linear’(me*approximaons*for* dominang*sets* and ... · linear’(me*approximaons*for*...
TRANSCRIPT
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Linear-‐(me Approxima(ons for Domina(ng Sets and Independent Domina(ng Sets
in Unit Disk Graphs
Celina Miraglia Herrera de Figueiredo Guilherme Dias da Fonseca
Raphael Carlos dos Santos Machado Vinícius Gusmão Pereira de Sá
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Domina(ng set
G (V, E)
D ⊆ V D dominating set ⟺ ∀w ∈ V \ D, ∃v ∈ D | vw ∈ E
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Domina(ng set
G (V, E)
D ⊆ V D dominating set ⟺ ∀w ∈ V \ D, ∃v ∈ D | vw ∈ E
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Domina(ng set
G (V, E)
D ⊆ V D dominating set ⟺ ∀w ∈ V \ D, ∃v ∈ D | vw ∈ E
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Domina(ng set
G (V, E)
D ⊆ V D dominating set ⟺ ∀w ∈ V \ D, ∃v ∈ D | vw ∈ E
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Domina(ng set
G (V, E)
D ⊆ V D dominating set ⟺ ∀w ∈ V \ D, ∃v ∈ D | vw ∈ E
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Domina(ng set
G (V, E)
D ⊆ V D dominating set ⟺ ∀w ∈ V \ D, ∃v ∈ D | vw ∈ E
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Domina(ng set
G (V, E)
D ⊆ V D dominating set ⟺ ∀w ∈ V \ D, ∃v ∈ D | vw ∈ E
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Minimum domina(ng set problem
Input: graph G (V, E) Output: dominating set D of G s.t. |D| is minimum
NP-hard
(1+log n)-approximation algorithm (Johnson 1974)
Not approximable within a (c log n) factor, for some c > 0 (Raz & Safra 1997)
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Unit disk graph
1
2
Model of Graph congruent disks G (V, E)
2
1
4 5
3 3
4 5
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Domina(ng sets in unit disk graphs
Several applications, e.g. ad-hoc wireless networks (Marathe, Breu, Hunt III, Ravi & Rosenkrantz 1995)
NP-hard nonetheless (Clark, Colbourn & Johnson 1990)
Constant factor approximations (breaking the log n barrier), and even PTAS
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Two simple facts 1st fact: Every maximal independent set is a domina(ng set.
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Two simple facts
3
4 5
2
6
1
2 1
3 6
4 5
K1,6
1st fact: Every maximal independent set is a domina(ng set. 2nd fact: A unit disk graph contains no K1,6 as an induced subgraph.
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5-‐approxima(on 1st fact: Every maximal independent set is a domina(ng set. 2nd fact: A unit disk graph contains no K1,6 as an induced subgraph.
Corollary: If G is a unit disk graph, then every maximal independent set S of G is a 5-‐approxima*on for the minimum independent set of G
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Algorithms for unit disk graphs
Graph-based algorithms
Input: a graph
Geometric algorithms
Input: a geometric model
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Domina(ng sets in unit disk graphs
Vast literature on approximation algorithms: (Marathe, Breu, Hunt III, Ravi & Rosenkrantz 1995)
O(n+m) graph-based 5-approximation (MBHRR’95)
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Domina(ng sets in unit disk graphs
Vast literature on approximation algorithms: (Marathe, Breu, Hunt III, Ravi & Rosenkrantz 1995) (Hunt III, Marathe, Radhakrishnan, Ravi, Rosenkrantz & Stearns 1998)* (Nieberg, Hurink & Kern 2008)* (Gibson & Pirwani 2010)*– (general) disk graphs
(Hurink & Nieberg 2011)*– independent dominating set version
O(n+m) graph-based 5-approximation (MBHRR’95) *PTAS
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Domina(ng sets in unit disk graphs
Vast literature on approximation algorithms: (Marathe, Breu, Hunt III, Ravi & Rosenkrantz 1995) (Hunt III, Marathe, Radhakrishnan, Ravi, Rosenkrantz & Stearns 1998)* (Nieberg, Hurink & Kern 2008)* (Gibson & Pirwani 2010)*– (general) disk graphs
(Hurink & Nieberg 2011)*– independent dominating set version
O(n+m) graph-based 5-approximation (MBHRR’95) *PTAS O(n225) graph-based 5-approximation (NHK’08)
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Domina(ng sets in unit disk graphs
Vast literature on approximation algorithms: (Marathe, Breu, Hunt III, Ravi & Rosenkrantz 1995) (Hunt III, Marathe, Radhakrishnan, Ravi, Rosenkrantz & Stearns 1998)* (Nieberg, Hurink & Kern 2008)* (Gibson & Pirwani 2010)*– (general) disk graphs (Erlebach & Mihalák 2010) – weighted version (Hurink & Nieberg 2011)*– independent dominating set version (Zou, Wang, Xu, Li, Du, Wan & Wu 2011) – weighted version
O(n+m) graph-based 5-approximation (MBHRR’95) *PTAS O(n225) graph-based 5-approximation (NHK’08)
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Domina(ng sets in unit disk graphs
Vast literature on approximation algorithms: (Marathe, Breu, Hunt III, Ravi & Rosenkrantz 1995) (Hunt III, Marathe, Radhakrishnan, Ravi, Rosenkrantz & Stearns 1998)* (Nieberg, Hurink & Kern 2008)* (Gibson & Pirwani 2010)*– (general) disk graphs (Erlebach & Mihalák 2010) – weighted version (Hurink & Nieberg 2011)*– independent dominating set version (Zou, Wang, Xu, Li, Du, Wan & Wu 2011) – weighted version (De, Das & Nandy 2011)
O(n+m) graph-based 5-approximation (MBHRR’95) *PTAS O(n225) graph-based 5-approximation (NHK’08)
O(n9) geometric 4-approximation (DDN’11) O(n18) geometric 3-approximation (DDN’11)
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Domina(ng sets in unit disk graphs
Vast literature on approximation algorithms: (Marathe, Breu, Hunt III, Ravi & Rosenkrantz 1995) (Hunt III, Marathe, Radhakrishnan, Ravi, Rosenkrantz & Stearns 1998)* (Nieberg, Hurink & Kern 2008)* (Gibson & Pirwani 2010)*– (general) disk graphs (Erlebach & Mihalák 2010) – weighted version (Hurink & Nieberg 2011)*– independent dominating set version (Zou, Wang, Xu, Li, Du, Wan & Wu 2011) – weighted version (De, Das & Nandy 2011)
O(n+m) graph-based 5-approximation (MBHRR’95) *PTAS O(n225) graph-based 5-approximation (NHK’08)
O(n9) geometric 4-approximation (DDN’11) O(n18) geometric 3-approximation (DDN’11)
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Domina(ng sets in unit disk graphs
Vast literature on approximation algorithms: (Marathe, Breu, Hunt III, Ravi & Rosenkrantz 1995) (Hunt III, Marathe, Radhakrishnan, Ravi, Rosenkrantz & Stearns 1998)* (Nieberg, Hurink & Kern 2008)* (Gibson & Pirwani 2010)*– (general) disk graphs (Erlebach & Mihalák 2010) – weighted version (Hurink & Nieberg 2011)*– independent dominating set version (Zou, Wang, Xu, Li, Du, Wan & Wu 2011) – weighted version (De, Das & Nandy 2011)
O(n+m) graph-based 5-approximation (MBHRR’95) *PTAS O(n225) graph-based 5-approximation (NHK’08)
O(n9) geometric 4-approximation (DDN’11) O(n18) geometric 3-approximation (DDN’11)
O(n+m) graph-based ,888 -approximation O(n log n) geometric ,888 -approximation (FFMS’12)
4.888… Our contribu(on: 4.888…
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1995 2012
100 meters world record
-‐ 2.7%
Leroy Burrell (USA) Usain Bolt (JAMAICA) 9.85 s 9.58 s
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Alexander Popov (RUSSIA) Cesar Cielo (BRASIL) 48.21 s 46.91 s
1995 2012
100 meters freestyle world record
-‐ 2.6%
Alexander Popov (RUSSIA) César Cielo (BRAZIL) 48.21 s 46.91 s
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Domina(ng sets in unit disk graphs
Vast literature on approximation algorithms: (Marathe, Breu, Hunt III, Ravi & Rosenkrantz 1995) (Hunt III, Marathe, Radhakrishnan, Ravi, Rosenkrantz & Stearns 1998)* (Nieberg, Hurink & Kern 2008)* (Gibson & Pirwani 2010)*– (general) disk graphs (Erlebach & Mihalák 2010) – weighted version (Hurink & Nieberg 2011)*– independent dominating set version (Zou, Wang, Xu, Li, Du, Wan & Wu 2011) – weighted version (De, Das & Nandy 2011)
O(n+m) graph-based 5-approximation (MBHRR’95) *PTAS O(n225) graph-based 5-approximation (NHK’08)
O(n9) geometric 4-approximation (DDN’11) O(n18) geometric 3-approximation (DDN’11)
O(n+m) graph-based ,888 -approximation O(n log n) geometric ,888 -approximation (FFMS’12)
4.888… Our contribu(on: 4.888…
-‐ 2.2%
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Coronas and cores
Let D be a maximal independent set of graph G(V,E). A corona consists of exactly 5 (five) vertices of D presenting a common neighbor in V \ D, called a .
corona
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Coronas and cores
Let D be a maximal independent set of graph G(V,E). A corona consists of exactly 5 (five) vertices of D presenting a common neighbor in V \ D, called a .
A corona C can be • reducible, if it has a core c s.t. D \ C U {c} is still a dominating set of G;
or • irreducible, otherwise.
corona
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Reducible and irreducible coronas
reducible corona
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Reducible and irreducible coronas
reducible corona
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Reducible and irreducible coronas
reducible corona irreducible corona
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reducible corona irreducible corona witness
Witnesses
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Witnesses
reducible corona irreducible corona witness
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Witnesses
Let C be a corona of graph G(V,E), and let c be a core of C. A vertex w is a witness of c iff cw ∉ E, and ND[w] ⊆ C.
reducible corona irreducible corona witness
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4.888…-‐approxima(on
1. Obtain a maximal independent set D 2. While there is a reducible corona C in D 3. Update D by reducing C 4. Return D
Input: adjacency lists (graph) Time: O(n+m)
Input: center coordinates in Real RAM Model Time: O(n log n)
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4.888…-‐approxima(on
1. Obtain a maximal independent set D 2. While there is a reducible corona C in D 3. Update D by reducing C 4. Return D
Input: adjacency lists (graph) Time: O(n+m)
Input: center coordinates in Real RAM Model Time: O(n log n)
Lemma: a maximal independent set D with no reducible coronas is a 4.888…-approximation for the minimum
(independent) dominating set.
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1. Obtain a maximal independent set D 2. While there is a reducible corona C in D 3. Update D by reducing C 4. Return D
4.888…-‐approxima(on
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1. Obtain a maximal independent set D 2. While there is a reducible corona C in D 3. Update D by reducing C 4. Return D
4.888…-‐approxima(on
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1. Obtain a maximal independent set D 2. While there is a reducible corona C in D 3. Update D by reducing C 4. Return D
4.888…-‐approxima(on
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1. Obtain a maximal independent set D 2. While there is a reducible corona C in D 3. Update D by reducing C 4. Return D
4.888…-‐approxima(on
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4.888…-‐approxima(on
D – a maximal independent set with no reducible coronas (algorithm output)
![Page 41: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/41.jpg)
4.888…-‐approxima(on
D – a maximal independent set with no reducible coronas (algorithm output) D* – a minimum dominating set (optimum solution)
![Page 42: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/42.jpg)
4.888…-‐approxima(on
D – a maximal independent set with no reducible coronas (algorithm output) D* – a minimum dominating set (optimum solution)
|D| |D*|
≤ ??
![Page 43: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/43.jpg)
4.888…-‐approxima(on
D – a maximal independent set with no reducible coronas (algorithm output) D* – a minimum dominating set (optimum solution)
|D| |D*|
≤ ??
1
1 1
1
1 1
1
![Page 44: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/44.jpg)
4.888…-‐approxima(on
D – a maximal independent set with no reducible coronas (algorithm output) D* – a minimum dominating set (optimum solution)
|D| |D*|
≤ ??
1
1 1
1
1 1
1
![Page 45: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/45.jpg)
4.888…-‐approxima(on
D – a maximal independent set with no reducible coronas (algorithm output) D* – a minimum dominating set (optimum solution)
|D| |D*|
≤ ??
1
1 1
1
1 1
1
![Page 46: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/46.jpg)
4.888…-‐approxima(on
D – a maximal independent set with no reducible coronas (algorithm output) D* – a minimum dominating set (optimum solution)
|D| |D*|
≤ ??
1
1 1
1
1 1
1
![Page 47: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/47.jpg)
4.888…-‐approxima(on
D – a maximal independent set with no reducible coronas (algorithm output) D* – a minimum dominating set (optimum solution)
|D| |D*|
≤ ??
1
1 1
1
1 1
1
![Page 48: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/48.jpg)
4.888…-‐approxima(on
D – a maximal independent set with no reducible coronas (algorithm output) D* – a minimum dominating set (optimum solution)
|D| |D*|
≤ ??
1
1 1
1
1 1
1
![Page 49: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/49.jpg)
4.888…-‐approxima(on
D – a maximal independent set with no reducible coronas (algorithm output) D* – a minimum dominating set (optimum solution)
|D| |D*|
≤ ??
1
1 1
1
1
1
1/2
1/2
![Page 50: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/50.jpg)
4.888…-‐approxima(on
D – a maximal independent set with no reducible coronas (algorithm output) D* – a minimum dominating set (optimum solution)
|D| |D*|
≤ ??
1
1 1
1
1
1
1/2
1/2
![Page 51: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/51.jpg)
4.888…-‐approxima(on
D – a maximal independent set with no reducible coronas (algorithm output) D* – a minimum dominating set (optimum solution)
|D| |D*|
≤ ??
1
1 1
1
1
1
1/2
1/2
![Page 52: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/52.jpg)
4.888…-‐approxima(on
D – a maximal independent set with no reducible coronas (algorithm output) D* – a minimum dominating set (optimum solution)
|D| |D*|
≤ ??
1 1
1
1
1
1/2
1/2
1
![Page 53: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/53.jpg)
4.888…-‐approxima(on
D – a maximal independent set with no reducible coronas (algorithm output) D* – a minimum dominating set (optimum solution)
|D| |D*|
≤ ??
1 1
1
1
1
1/2
1/2
1
![Page 54: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/54.jpg)
4.888…-‐approxima(on
D – a maximal independent set with no reducible coronas (algorithm output) D* – a minimum dominating set (optimum solution)
|D| |D*|
≤ ??
1 1
1
1
1
1/2
1/2
1
![Page 55: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/55.jpg)
4.888…-‐approxima(on
D – a maximal independent set with no reducible coronas (algorithm output) D* – a minimum dominating set (optimum solution)
|D| |D*|
≤ ??
1 1
1
1
1/2
1/2
1
1
![Page 56: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/56.jpg)
4.888…-‐approxima(on
D – a maximal independent set with no reducible coronas (algorithm output) D* – a minimum dominating set (optimum solution)
|D| |D*|
≤ ??
1 1
1
1
1/2
1/2
1
1
![Page 57: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/57.jpg)
4.888…-‐approxima(on
D – a maximal independent set with no reducible coronas (algorithm output) D* – a minimum dominating set (optimum solution)
|D| |D*|
≤ ??
1 1
1
1
1/2
1/2
1
1
![Page 58: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/58.jpg)
4.888…-‐approxima(on
D – a maximal independent set with no reducible coronas (algorithm output) D* – a minimum dominating set (optimum solution)
|D| |D*|
≤ ??
1 1
1
1/2
1/2
1
1
1
![Page 59: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/59.jpg)
4.888…-‐approxima(on
D – a maximal independent set with no reducible coronas (algorithm output) D* – a minimum dominating set (optimum solution)
|D| |D*|
≤ ??
1 1
1
1/2
1/2
1
1
1
![Page 60: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/60.jpg)
4.888…-‐approxima(on
D – a maximal independent set with no reducible coronas (algorithm output) D* – a minimum dominating set (optimum solution)
|D| |D*|
≤ ??
1 1
1
1/2
1/2
1
1
1
![Page 61: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/61.jpg)
4.888…-‐approxima(on
D – a maximal independent set with no reducible coronas (algorithm output) D* – a minimum dominating set (optimum solution)
|D| |D*|
≤ ??
1
1
1/2
1/2
1
1
1
1
![Page 62: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/62.jpg)
4.888…-‐approxima(on
D – a maximal independent set with no reducible coronas (algorithm output) D* – a minimum dominating set (optimum solution)
|D| |D*|
≤ ??
1
1
1/2
1/2
1
1
1
1
![Page 63: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/63.jpg)
4.888…-‐approxima(on
D – a maximal independent set with no reducible coronas (algorithm output) D* – a minimum dominating set (optimum solution)
|D| |D*|
≤ ??
1
1
1/2
1/2
1
1
1
1
![Page 64: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/64.jpg)
4.888…-‐approxima(on
D – a maximal independent set with no reducible coronas (algorithm output) D* – a minimum dominating set (optimum solution)
|D| |D*|
≤ ??
1
1/2
1/2
1
1
1
1
1/3
1/3 1/3
![Page 65: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/65.jpg)
4.888…-‐approxima(on
D – a maximal independent set with no reducible coronas (algorithm output) D* – a minimum dominating set (optimum solution)
|D| |D*|
≤ ??
![Page 66: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/66.jpg)
4.888…-‐approxima(on
D – a maximal independent set with no reducible coronas (algorithm output) D* – a minimum dominating set (optimum solution)
|D| |D*|
≤ ??
1
1/2
1/2
1
1
1
1
1/3
1/3 1/3
![Page 67: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/67.jpg)
4.888…-‐approxima(on
D – a maximal independent set with no reducible coronas (algorithm output) D* – a minimum dominating set (optimum solution)
|D| |D*|
≤ ??
1
1 1
1
1 1
1
![Page 68: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/68.jpg)
4.888…-‐approxima(on
D – a maximal independent set with no reducible coronas (algorithm output) D* – a minimum dominating set (optimum solution)
|D| |D*|
≤ ??
1
1/2
1/2
1
1
1
1
1/3
1/3 1/3
![Page 69: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/69.jpg)
4.888…-‐approxima(on
D – a maximal independent set with no reducible coronas (algorithm output) D* – a minimum dominating set (optimum solution)
|D| |D*|
≤ ??
![Page 70: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/70.jpg)
4.888…-‐approxima(on
D – a maximal independent set with no reducible coronas (algorithm output) D* – a minimum dominating set (optimum solution)
|D| |D*|
≤ ??
![Page 71: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/71.jpg)
4.888…-‐approxima(on
D – a maximal independent set with no reducible coronas (algorithm output) D* – a minimum dominating set (optimum solution)
|D| |D*|
≤ ??
average of f(.)
![Page 72: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/72.jpg)
f : D* ⟶ (0, 5]
|D| |D*|
4.888…-‐approxima(on
= average of f (.) over D* ≤ 4,888…
D
D*
![Page 73: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/73.jpg)
4,888… < f (c*) ≤ 5
4.888…-‐approxima(on
C*
f : D* ⟶ (0, 5]
|D| |D*| = average of f (.) over D* ≤ 4,888…
D
D*
![Page 74: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/74.jpg)
4,888… < f (c*) ≤ 5 f (r*) ≤ 4
4.888…-‐approxima(on
C*
f : D* ⟶ (0, 5]
|D| |D*| = average of f (.) over D* ≤ 4,888…
D
D*
r* reliever
![Page 75: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/75.jpg)
4,888… < f (c*) ≤ 5 f (r*) ≤ 4
f (c*) = 5
4.888…-‐approxima(on
C*
f : D* ⟶ (0, 5]
|D| |D*| = average of f (.) over D* ≤ 4,888…
D
D*
r* reliever
![Page 76: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/76.jpg)
f (c*) = 5
4.888…-‐approxima(on
f : D* ⟶ (0, 5]
|D| |D*| = average of f (.) over D* ≤ 4,888…
D
D*
C*
1
1
1
1
1
![Page 77: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/77.jpg)
4.888…-‐approxima(on
C*
corona C
|D| |D*| = average of f (.) over D* ≤ 4,888…
D
D*
f (c*) = 5
f : D* ⟶ (0, 5]
![Page 78: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/78.jpg)
4.888…-‐approxima(on
C*
corona C
|D| |D*| = average of f (.) over D* ≤ 4,888…
D
D*
f (c*) = 5 ND[w] ⊆ C
f : D* ⟶ (0, 5]
w
![Page 79: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/79.jpg)
4.888…-‐approxima(on
C*
corona C
|D| |D*| = average of f (.) over D* ≤ 4,888…
D
D*
f (c*) = 5 ND[w] ⊆ C
f : D* ⟶ (0, 5]
w
r*
![Page 80: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/80.jpg)
4,888… < f (c*) ≤ 5 f (r*) ≤ 4
f (c*) = 5
4.888…-‐approxima(on
C*
corona C
|D| |D*| = average of f (.) over D* ≤ 4,888…
D
D*
f (c*) = 5 ND[w] ⊆ C
f : D* ⟶ (0, 5]
w
r* reliever
![Page 81: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/81.jpg)
4,888… < f (c*) ≤ 5 f (r*) ≤ 4
f(r*) > 4 (hypothesis) f (c*) = 5
4.888…-‐approxima(on
C*
corona C
|D| |D*| = average of f (.) over D* ≤ 4,888…
D
D*
f (c*) = 5 ND[w] ⊆ C
f : D* ⟶ (0, 5]
w
r* reliever
![Page 82: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/82.jpg)
4,888… < f (c*) ≤ 5 f (r*) ≤ 4
f(r*) > 4 (hypothesis) f (c*) = 5 ND[r*] ∩ C = ∅
4.888…-‐approxima(on
C*
corona C
|D| |D*| = average of f (.) over D* ≤ 4,888…
D
D*
f (c*) = 5 ND[w] ⊆ C
f : D* ⟶ (0, 5]
w
r* reliever
![Page 83: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/83.jpg)
4,888… < f (c*) ≤ 5 f (r*) ≤ 4
f(r*) > 4 (hypothesis) f (c*) = 5 ND[r*] ∩ C = ∅
4.888…-‐approxima(on
C*
corona C
|D| |D*| = average of f (.) over D* ≤ 4,888…
D
D*
f (c*) = 5 ND[w] ⊆ C
f : D* ⟶ (0, 5]
w
r* reliever
K1,6
![Page 84: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/84.jpg)
4,888… < f (c*) ≤ 5 f (r*) ≤ 4
4.888…-‐approxima(on
C*
corona C
|D| |D*| = average of f (.) over D* ≤ 4,888…
D
D*
f (c*) = 5 ND[w] ⊆ C
f : D* ⟶ (0, 5]
w
r* reliever
![Page 85: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/85.jpg)
f (c*) = 5 f (r*) ≤ 4
4.888…-‐approxima(on
C*
f : D* ⟶ (0, 5]
|D| |D*| = average of f (.) over D* ≤ 4,888…
D
D*
r* reliever
![Page 86: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/86.jpg)
f (ci*) = 5, i = 1, 2, … ?
f (c*) = 5 f (r*) ≤ 4
4.888…-‐approxima(on
C*
f : D* ⟶ (0, 5]
|D| |D*| = average of f (.) over D* ≤ 4,888…
D
D*
r* reliever
C1* r*
C3*
C2*
C4* . . .
![Page 87: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/87.jpg)
4.888…-‐approxima(on
r*
C1* C4*
C2* C3*
w1 w4
w3 w2
f (c1*) = 5
f (c2*) = 5
f (c3*) = 5
f (c4*) = 5
f (r*) ≤ 4 reliever
![Page 88: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/88.jpg)
Some geometric lemmas Lemma 1 (Pál 1921): If a set of points P has diameter 1, then P can be enclosed by a circle of radius 1 / .
Lemma 2 (Fodor 2007): The radius of the smallest circle enclosing 13 points with mutual distance ≥ 1 is (1 + ) / 2.
Lemma 3 (Fejes Tóth 1953): Every packing of two or more congruent disks in a convex region has density at most / .
€
3
€
5
€
12
€
π
![Page 89: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/89.jpg)
Some geometric lemmas Lemma 1 (Pál 1921): If a set of points P has diameter 1, then P can be enclosed by a circle of radius 1 / .
Lemma 2 (Fodor 2007): The radius of the smallest circle enclosing 13 points with mutual distance ≥ 1 is (1 + ) / 2.
Lemma 3 (Fejes Tóth 1953): Every packing of two or more congruent disks in a convex region has density at most / .
Lemma 4 (FFMS 2012): The closed neighborhood of a clique in a unit disk graph contains at most 12 independent vertices.
Lemma 5 (FFMS 2012): The closed d-neighborhood of a vertex in a unit disk graph contains at most (2d + 1)2 / independent vertices, for integer d ≥ 1.
Lemma 6 (FFMS 2012): If G is a (4,L)-pendant unit disk graph, then L ≤ 8.
€
3
€
5
€
12
€
π €
π
€
12
![Page 90: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/90.jpg)
r*
C1* C4*
C2* C3*
w1 w4
w3 w2
f (c1*) = 5
f (c2*) = 5
f (c3*) = 5
f (c4*) = 5
f (r*) ≤ 4
Establishing the approxima(on factor
reliever
![Page 91: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/91.jpg)
r*
C1* C4*
C2* C3*
w1 w4
w3 w2
f (c1*) = 5
f (c2*) = 5
f (c3*) = 5
f (c4*) = 5
f (r*) ≤ 4
ND[r*] = 4
Establishing the approxima(on factor
reliever
![Page 92: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/92.jpg)
Lemma 1 (Pál 1921): If a set of points P has diameter 1, then P can be enclosed by a circle of radius 1 / .
Lemma 2 (Fodor 2007): The radius of the smallest circle enclosing 13 points with mutual distance ≥ 1 is (1 + ) / 2.
Lemma 3 (Fejes Tóth 1953): Every packing of two or more congruent disks in a convex region has density at most / .
Lemma 4 (FFMS 2012): The closed neighborhood of a clique in a unit disk graph contains at most 12 independent vertices.
Lemma 5 (FFMS 2012): The closed d-neighborhood of a vertex in a unit disk graph contains at most (2d + 1)2 / independent vertices, for integer d ≥ 1.
Lemma 6 (FFMS 2012): If G is a (4,L)-pendant unit disk graph, Then L ≤ 8.
€
3
€
5
€
12
€
π €
π
€
12
Establishing the approxima(on factor
![Page 93: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/93.jpg)
r*
C1* C4*
C2* C3*
w1 w4
w3 w2
f (c1*) = 5
f (c2*) = 5
f (c3*) = 5
f (c4*) = 5
f (r*) ≤ 4
ND[r*] = 4
reliever
Establishing the approxima(on factor
![Page 94: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/94.jpg)
r*
C1* C4*
C2* C3*
w1 w4
w3 w2
f (c1*) = 5
f (c2*) = 5
f (c3*) = 5
f (c4*) = 5
f (r*) ≤ 4
ND[r*] = 4
Establishing the approxima(on factor
![Page 95: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/95.jpg)
r*
w1 w4
w3 w2
ND[r*] = 4
Establishing the approxima(on factor
![Page 96: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/96.jpg)
r*
C1* C4*
C2* C3*
w1 w4
w3 w2
f (c1*) = 5
f (c2*) = 5
f (c3*) = 5
f (c4*) = 5
f (r*) ≤ 4
ND[r*] = 4
Establishing the approxima(on factor
![Page 97: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/97.jpg)
r*
C1* C4*
C2* C3*
w1 w4
w3 w2
f (c1*) = 5
f (c2*) = 5
f (c3*) = 5
f (c4*) = 5
f (r*) ≤ 4
ND[r*] = 4 ⇒ at most 8 cores per reliever
reliever
Establishing the approxima(on factor
![Page 98: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/98.jpg)
r*
C1* C4*
C2* C3*
w1 w4
w3 w2
f (c1*) = 5
f (c2*) = 5
f (c3*) = 5
f (c4*) = 5
f (r*) ≤ 4
ND[r*] ∈ {3}
reliever
Establishing the approxima(on factor
![Page 99: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/99.jpg)
r*
C1* C4*
C2* C3*
w1 w4
w3 w2
f (c1*) = 5
f (c2*) = 5
f (c3*) = 5
f (c4*) = 5
f (r*) ≤ 4
ND[r*] ∈ {2,3}
reliever
Establishing the approxima(on factor
![Page 100: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/100.jpg)
r*
C1* C4*
C2* C3*
w1 w4
w3 w2
f (c1*) = 5
f (c2*) = 5
f (c3*) = 5
f (c4*) = 5
f (r*) ≤ 4
ND[r*] ∈ {1,2,3}
reliever
Establishing the approxima(on factor
![Page 101: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/101.jpg)
Lemma 1 (Pál 1921): If a set of points P has diameter 1, then P can be enclosed by a circle of radius 1 / .
Lemma 2 (Fodor 2007): The radius of the smallest circle enclosing 13 points with mutual distance ≥ 1 is (1 + ) / 2.
Lemma 3 (Fejes Tóth 1953): Every packing of two or more congruent disks in a convex region has density at most / .
Lemma 4 (FFMS 2012): The closed neighborhood of a clique in a unit disk graph contains at most 12 independent vertices.
Lemma 5 (FFMS 2012): The closed d-neighborhood of a vertex in a unit disk graph contains at most (2d + 1)2 / independent vertices, for integer d ≥ 1.
Lemma 6 (FFMS 2012): If G is a (4,L)-pendant unit disk graph, Then L ≤ 8.
€
3
€
5
€
12
€
π €
π
€
12
Establishing the approxima(on factor
![Page 102: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/102.jpg)
r*
C1* C4*
C2* C3*
w1 w4
w3 w2
f (c1*) = 5
f (c2*) = 5
f (c3*) = 5
f (c4*) = 5
f (r*) ≤ 4
ND[r*] ∈ {1,2,3}
reliever
Establishing the approxima(on factor
![Page 103: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/103.jpg)
r*
C1* C4*
C2* C3*
w1 w4
w3 w2
f (c1*) = 5
f (c2*) = 5
f (c3*) = 5
f (c4*) = 5
f (r*) ≤ 4
ND[r*] ∈ {1,2,3}
r*
Establishing the approxima(on factor
![Page 104: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/104.jpg)
r*
C1*
w1
ND[r*] ∈ {1,2,3}
r*
Establishing the approxima(on factor
![Page 105: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/105.jpg)
r*
C1* C4*
C2* C3*
w1 w4
w3 w2
f (c1*) = 5
f (c2*) = 5
f (c3*) = 5
f (c4*) = 5
f (r*) ≤ 4
ND[r*] ∈ {1,2,3}
r*
Establishing the approxima(on factor
![Page 106: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/106.jpg)
r*
C1* C4*
C2* C3*
w1 w4
w3 w2
f (c1*) = 5
f (c2*) = 5
f (c3*) = 5
f (c4*) = 5
f (r*) ≤ 4
ND[r*] ∈ {1,2,3} ⇒ at most 14 cores per reliever
Establishing the approxima(on factor
reliever
![Page 107: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/107.jpg)
| ND[r*] | Maximum number of cores ci* per reliever
Upper bound for |D| / |D*|
1 (1 x 1 + x 5) / 15 = 4,7333…
2 (1 x 2 + x 5) / 15 = 4,8
3 (1 x 3 + x 5) / 15 = 4,8666…
4 (1 x 4 + x 5) / 9 = 4,888…
Establishing the approxima(on factor
1 reliever
f(reliever) f(core)
![Page 108: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/108.jpg)
r*
C1* C4*
C2* C3*
w1 w4
w3 w2
f (c1*) = 5
f (c2*) = 5
f (c3*) = 5
f (c4*) = 5
f (r*) ≤ 4
Lower bound
reliever
(1 x 4 + x 5) / 5 = 4.8
![Page 109: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/109.jpg)
Thank you.
![Page 110: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/110.jpg)
Linear-‐(me Approxima(ons for Domina(ng Sets and Independent Domina(ng Sets
in Unit Disk Graphs
Celina Miraglia Herrera de Figueiredo Guilherme Dias da Fonseca
Raphael Carlos dos Santos Machado Vinícius Gusmão Pereira de Sá
![Page 111: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/111.jpg)
Future direc(ons
• Improve the algorithms (par(al reduc(ons)
Irreducible corona
![Page 112: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/112.jpg)
Future direc(ons
• Improve the algorithms (par(al reduc(ons)
Irreducible corona
![Page 113: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/113.jpg)
Future direc(ons
• Improve the algorithms (par(al reduc(ons)
Irreducible corona
![Page 114: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/114.jpg)
Future direc(ons
• Improve the analysis of the approxima(on factor (geometric/computa(onal proofs)
Show that Lemmas 5 and 6 are not (ght
and that some graphs that sa(sfy them
are actually not unit disk graphs.
![Page 115: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/115.jpg)
r*
C1* C4*
C2* C3*
w1 w4
w3 w2
f (c1*) = 5
f (c2*) = 5
f (c3*) = 5
f (c4*) = 5
f (r*) ≤ 4
Lower bound
reliever
(1 x 4 + x 5) / 5 = 4.8
![Page 116: Linear’(me*Approximaons*for* Dominang*Sets* and ... · Linear’(me*Approximaons*for* Dominang*Sets* and*Independent*Dominang*Sets* in*UnitDisk*Graphs* CelinaMiragliaHerrerade*Figueiredo*](https://reader035.vdocuments.mx/reader035/viewer/2022071213/602ac467f14d8824464cd541/html5/thumbnails/116.jpg)
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Future direc(ons
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Thank you.
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Linear-‐(me Approxima(ons for Domina(ng Sets and Independent Domina(ng Sets
in Unit Disk Graphs
Celina Miraglia Herrera de Figueiredo Guilherme Dias da Fonseca
Raphael Carlos dos Santos Machado Vinícius Gusmão Pereira de Sá
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(k,l)-‐pendant graphs
A (k,l)-pendant graph is a graph containing a vertex v with k pendant vertices in its open neighborhood and l pendant vertices in its open 2-neighborhood.
V
a (4,5)-‐pendant graph