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On the complexity of the upper r-tolerant edge cover problem
Mehdi Khosravian
Joint work with: Ararat Harutyunyan, Nikolaos Melissinos, Jérôme Monnot and Aris Pagourtzis
July 2, 2020
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Tolerant Edge Cover Problems
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On the complexity of the upper r -tolerant edge cover problem Mehdi - - PowerPoint PPT Presentation
On the complexity of the upper r -tolerant edge cover problem Mehdi - - PowerPoint PPT Presentation
On the complexity of the upper r -tolerant edge cover problem Mehdi Khosravian Joint work with: Ararat Harutyunyan, Nikolaos Melissinos, Jrme Monnot and Aris Pagourtzis July 2, 2020 1/10 Tolerant Edge Cover Problems 2/10 Tolerant Edge
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Tolerant Edge Cover Problems
An edge subset S ⊆ E of G = (V, E) is a tolerant edge cover, if:
◮ S is an edge cover i.e., each vertex of G is an endpoint of at least
- ne edge in S,
◮ S is minimal (with respect to inclusion) i.e., no proper subset of
S is an edge cover.
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Tolerant Edge Cover Problems
An edge subset S ⊆ E of G = (V, E) is a tolerant edge cover, if:
◮ S is an edge cover i.e., each vertex of G is an endpoint of at least
- ne edge in S,
◮ S is minimal (with respect to inclusion) i.e., no proper subset of
S is an edge cover.
Min Edge Cover
Goal: Finding a tolerant edge cover of minimum size.
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Tolerant Edge Cover Problems
An edge subset S ⊆ E of G = (V, E) is a tolerant edge cover, if:
◮ S is an edge cover i.e., each vertex of G is an endpoint of at least
- ne edge in S,
◮ S is minimal (with respect to inclusion) i.e., no proper subset of
S is an edge cover.
Min Edge Cover → Polynomial Solvable
Goal: Finding a tolerant edge cover of minimum size.
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Tolerant Edge Cover Problems
An edge subset S ⊆ E of G = (V, E) is a tolerant edge cover, if:
◮ S is an edge cover i.e., each vertex of G is an endpoint of at least
- ne edge in S,
◮ S is minimal (with respect to inclusion) i.e., no proper subset of
S is an edge cover.
Min Edge Cover → Polynomial Solvable
Goal: Finding a tolerant edge cover of minimum size.
Upper Edge Cover
Goal: Finding a tolerant edge cover of maximum size.
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Tolerant Edge Cover Problems
An edge subset S ⊆ E of G = (V, E) is a tolerant edge cover, if:
◮ S is an edge cover i.e., each vertex of G is an endpoint of at least
- ne edge in S,
◮ S is minimal (with respect to inclusion) i.e., no proper subset of
S is an edge cover.
Min Edge Cover → Polynomial Solvable
Goal: Finding a tolerant edge cover of minimum size.
Upper Edge Cover → NP-hard [Manlove - 1999]
Goal: Finding a tolerant edge cover of maximum size.
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r-Tolerant Edge Cover Problems
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r-Tolerant Edge Cover Problems
Given an integer r ≥ 1, an edge subset S ⊆ E of G = (V, E) is a r-tolerant edge cover, if:
◮ S is an r-edge cover i.e., deletion of any set of at most r − 1
edges from S maintains an edge cover,
◮ removing of any edge from S yields a set which is not an r-edge
cover.
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r-Tolerant Edge Cover Problems
Given an integer r ≥ 1, an edge subset S ⊆ E of G = (V, E) is a r-tolerant edge cover, if:
◮ S is an r-edge cover i.e., deletion of any set of at most r − 1
edges from S maintains an edge cover,
◮ removing of any edge from S yields a set which is not an r-edge
cover.
r- Edge Cover (r-EC)
Input: A graph G = (V, E) of minimum degree r. Output: An r-tolerance edge cover S ⊆ E of minimum size.
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r-Tolerant Edge Cover Problems
Given an integer r ≥ 1, an edge subset S ⊆ E of G = (V, E) is a r-tolerant edge cover, if:
◮ S is an r-edge cover i.e., deletion of any set of at most r − 1
edges from S maintains an edge cover,
◮ removing of any edge from S yields a set which is not an r-edge
cover.
r- Edge Cover (r-EC)
Input: A graph G = (V, E) of minimum degree r. Output: An r-tolerance edge cover S ⊆ E of minimum size.
Upper r- Edge Cover (Upper r-EC)
Input: A graph G = (V, E) of minimum degree r. Output: An r-tolerance edge cover S ⊆ E of maximum size.
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r-Tolerant Edge Cover Problems
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r-Tolerant Edge Cover Problems
G = (V, E) r = 2 A B C D E F
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r-Tolerant Edge Cover Problems
G = (V, E) r = 2 A B C D E F 2-EC A B C D E F
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r-Tolerant Edge Cover Problems
G = (V, E) r = 2 A B C D E F 2-EC A B C D E F Upper 2-EC A B C D E F
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Basic properties of r-tolerant solutions
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Basic properties of r-tolerant solutions
Let r ≥ 1. S is an r-tec solution of G = (V, E) if and only if the following conditions meet on GS = (V, S):
◮ V = V1(S) ∪ V2(S) where V1(S) = {v ∈ V : dGS(v) = r} and
V2(S) = {v ∈ V : dGS(v) > r}.
◮ V2(S) is an independent set of GS.
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Basic properties of r-tolerant solutions
Let r ≥ 1. S is an r-tec solution of G = (V, E) if and only if the following conditions meet on GS = (V, S):
◮ V = V1(S) ∪ V2(S) where V1(S) = {v ∈ V : dGS(v) = r} and
V2(S) = {v ∈ V : dGS(v) > r}.
◮ V2(S) is an independent set of GS.
V1(S) V1(S) V1(S)
. . . . . .
V2(S) V2(S) V2(S)
Contradicts to r-tolerancy
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Basic properties of r-tolerant solutions
Let r ≥ 1. S is an r-tec solution of G = (V, E) if and only if the following conditions meet on GS = (V, S):
◮ V = V1(S) ∪ V2(S) where V1(S) = {v ∈ V : dGS(v) = r} and
V2(S) = {v ∈ V : dGS(v) > r}.
◮ V2(S) is an independent set of GS.
V1(S) V1(S) V1(S)
. . . . . .
V2(S) V2(S) V2(S)
Contradicts to r-tolerancy
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Basic properties of r-tolerant solutions
Let r ≥ 1. S is an r-tec solution of G = (V, E) if and only if the following conditions meet on GS = (V, S):
◮ V = V1(S) ∪ V2(S) where V1(S) = {v ∈ V : dGS(v) = r} and
V2(S) = {v ∈ V : dGS(v) > r}.
◮ V2(S) is an independent set of GS.
V1(S) V1(S) V1(S)
. . . . . .
V2(S) V2(S) V2(S)
Contradicts to r-tolerancy
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Basic properties of r-tolerant solutions
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Basic properties of r-tolerant solutions
Let r ≥ 1, for all graphs G = (V, E) of minimum degree at least r, 2ecr(G) ≥ uecr(G).
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Basic properties of r-tolerant solutions
Let r ≥ 1, for all graphs G = (V, E) of minimum degree at least r, 2ecr(G) ≥ uecr(G). 1− 2ecr(G) =
v∈V dGS(v) ≥ rn where |V | = n.
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Basic properties of r-tolerant solutions
Let r ≥ 1, for all graphs G = (V, E) of minimum degree at least r, 2ecr(G) ≥ uecr(G). 1− 2ecr(G) =
v∈V dGS(v) ≥ rn where |V | = n. V1(S) V1(S) V1(S)
. . . . . .
V2(S) V2(S) V2(S)
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Basic properties of r-tolerant solutions
Let r ≥ 1, for all graphs G = (V, E) of minimum degree at least r, 2ecr(G) ≥ uecr(G). 1− 2ecr(G) =
v∈V dGS(v) ≥ rn where |V | = n.
2− uecr(G) ≤ r|V1(S)| ≤ rn.
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Hardness of Exact Computation
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Hardness of Exact Computation
Theorem
Let G = (V, E) be an (r + 1)-regular graph with r ≥ 2. Then, uecr(G) = |E| − eds(G).
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Hardness of Exact Computation
Theorem
Let G = (V, E) be an (r + 1)-regular graph with r ≥ 2. Then, uecr(G) = |E| − eds(G).
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Hardness of Exact Computation
Theorem
Let G = (V, E) be an (r + 1)-regular graph with r ≥ 2. Then, uecr(G) = |E| − eds(G). eds(G) = 2
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Hardness of Exact Computation
Theorem
Let G = (V, E) be an (r + 1)-regular graph with r ≥ 2. Then, uecr(G) = |E| − eds(G). Finding an edge dominating set of minimum size in cubic planar graphs is NP-hard. [Demange et. al. - 2014]
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Hardness of Exact Computation
Theorem
Let G = (V, E) be an (r + 1)-regular graph with r ≥ 2. Then, uecr(G) = |E| − eds(G). Finding an edge dominating set of minimum size in cubic planar graphs is NP-hard. [Demange et. al. - 2014]
Theorem
Let Upper (r + 1)-EC is NP-hard in graphs of maximum degree ∆ + 1 if Upper r-EC is NP-hard in graphs of maximum degree ∆, and this holds even for bipartite graphs.
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Hardness of Exact Computation
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Hardness of Exact Computation
Theorem
Double Upper EC is NP-hard in split graphs.
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Hardness of Exact Computation
Theorem
Double Upper EC is NP-hard in split graphs.
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Hardness of Exact Computation
Theorem
Double Upper EC is NP-hard in split graphs.
A polynomial reduction from 2-tuple Dominating Set problem.
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Hardness of Exact Computation
Theorem
Double Upper EC is NP-hard in split graphs.
A polynomial reduction from 2-tuple Dominating Set problem. let G = (V, E) be an instance of 2-tuple DS, where V = {v1, ..., vn}:
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Hardness of Exact Computation
. . . V ∗ . . . V ′ . . . V ′′ . . . V ′′′ add four copies of V
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Hardness of Exact Computation
. . . V ∗ . . . V ′ . . . V ′′ . . . V ′′′ if vj ∈ NG(vi), then add three edges v∗
i v′ j, v∗ i v′′ j , v∗ i v′′ j
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Hardness of Exact Computation
. . . V ∗ . . . V ′ . . . V ′′ . . . V ′′′ if vj ∈ NG(vi), then add three edges v∗
i v′ j, v∗ i v′′ j , v∗ i v′′ j
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Hardness of Exact Computation
. . . V ∗ . . . V ′ . . . V ′′ . . . V ′′′ add a K2,3 component
u1 u2
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Hardness of Exact Computation
. . . . . . V ′ . . . V ′′ . . . V ′′′ make a Kn+2 by adding edges
u1 u2
Kn+2
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Hardness of Exact Computation
. . . . . . V ′ . . . V ′′ . . . V ′′′ 2-t DS of size k in G iff Double UEC of size 8n − 2k + 6 in G′
u1 u2
Kn+2
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Hardness of Exact Computation
. . . . . . V ′ . . . V ′′ . . . V ′′′ Suppose S is a 2-t DS of size k
u1 u2
Kn+2
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Hardness of Exact Computation
. . . . . . V ′ . . . V ′′ . . . V ′′′ Suppose S is a 2-t DS of size k
u1 u2
Kn+2
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Hardness of Exact Computation
. . . . . . V ′ . . . V ′′ . . . V ′′′ Suppose S is a 2-t DS of size k
u1 u2
Kn+2
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Hardness of Exact Computation
. . . . . . V ′ . . . V ′′ . . . V ′′′ Suppose S′ is a Double UEC of size k′
u1 u2
Kn+2
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Hardness of Exact Computation
. . . . . . V ′ . . . V ′′ . . . V ′′′ 1- dS′(v) = 2 for all v ∈ V ′, V ′′, V ′′′
u1 u2
Kn+2
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Hardness of Exact Computation
. . . . . . V ′ . . . V ′′ . . . V ′′′ 2- if dS′(v) = 2 for v ∈ V ∗, then NS(v) = {u1, u2}
u1 u2
Kn+2
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Hardness of Approximation
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Hardness of Approximation
Theorem
It is NP-hard to approximate the solution of Upper EC to within 593
594 in graphs of max degree 4.
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Hardness of Approximation
Theorem
It is NP-hard to approximate the solution of Upper EC to within 593
594 in graphs of max degree 4.
An approximation preserving reduction from Min Vertex Cover problem.
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Hardness of Approximation
1 2 3 4
G = (V, E) an instance of Min VC
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Hardness of Approximation
1 2 3 4
G = (V, E) an instance of Min VC
1 2 3 4
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Hardness of Approximation
1 2 3 4
G = (V, E) an instance of Min VC
1 2 3 4
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Hardness of Approximation
1 2 3 4
G = (V, E) an instance of Min VC
1 2 3 4
G′ = (V ′, E′) an instance of Upper EC vertex cover of G of size k iff Upper EC of size 2|V | + |E| − k
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Hardness of Approximation
1 2 3 4 1 2 3 4
Suppose S is a vertex cover of G
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Hardness of Approximation
1 2 3 4 1 2 3 4
Suppose S is a vertex cover of G
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Hardness of Approximation
1 2 3 4 1 2 3 4
Suppose S is a vertex cover of G
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Hardness of Approximation
1 2 3 4 1 2 3 4
Suppose S is a vertex cover of G
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Hardness of Approximation
1 2 3 4 1 2 3 4
Suppose S′ is an Upper EC of G′ S′ does not contain any P4 (a collection of stars)
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Hardness of Approximation
1 2 3 4 1 2 3 4
Suppose S′ is an Upper EC of G′ S′ does not contain any P4 (a collection of stars)
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Hardness of Approximation
1 2 3 4 1 2 3 4
Suppose S′ is an Upper EC of G′ S′ does not contain any P4 (a collection of stars)
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Hardness of Approximation
Theorem
It is NP-hard to approximate the solution of Upper EC to within 593
594 in graphs of max degree 4.
An approximation preserving reduction from Min Vertex Cover problem. Min VC can not be approximated to within a factor 100
99 in
3-regular graphs. [Chlebik et. al. - 2006]
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Hardness of Approximation
Theorem
It is NP-hard to approximate the solution of Upper EC to within 593
594 in graphs of max degree 4.
An approximation preserving reduction from Min Vertex Cover problem. Min VC can not be approximated to within a factor 100
99 in
3-regular graphs. [Chlebik et. al. - 2006]
Theorem
It is NP-hard to approximate the solution of Double Upper EC to within 883
884 in graphs of max degree 6.
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Thanks for your attention!
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