Balanced Independent Sets on Colored Interval Graphs Sujoy Bhore, - - PowerPoint PPT Presentation

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Balanced Independent Sets on Colored Interval Graphs Sujoy Bhore, - - PowerPoint PPT Presentation

Aug 06, 2023 239 likes •602 views

Balanced Independent Sets on Colored Interval Graphs Sujoy Bhore, Jan-Henrik Haunert, Fabian Klute, Guangping Li , Martin N ollenburg 1/7 Boundary labeling Top Pot Doughnuts I Paragon Restaurant Lola Vios Cafe Tulio Queen City Grill

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Balanced Independent Sets on Colored Interval Graphs

Sujoy Bhore, Jan-Henrik Haunert, Fabian Klute, Guangping Li, Martin N¨

  • llenburg
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Sujoy Bhore, Jan-Henrik Haunert, Fabian Klute, Guangping Li, Martin N¨

  • llenburg · Balanced Independent and Dominating Sets on Colored Interval Graphs

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Boundary labeling

labels are at the boundary of the focus region

Tulio Daniel’s Broiler Vios Cafe Metropolitan Grill Sodo Deli Top Pot Doughnuts I Queen City Grill Paragon Restaurant Lola Maximilien Circa

a leader connects a label with its corresponding POI task: select a large conflict-free labeling

  • M. Fink, J.-H. Haunert, A. Schulz, J. Spoerhase und A. Wolff.

Algorithms for labeling focus regions. IEEE Transactions on Visualization and Computer Graphics (Proc. InfoVis’12), 18(12):2583–2592, 2012.

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Sujoy Bhore, Jan-Henrik Haunert, Fabian Klute, Guangping Li, Martin N¨

  • llenburg · Balanced Independent and Dominating Sets on Colored Interval Graphs

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Boundary labeling

labels represent objects of multiple categories task: select a good mixture of different object types

c by Jan-Henrik Haunert

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Sujoy Bhore, Jan-Henrik Haunert, Fabian Klute, Guangping Li, Martin N¨

  • llenburg · Balanced Independent and Dominating Sets on Colored Interval Graphs

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Model

input: a set of n colored axis-parallel unit squres touching a disk D rectangle: icon D

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Sujoy Bhore, Jan-Henrik Haunert, Fabian Klute, Guangping Li, Martin N¨

  • llenburg · Balanced Independent and Dominating Sets on Colored Interval Graphs

3/7

Model

input: a set of n colored axis-parallel unit squres touching a disk D interval representation of its intersection model rectangle: icon D

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SLIDE 6

Sujoy Bhore, Jan-Henrik Haunert, Fabian Klute, Guangping Li, Martin N¨

  • llenburg · Balanced Independent and Dominating Sets on Colored Interval Graphs

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Model

input: a set I of n intervals on the real line each interval is colored by a coloring c: I → {1, . . . , k}

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Sujoy Bhore, Jan-Henrik Haunert, Fabian Klute, Guangping Li, Martin N¨

  • llenburg · Balanced Independent and Dominating Sets on Colored Interval Graphs

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Model

input: goal: f-Balanced Independent Set (f-BIS) a set I of n intervals on the real line an independent set M ⊆ I M contains exactly f elements from each of k color classes each interval is colored by a coloring c: I → {1, . . . , k}

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SLIDE 8

Sujoy Bhore, Jan-Henrik Haunert, Fabian Klute, Guangping Li, Martin N¨

  • llenburg · Balanced Independent and Dominating Sets on Colored Interval Graphs

4/7

Model

input: goal: f-Balanced Independent Set (f-BIS) a set I of n intervals on the real line an independent set M ⊆ I M contains exactly f elements from each of k color classes each interval is colored by a coloring c: I → {1, . . . , k} 1-BIS

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Sujoy Bhore, Jan-Henrik Haunert, Fabian Klute, Guangping Li, Martin N¨

  • llenburg · Balanced Independent and Dominating Sets on Colored Interval Graphs

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1-BIS Problem: NP hardness

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Sujoy Bhore, Jan-Henrik Haunert, Fabian Klute, Guangping Li, Martin N¨

  • llenburg · Balanced Independent and Dominating Sets on Colored Interval Graphs

5/7

1-BIS Problem: NP hardness

C1 (x1 ∨ x2 ∨ x4) ∧ (x1 ∨ x3 ∨ x4) ∧ (x3 ∨ x4) ∧ (x1 ∨ x2 ∨ x3) reduction from 3-bounded 3SAT C2 C3 C4 each variable xi appears in ≤ 3 clauses each clause Cj has 2 or 3 literals

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Sujoy Bhore, Jan-Henrik Haunert, Fabian Klute, Guangping Li, Martin N¨

  • llenburg · Balanced Independent and Dominating Sets on Colored Interval Graphs

5/7

1-BIS Problem: NP hardness

C1 (x1 ∨ x2 ∨ x4) ∧ (x1 ∨ x3 ∨ x4) ∧ (x3 ∨ x4) ∧ (x1 ∨ x2 ∨ x3) reduction from 3-bounded 3SAT C2 C3 C4 gadgets: each variable xi appears in ≤ 3 clauses each clause Cj has 2 or 3 literals

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SLIDE 12

Sujoy Bhore, Jan-Henrik Haunert, Fabian Klute, Guangping Li, Martin N¨

  • llenburg · Balanced Independent and Dominating Sets on Colored Interval Graphs

5/7

1-BIS Problem: NP hardness

C1 (x1 ∨ x2 ∨ x4) ∧ (x1 ∨ x3 ∨ x4) ∧ (x3 ∨ x4) ∧ (x1 ∨ x2 ∨ x3) reduction from 3-bounded 3SAT clause: color C2 C3 C4 gadgets: each variable xi appears in ≤ 3 clauses each clause Cj has 2 or 3 literals

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SLIDE 13

Sujoy Bhore, Jan-Henrik Haunert, Fabian Klute, Guangping Li, Martin N¨

  • llenburg · Balanced Independent and Dominating Sets on Colored Interval Graphs

5/7

1-BIS Problem: NP hardness

C1 (x1 ∨ x2 ∨ x4) ∧ (x1 ∨ x3 ∨ x4) ∧ (x3 ∨ x4) ∧ (x1 ∨ x2 ∨ x3) reduction from 3-bounded 3SAT clause: color C2 C3 C4 variable: one (colored) interval for each occurence intersection: each pair of opposite literals gadgets: x1 C1 C2 C3 C4 + − each variable xi appears in ≤ 3 clauses each clause Cj has 2 or 3 literals

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SLIDE 14

Sujoy Bhore, Jan-Henrik Haunert, Fabian Klute, Guangping Li, Martin N¨

  • llenburg · Balanced Independent and Dominating Sets on Colored Interval Graphs

5/7

1-BIS Problem: NP hardness

C1 (x1 ∨ x2 ∨ x4) ∧ (x1 ∨ x3 ∨ x4) ∧ (x3 ∨ x4) ∧ (x1 ∨ x2 ∨ x3) reduction from 3-bounded 3SAT clause: color C2 C3 C4 variable: one (colored) interval for each occurence intersection: each pair of opposite literals gadgets: + − x3 C1 C2 C3 C4 each variable xi appears in ≤ 3 clauses each clause Cj has 2 or 3 literals

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SLIDE 15

Sujoy Bhore, Jan-Henrik Haunert, Fabian Klute, Guangping Li, Martin N¨

  • llenburg · Balanced Independent and Dominating Sets on Colored Interval Graphs

5/7

1-BIS Problem: NP hardness

C1 (x1 ∨ x2 ∨ x4) ∧ (x1 ∨ x3 ∨ x4) ∧ (x3 ∨ x4) ∧ (x1 ∨ x2 ∨ x3) reduction from 3-bounded 3SAT clause: color C2 C3 C4 variable: one (colored) interval for each occurence intersection: each pair of opposite literals gadgets: x1 C1 C2 C3 C4 + − x3 C1 C2 C3 C4 x2 x4 each variable xi appears in ≤ 3 clauses each clause Cj has 2 or 3 literals

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SLIDE 16

Sujoy Bhore, Jan-Henrik Haunert, Fabian Klute, Guangping Li, Martin N¨

  • llenburg · Balanced Independent and Dominating Sets on Colored Interval Graphs

5/7

1-BIS Problem: NP hardness

C1 (x1 ∨ x2 ∨ x4) ∧ (x1 ∨ x3 ∨ x4) ∧ (x3 ∨ x4) ∧ (x1 ∨ x2 ∨ x3) C2 C3 C4 C1 C2 C3 C4 C1 C2 C3 C4 1-BIS ⇒: Correctness x1 + − x3 x2 x4

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SLIDE 17

Sujoy Bhore, Jan-Henrik Haunert, Fabian Klute, Guangping Li, Martin N¨

  • llenburg · Balanced Independent and Dominating Sets on Colored Interval Graphs

5/7

1-BIS Problem: NP hardness

C1 (x1 ∨ x2 ∨ x4) ∧ (x1 ∨ x3 ∨ x4) ∧ (x3 ∨ x4) ∧ (x1 ∨ x2 ∨ x3) C2 C3 C4 C1 C2 C3 C4 C1 C2 C3 C4 1-BIS ⇒: Correctness x1 + − x3 x2 x4

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SLIDE 18

Sujoy Bhore, Jan-Henrik Haunert, Fabian Klute, Guangping Li, Martin N¨

  • llenburg · Balanced Independent and Dominating Sets on Colored Interval Graphs

5/7

1-BIS Problem: NP hardness

C1 (x1 ∨ x2 ∨ x4) ∧ (x1 ∨ x3 ∨ x4) ∧ (x3 ∨ x4) ∧ (x1 ∨ x2 ∨ x3) C2 C3 C4 C1 C2 C3 C4 C1 C2 C3 C4 1-BIS ⇒: evaluate the chosen literals as true Correctness x1 + − x3 x2 x4

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SLIDE 19

Sujoy Bhore, Jan-Henrik Haunert, Fabian Klute, Guangping Li, Martin N¨

  • llenburg · Balanced Independent and Dominating Sets on Colored Interval Graphs

5/7

1-BIS Problem: NP hardness

C1 (x1 ∨ x2 ∨ x4) ∧ (x1 ∨ x3 ∨ x4) ∧ (x3 ∨ x4) ∧ (x1 ∨ x2 ∨ x3) C2 C3 C4 C1 C2 C3 C4 C1 C2 C3 C4 1-BIS ⇒: ⇐ assignment: evaluate the chosen literals as true Correctness x1 + − x3 x2 x4

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SLIDE 20

Sujoy Bhore, Jan-Henrik Haunert, Fabian Klute, Guangping Li, Martin N¨

  • llenburg · Balanced Independent and Dominating Sets on Colored Interval Graphs

5/7

1-BIS Problem: NP hardness

C1 (x1 ∨ x2 ∨ x4) ∧ (x1 ∨ x3 ∨ x4) ∧ (x3 ∨ x4) ∧ (x1 ∨ x2 ∨ x3) C2 C3 C4 C1 C2 C3 C4 C1 C2 C3 C4 1-BIS ⇒: ⇐ assignment: evaluate the chosen literals as true Correctness {x1: T, x2: F, x3: T, x4: F} x1 + − x3 x2 x4

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SLIDE 21

Sujoy Bhore, Jan-Henrik Haunert, Fabian Klute, Guangping Li, Martin N¨

  • llenburg · Balanced Independent and Dominating Sets on Colored Interval Graphs

5/7

1-BIS Problem: NP hardness

C1 (x1 ∨ x2 ∨ x4) ∧ (x1 ∨ x3 ∨ x4) ∧ (x3 ∨ x4) ∧ (x1 ∨ x2 ∨ x3) C2 C3 C4 C1 C2 C3 C4 C1 C2 C3 C4 1-BIS ⇒: ⇐ assignment: evaluate the chosen literals as true Correctness choose a positive evaluated literal in each Ci {x1: T, x2: F, x3: T, x4: F} x1 + − x3 x2 x4

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SLIDE 22

Sujoy Bhore, Jan-Henrik Haunert, Fabian Klute, Guangping Li, Martin N¨

  • llenburg · Balanced Independent and Dominating Sets on Colored Interval Graphs

5/7

1-BIS Problem: NP hardness

C1 (x1 ∨ x2 ∨ x4) ∧ (x1 ∨ x3 ∨ x4) ∧ (x3 ∨ x4) ∧ (x1 ∨ x2 ∨ x3) C2 C3 C4 C1 C2 C3 C4 C1 C2 C3 C4 1-BIS ⇒: ⇐ assignment: evaluate the chosen literals as true Correctness choose a positive evaluated literal in each Ci {x1: T, x2: F, x3: T, x4: F} x1 + − x3 x2 x4

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Sujoy Bhore, Jan-Henrik Haunert, Fabian Klute, Guangping Li, Martin N¨

  • llenburg · Balanced Independent and Dominating Sets on Colored Interval Graphs

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f-BIS: An FPT Algorithm by (f, k)

sorted set of intervals I = {I1, . . . , In} sorted by right-endpoints

1 2 3 4 5 6

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Sujoy Bhore, Jan-Henrik Haunert, Fabian Klute, Guangping Li, Martin N¨

  • llenburg · Balanced Independent and Dominating Sets on Colored Interval Graphs

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f-BIS: An FPT Algorithm by (f, k)

sorted set of intervals I = {I1, . . . , In} pred(Ij): rightmost interval completely left to Ij (if it exists)

1 2 3 4 5 6

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Sujoy Bhore, Jan-Henrik Haunert, Fabian Klute, Guangping Li, Martin N¨

  • llenburg · Balanced Independent and Dominating Sets on Colored Interval Graphs

6/7

f-BIS: An FPT Algorithm by (f, k)

sorted set of intervals I = {I1, . . . , In} CI′ is valid: I′ is independent and ci ≤ f cardinality vector CI′: k-dimensional vector (c1, . . . , ci, . . . , ck) cardinality of intervals of color i in I′ pred(Ij): rightmost interval completely left to Ij (if it exists)

1 2 3 4 5 6

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SLIDE 26

Sujoy Bhore, Jan-Henrik Haunert, Fabian Klute, Guangping Li, Martin N¨

  • llenburg · Balanced Independent and Dominating Sets on Colored Interval Graphs

6/7

f-BIS: An FPT Algorithm by (f, k)

sorted set of intervals I = {I1, . . . , In} CI′ is valid: I′ is independent and ci ≤ f cardinality vector CI′: k-dimensional vector (c1, . . . , ci, . . . , ck) cardinality of intervals of color i in I′ pred(Ij): rightmost interval completely left to Ij (if it exists) Uj: union of valid cardinarlity vectors of {I1, . . . , Ij} U0 = {(0, . . . , 0)} Uj = Uj−1 ∪ {u ⊕ ˆ ec(Ij) | u ∈ Upred(Ij) }

1 2 3 4 5 6

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SLIDE 27

Sujoy Bhore, Jan-Henrik Haunert, Fabian Klute, Guangping Li, Martin N¨

  • llenburg · Balanced Independent and Dominating Sets on Colored Interval Graphs

6/7

f-BIS: An FPT Algorithm by (f, k)

sorted set of intervals I = {I1, . . . , In} CI′ is valid: I′ is independent and ci ≤ f cardinality vector CI′: k-dimensional vector (c1, . . . , ci, . . . , ck) cardinality of intervals of color i in I′ pred(Ij): rightmost interval completely left to Ij (if it exists) Uj: union of valid cardinarlity vectors of {I1, . . . , Ij} U0 = {(0, . . . , 0)} Uj = Uj−1 ∪ {u ⊕ ˆ ec(Ij) | u ∈ Upred(Ij) }

1 2 3 4 5 6 (0 . . . , 1, . . . , 0) c(Ij )

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SLIDE 28

Sujoy Bhore, Jan-Henrik Haunert, Fabian Klute, Guangping Li, Martin N¨

  • llenburg · Balanced Independent and Dominating Sets on Colored Interval Graphs

6/7

f-BIS: An FPT Algorithm by (f, k)

sorted set of intervals I = {I1, . . . , In} CI′ is valid: I′ is independent and ci ≤ f cardinality vector CI′: k-dimensional vector (c1, . . . , ci, . . . , ck) cardinality of intervals of color i in I′ pred(Ij): rightmost interval completely left to Ij (if it exists) Uj: union of valid cardinarlity vectors of {I1, . . . , Ij} U0 = {(0, . . . , 0)} Uj = Uj−1 ∪ {u ⊕ ˆ ec(Ij) | u ∈ Upred(Ij) }

1 2 3 4 5 6

O(n log n)

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SLIDE 29

Sujoy Bhore, Jan-Henrik Haunert, Fabian Klute, Guangping Li, Martin N¨

  • llenburg · Balanced Independent and Dominating Sets on Colored Interval Graphs

6/7

f-BIS: An FPT Algorithm by (f, k)

sorted set of intervals I = {I1, . . . , In} CI′ is valid: I′ is independent and ci ≤ f cardinality vector CI′: k-dimensional vector (c1, . . . , ci, . . . , ck) cardinality of intervals of color i in I′ pred(Ij): rightmost interval completely left to Ij (if it exists) Uj: union of valid cardinarlity vectors of {I1, . . . , Ij} U0 = {(0, . . . , 0)} Uj = Uj−1 ∪ {u ⊕ ˆ ec(Ij) | u ∈ Upred(Ij) }

1 2 3 4 5 6

O(n log n) O(|Un| × α(|Un|))

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SLIDE 30

Sujoy Bhore, Jan-Henrik Haunert, Fabian Klute, Guangping Li, Martin N¨

  • llenburg · Balanced Independent and Dominating Sets on Colored Interval Graphs

6/7

f-BIS: An FPT Algorithm by (f, k)

sorted set of intervals I = {I1, . . . , In} CI′ is valid: I′ is independent and ci ≤ f cardinality vector CI′: k-dimensional vector (c1, . . . , ci, . . . , ck) cardinality of intervals of color i in I′ pred(Ij): rightmost interval completely left to Ij (if it exists) Uj: union of valid cardinarlity vectors of {I1, . . . , Ij} U0 = {(0, . . . , 0)} Uj = Uj−1 ∪ {u ⊕ ˆ ec(Ij) | u ∈ Upred(Ij) }

1 2 3 4 5 6

O(n log n) |Uj| = O(f k) O(|Un| × α(|Un|))

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SLIDE 31

Sujoy Bhore, Jan-Henrik Haunert, Fabian Klute, Guangping Li, Martin N¨

  • llenburg · Balanced Independent and Dominating Sets on Colored Interval Graphs

6/7

f-BIS: An FPT Algorithm by (f, k)

sorted set of intervals I = {I1, . . . , In} CI′ is valid: I′ is independent and ci ≤ f cardinality vector CI′: k-dimensional vector (c1, . . . , ci, . . . , ck) cardinality of intervals of color i in I′ pred(Ij): rightmost interval completely left to Ij (if it exists) Uj: union of valid cardinarlity vectors of {I1, . . . , Ij} U0 = {(0, . . . , 0)} Uj = Uj−1 ∪ {u ⊕ ˆ ec(Ij) | u ∈ Upred(Ij) }

1 2 3 4 5 6

O(n log n) |Uj| = O(f k) O(|Un| × α(|Un|))

runtime: O(n log n + nf kα(f k))

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SLIDE 32

Sujoy Bhore, Jan-Henrik Haunert, Fabian Klute, Guangping Li, Martin N¨

  • llenburg · Balanced Independent and Dominating Sets on Colored Interval Graphs

7/7

Conclusion

f-Balanced Independent Set: NP-hardness FPT by (f, k)

  • ur results
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SLIDE 33

Sujoy Bhore, Jan-Henrik Haunert, Fabian Klute, Guangping Li, Martin N¨

  • llenburg · Balanced Independent and Dominating Sets on Colored Interval Graphs

7/7

Conclusion

f-Balanced Independent Set: NP-hardness FPT by (f, k)

  • ur results

FPT by the Vertex Cover Number NP-hardness of f-Balanced Dominating Set relevant problems: 2-approximation for 1-Max-Colored Independent Sets

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SLIDE 34

Sujoy Bhore, Jan-Henrik Haunert, Fabian Klute, Guangping Li, Martin N¨

  • llenburg · Balanced Independent and Dominating Sets on Colored Interval Graphs

7/7

Conclusion

f-Balanced Independent Set: NP-hardness FPT by (f, k)

  • ur results
  • pen problems:

balanced set on intersection graphs (e.g. boxicity graphs) FPT by the Vertex Cover Number NP-hardness of f-Balanced Dominating Set relevant problems: 2-approximation for 1-Max-Colored Independent Sets

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SLIDE 35

Sujoy Bhore, Jan-Henrik Haunert, Fabian Klute, Guangping Li, Martin N¨

  • llenburg · Balanced Independent and Dominating Sets on Colored Interval Graphs

7/7

Conclusion

f-Balanced Independent Set: NP-hardness FPT by (f, k)

  • ur results
  • pen problems:

balanced set on intersection graphs (e.g. boxicity graphs) FPT by the Vertex Cover Number NP-hardness of f-Balanced Dominating Set relevant problems: 2-approximation for 1-Max-Colored Independent Sets

guangping@ac.tuwien.ac.at