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, Jérôme Monnot and Aris Pagourtzis July 2, 2020 1/10

Tolerant Edge Cover Problems 2/10

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 one edge in S , ◮ S is minimal (with respect to inclusion) i.e., no proper subset of S is an edge cover. 2/10

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 one 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. 2/10

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 one 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. 2/10

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 one 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. 2/10

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 one 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. 2/10

r -Tolerant Edge Cover Problems 3/10

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. 3/10

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. 3/10

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. 3/10

r -Tolerant Edge Cover Problems 4/10

r -Tolerant Edge Cover Problems A F B E C D G = ( V, E ) r = 2 4/10

r -Tolerant Edge Cover Problems A F A F B E B E C D C D G = ( V, E ) 2-EC r = 2 4/10

r -Tolerant Edge Cover Problems A F A F A F B E B E B E C D C D C D G = ( V, E ) Upper 2-EC 2-EC r = 2 4/10

Basic properties of r -tolerant solutions 5/10

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 G S = ( V, S ): ◮ V = V 1 ( S ) ∪ V 2 ( S ) where V 1 ( S ) = { v ∈ V : d G S ( v ) = r } and V 2 ( S ) = { v ∈ V : d G S ( v ) > r } . ◮ V 2 ( S ) is an independent set of G S . 5/10

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 G S = ( V, S ): ◮ V = V 1 ( S ) ∪ V 2 ( S ) where V 1 ( S ) = { v ∈ V : d G S ( v ) = r } and V 2 ( S ) = { v ∈ V : d G S ( v ) > r } . ◮ V 2 ( S ) is an independent set of G S . Contradicts to r -tolerancy . . . . . . V 1 ( S ) V 2 ( S ) V 1 ( S ) V 1 ( S ) V 2 ( S ) V 2 ( S ) 5/10

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 G S = ( V, S ): ◮ V = V 1 ( S ) ∪ V 2 ( S ) where V 1 ( S ) = { v ∈ V : d G S ( v ) = r } and V 2 ( S ) = { v ∈ V : d G S ( v ) > r } . ◮ V 2 ( S ) is an independent set of G S . Contradicts to r -tolerancy . . . . . . V 1 ( S ) V 2 ( S ) V 1 ( S ) V 1 ( S ) V 2 ( S ) V 2 ( S ) 5/10

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 G S = ( V, S ): ◮ V = V 1 ( S ) ∪ V 2 ( S ) where V 1 ( S ) = { v ∈ V : d G S ( v ) = r } and V 2 ( S ) = { v ∈ V : d G S ( v ) > r } . ◮ V 2 ( S ) is an independent set of G S . Contradicts to r -tolerancy . . . . . . V 1 ( S ) V 2 ( S ) V 1 ( S ) V 1 ( S ) V 2 ( S ) V 2 ( S ) 5/10

Basic properties of r -tolerant solutions 6/10

Basic properties of r -tolerant solutions Let r ≥ 1, for all graphs G = ( V, E ) of minimum degree at least r , 2 ec r ( G ) ≥ uec r ( G ). 6/10

Basic properties of r -tolerant solutions Let r ≥ 1, for all graphs G = ( V, E ) of minimum degree at least r , 2 ec r ( G ) ≥ uec r ( G ). 1 − 2 ec r ( G ) = � v ∈ V d G S ( v ) ≥ rn where | V | = n . 6/10

Basic properties of r -tolerant solutions Let r ≥ 1, for all graphs G = ( V, E ) of minimum degree at least r , 2 ec r ( G ) ≥ uec r ( G ). 1 − 2 ec r ( G ) = � v ∈ V d G S ( v ) ≥ rn where | V | = n . . . . . . . V 1 ( S ) V 1 ( S ) V 1 ( S ) V 2 ( S ) V 2 ( S ) V 2 ( S ) 6/10

Basic properties of r -tolerant solutions Let r ≥ 1, for all graphs G = ( V, E ) of minimum degree at least r , 2 ec r ( G ) ≥ uec r ( G ). 1 − 2 ec r ( G ) = � v ∈ V d G S ( v ) ≥ rn where | V | = n . 2 − uec r ( G ) ≤ r | V 1 ( S ) | ≤ rn . 6/10

Hardness of Exact Computation 7/10

Hardness of Exact Computation Theorem Let G = ( V, E ) be an ( r + 1)-regular graph with r ≥ 2. Then, uec r ( G ) = | E | − eds ( G ). 7/10

Hardness of Exact Computation Theorem Let G = ( V, E ) be an ( r + 1)-regular graph with r ≥ 2. Then, uec r ( G ) = | E | − eds ( G ). 7/10

Hardness of Exact Computation Theorem Let G = ( V, E ) be an ( r + 1)-regular graph with r ≥ 2. Then, uec r ( G ) = | E | − eds ( G ). eds ( G ) = 2 7/10

Hardness of Exact Computation Theorem Let G = ( V, E ) be an ( r + 1)-regular graph with r ≥ 2. Then, uec r ( G ) = | E | − eds ( G ). Finding an edge dominating set of minimum size in cubic planar graphs is NP-hard. [Demange et. al. - 2014] 7/10

Hardness of Exact Computation Theorem Let G = ( V, E ) be an ( r + 1)-regular graph with r ≥ 2. Then, uec r ( 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. 7/10

Hardness of Exact Computation 8/10

Hardness of Exact Computation Theorem Double Upper EC is NP-hard in split graphs. 8/10

Hardness of Exact Computation Theorem Double Upper EC is NP-hard in split graphs. 8/10

Hardness of Exact Computation Theorem Double Upper EC is NP-hard in split graphs. A polynomial reduction from 2-tuple Dominating Set problem. 8/10

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 = { v 1 , ..., v n } : 8/10

Hardness of Exact Computation V ′ V ′′ V ′′′ . . . . . . . . . . . . V ∗ add four copies of V 8/10

Hardness of Exact Computation V ′ V ′′ V ′′′ . . . . . . . . . . . . V ∗ if v j ∈ N G ( v i ), then add three edges v ∗ i v ′ j , v ∗ i v ′′ j , v ∗ i v ′′ j 8/10

Hardness of Exact Computation V ′ V ′′ V ′′′ . . . . . . . . . . . . V ∗ if v j ∈ N G ( v i ), then add three edges v ∗ i v ′ j , v ∗ i v ′′ j , v ∗ i v ′′ j 8/10

Hardness of Exact Computation V ′ V ′′ V ′′′ . . . . . . . . . . . . V ∗ u 1 u 2 add a K 2 , 3 component 8/10

Hardness of Exact Computation V ′ V ′′ V ′′′ . . . . . . . . . . . . K n +2 u 1 u 2 make a K n +2 by adding edges 8/10

Hardness of Exact Computation V ′ V ′′ V ′′′ . . . . . . . . . . . . K n +2 u 1 u 2 2-t DS of size k in G iff Double UEC of size 8 n − 2 k + 6 in G ′ 8/10