|/n
if we plug d = sh2 it is follow that there is a constant c such that d(b) ≤ (ch)h
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22.12.2008Linear Time Algorithms for Finding a Dominating Set of Fixed Size in Degenerated Graphs
Theorem 3
There exists a constant c > 0, such that for every reduced black and white graph G = (B ⊎ W,E), if G is Kh- minor-free, then there exists a vertex b ∊ B of degree at most (clogh)h/2
Proof - We proceed as in the proof of Theorem 2 using Proposition 2 instead of Proposition 1
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22.12.2008Linear Time Algorithms for Finding a Dominating Set of Fixed Size in Degenerated Graphs
Theorem 4
There is an (O(h))hkn time algorithm for finding a dominating set of size at most k in a black and white graph with n vertices and no Kh as a topological minor
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22.12.2008Linear Time Algorithms for Finding a Dominating Set of Fixed Size in Degenerated Graphs
Algorithm 2 DominatingSetNoMinor(G, k)
Input: Black and white (Kh -minor-free) graph G = (B ⊎ W,E), integer k Output: A set dominating all vertices of B of size at most k or NONE if no
such set exists if B = ∅ then
– return ∅ else if k = 0 then
– return NONE else
– Remove all edges of G whose two endpoints are in W– Remove all white vertices of G of degree 0 or 1– As long as there are two different vertices w1, w2 ∊W with
N(w1) = N(w2), |N(w1)| < h−1, remove one of them from the graph– Let b ∊ B be a vertex of minimum degree among all vertices in B– forall v ∊ N(b) ∪ {b} do
Create a new graph G’ from G by marking all the elements of N(v) aswhite and removing v from the graph
D←DominatingSetNoMinor(G, k −1) if D ≠ NONE then
– return D {∪ v}– return NONE
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22.12.2008Linear Time Algorithms for Finding a Dominating Set of Fixed Size in Degenerated Graphs
Algorithm explanation
Removes vertices and edges in order to get a (nearly) reduced black and white graph
Edges between white vertices as well as white vertices of degree at most 1 can be removed
We can eliminate all duplicates, that is, ensure that N(w1) ≠ N(w2) for every two different vertices w1, w2 ∊ W
It follows from the proof of Theorem 2 that it is enough to do so for vertices with at most h−2 neighbors
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22.12.2008Linear Time Algorithms for Finding a Dominating Set of Fixed Size in Degenerated Graphs
Running time
By theorem 2, d(b) ≤ (ch)h We can remove all duplicates in O(hn)
time using radix sort Number of recursive calls - (ch)hk
Each one - O(hn) time Running time - (ch)hk ∙ O(hn) = O(h)hkn
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22.12.2008Linear Time Algorithms for Finding a Dominating Set of Fixed Size in Degenerated Graphs
Theorem 5
There is an (O(log h))hk/2n time algorithm for finding a dominating set of size at most k in a black and white graph with n vertices which is Kh -minor-free
Proof - The proof is analogues to that of Theorem 4 using Theorem 3 instead of Theorem 2
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22.12.2008Linear Time Algorithms for Finding a Dominating Set of Fixed Size in Degenerated Graphs
The Weighted Case
each vertex of the graph has some positive real weight
The goal is to find a dominating set of size at most k, such that the sum of the weights of all the vertices of the dominating set is as small as possible
The algorithms we presented can be generalized to deal with the weighted case
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22.12.2008Linear Time Algorithms for Finding a Dominating Set of Fixed Size in Degenerated Graphs
Weighted Case modifications
In Algorithm 1 for degenerated graphs,we need to address the case where |B| ≤ (4d + 2)k - always choose a vertex with minimum weight that dominates each piece
In Algorithm 2 for graphs with an excluded minor, the criterion for removing white vertices from the graph is modified so that whenever two vertices w1, w2 ∊ W satisfy N(w1) = N(w2), the vertex with the bigger weight is removed
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22.12.2008Linear Time Algorithms for Finding a Dominating Set of Fixed Size in Degenerated Graphs
Concluding Remarks
similar techniques may be useful in improving and simplifying other known fixed-parameter algorithms for graphs with an excluded minor
An interesting open problem - is there a 2O(√k)nc time algorithm for finding a dominating set of size k in graphs with n vertices and an excluded minor, where c is some absolute constant that does not depend on the excluded graph
Maybe even a 2O(√k)n time algorithm can be achieved