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

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 Daniel’s Broiler Circa Metropolitan Grill Maximilien Sodo Deli labels are at the boundary of the focus region 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. 2/7 Sujoy Bhore, Jan-Henrik Haunert, Fabian Klute, Guangping Li , Martin N¨ ollenburg · Balanced Independent and Dominating Sets on Colored Interval Graphs

Boundary labeling � by Jan-Henrik Haunert c labels represent objects of multiple categories task: select a good mixture of different object types 2/7 Sujoy Bhore, Jan-Henrik Haunert, Fabian Klute, Guangping Li , Martin N¨ ollenburg · Balanced Independent and Dominating Sets on Colored Interval Graphs

Model input: a set of n colored axis-parallel unit squres touching a disk D rectangle: icon D 3/7 Sujoy Bhore, Jan-Henrik Haunert, Fabian Klute, Guangping Li , Martin N¨ ollenburg · Balanced Independent and Dominating Sets on Colored Interval Graphs

Model input: a set of n colored axis-parallel unit squres touching a disk D rectangle: icon D interval representation of its intersection model 3/7 Sujoy Bhore, Jan-Henrik Haunert, Fabian Klute, Guangping Li , Martin N¨ ollenburg · Balanced Independent and Dominating Sets on Colored Interval Graphs

Model input: a set I of n intervals on the real line each interval is colored by a coloring c : I → { 1 , . . . , k } 4/7 Sujoy Bhore, Jan-Henrik Haunert, Fabian Klute, Guangping Li , Martin N¨ ollenburg · Balanced Independent and Dominating Sets on Colored Interval Graphs

Model input: a set I of n intervals on the real line each interval is colored by a coloring c : I → { 1 , . . . , k } goal: f -Balanced Independent Set ( f - BIS ) an independent set M ⊆ I M contains exactly f elements from each of k color classes 4/7 Sujoy Bhore, Jan-Henrik Haunert, Fabian Klute, Guangping Li , Martin N¨ ollenburg · Balanced Independent and Dominating Sets on Colored Interval Graphs

Model input: a set I of n intervals on the real line each interval is colored by a coloring c : I → { 1 , . . . , k } goal: f -Balanced Independent Set ( f - BIS ) an independent set M ⊆ I M contains exactly f elements from each of k color classes 1 - BIS 4/7 Sujoy Bhore, Jan-Henrik Haunert, Fabian Klute, Guangping Li , Martin N¨ ollenburg · Balanced Independent and Dominating Sets on Colored Interval Graphs

1 -BIS Problem: NP hardness 5/7 Sujoy Bhore, Jan-Henrik Haunert, Fabian Klute, Guangping Li , Martin N¨ ollenburg · Balanced Independent and Dominating Sets on Colored Interval Graphs

1 -BIS Problem: NP hardness reduction from 3-bounded 3SAT each variable x i appears in ≤ 3 clauses each clause C j has 2 or 3 literals ( x 1 ∨ x 2 ∨ x 4 ) ∧ ( x 1 ∨ x 3 ∨ x 4 ) ∧ ( x 3 ∨ x 4 ) ∧ ( x 1 ∨ x 2 ∨ x 3 ) C 2 C 3 C 4 C 1 5/7 Sujoy Bhore, Jan-Henrik Haunert, Fabian Klute, Guangping Li , Martin N¨ ollenburg · Balanced Independent and Dominating Sets on Colored Interval Graphs

1 -BIS Problem: NP hardness reduction from 3-bounded 3SAT each variable x i appears in ≤ 3 clauses each clause C j has 2 or 3 literals gadgets: ( x 1 ∨ x 2 ∨ x 4 ) ∧ ( x 1 ∨ x 3 ∨ x 4 ) ∧ ( x 3 ∨ x 4 ) ∧ ( x 1 ∨ x 2 ∨ x 3 ) C 2 C 3 C 4 C 1 5/7 Sujoy Bhore, Jan-Henrik Haunert, Fabian Klute, Guangping Li , Martin N¨ ollenburg · Balanced Independent and Dominating Sets on Colored Interval Graphs

1 -BIS Problem: NP hardness reduction from 3-bounded 3SAT each variable x i appears in ≤ 3 clauses each clause C j has 2 or 3 literals gadgets: clause: color ( x 1 ∨ x 2 ∨ x 4 ) ∧ ( x 1 ∨ x 3 ∨ x 4 ) ∧ ( x 3 ∨ x 4 ) ∧ ( x 1 ∨ x 2 ∨ x 3 ) C 2 C 3 C 4 C 1 5/7 Sujoy Bhore, Jan-Henrik Haunert, Fabian Klute, Guangping Li , Martin N¨ ollenburg · Balanced Independent and Dominating Sets on Colored Interval Graphs

1 -BIS Problem: NP hardness reduction from 3-bounded 3SAT each variable x i appears in ≤ 3 clauses each clause C j has 2 or 3 literals gadgets: clause: color variable: one (colored) interval for each occurence intersection: each pair of opposite literals x 1 + − ( x 1 ∨ x 2 ∨ x 4 ) ∧ ( x 1 ∨ x 3 ∨ x 4 ) ∧ ( x 3 ∨ x 4 ) ∧ ( x 1 ∨ x 2 ∨ x 3 ) C 2 C 2 C 3 C 3 C 4 C 4 C 1 C 1 5/7 Sujoy Bhore, Jan-Henrik Haunert, Fabian Klute, Guangping Li , Martin N¨ ollenburg · Balanced Independent and Dominating Sets on Colored Interval Graphs

1 -BIS Problem: NP hardness reduction from 3-bounded 3SAT each variable x i appears in ≤ 3 clauses each clause C j has 2 or 3 literals gadgets: clause: color variable: one (colored) interval for each occurence intersection: each pair of opposite literals x 3 + − ( x 1 ∨ x 2 ∨ x 4 ) ∧ ( x 1 ∨ x 3 ∨ x 4 ) ∧ ( x 3 ∨ x 4 ) ∧ ( x 1 ∨ x 2 ∨ x 3 ) C 2 C 3 C 4 C 2 C 3 C 4 C 1 C 1 5/7 Sujoy Bhore, Jan-Henrik Haunert, Fabian Klute, Guangping Li , Martin N¨ ollenburg · Balanced Independent and Dominating Sets on Colored Interval Graphs

1 -BIS Problem: NP hardness reduction from 3-bounded 3SAT each variable x i appears in ≤ 3 clauses each clause C j has 2 or 3 literals gadgets: clause: color variable: one (colored) interval for each occurence intersection: each pair of opposite literals x 1 x 2 x 3 x 4 + − ( x 1 ∨ x 2 ∨ x 4 ) ∧ ( x 1 ∨ x 3 ∨ x 4 ) ∧ ( x 3 ∨ x 4 ) ∧ ( x 1 ∨ x 2 ∨ x 3 ) C 2 C 2 C 3 C 3 C 4 C 4 C 2 C 3 C 4 C 1 C 1 C 1 5/7 Sujoy Bhore, Jan-Henrik Haunert, Fabian Klute, Guangping Li , Martin N¨ ollenburg · Balanced Independent and Dominating Sets on Colored Interval Graphs

1 -BIS Problem: NP hardness Correctness x 1 x 2 x 3 x 4 + − ( x 1 ∨ x 2 ∨ x 4 ) ∧ ( x 1 ∨ x 3 ∨ x 4 ) ∧ ( x 3 ∨ x 4 ) ∧ ( x 1 ∨ x 2 ∨ x 3 ) C 2 C 2 C 2 C 3 C 3 C 3 C 4 C 4 C 4 C 1 C 1 C 1 1-BIS ⇒ : 5/7 Sujoy Bhore, Jan-Henrik Haunert, Fabian Klute, Guangping Li , Martin N¨ ollenburg · Balanced Independent and Dominating Sets on Colored Interval Graphs

1 -BIS Problem: NP hardness Correctness x 1 x 2 x 3 x 4 + − ( x 1 ∨ x 2 ∨ x 4 ) ∧ ( x 1 ∨ x 3 ∨ x 4 ) ∧ ( x 3 ∨ x 4 ) ∧ ( x 1 ∨ x 2 ∨ x 3 ) C 2 C 2 C 2 C 3 C 3 C 3 C 4 C 4 C 4 C 1 C 1 C 1 1-BIS ⇒ : 5/7 Sujoy Bhore, Jan-Henrik Haunert, Fabian Klute, Guangping Li , Martin N¨ ollenburg · Balanced Independent and Dominating Sets on Colored Interval Graphs

1 -BIS Problem: NP hardness Correctness x 1 x 2 x 3 x 4 + − ( x 1 ∨ x 2 ∨ x 4 ) ∧ ( x 1 ∨ x 3 ∨ x 4 ) ∧ ( x 3 ∨ x 4 ) ∧ ( x 1 ∨ x 2 ∨ x 3 ) C 2 C 2 C 2 C 3 C 3 C 3 C 4 C 4 C 4 C 1 C 1 C 1 1-BIS ⇒ : evaluate the chosen literals as true 5/7 Sujoy Bhore, Jan-Henrik Haunert, Fabian Klute, Guangping Li , Martin N¨ ollenburg · Balanced Independent and Dominating Sets on Colored Interval Graphs

1 -BIS Problem: NP hardness Correctness x 1 x 2 x 3 x 4 + − ( x 1 ∨ x 2 ∨ x 4 ) ∧ ( x 1 ∨ x 3 ∨ x 4 ) ∧ ( x 3 ∨ x 4 ) ∧ ( x 1 ∨ x 2 ∨ x 3 ) C 2 C 2 C 2 C 3 C 3 C 3 C 4 C 4 C 4 C 1 C 1 C 1 evaluate the chosen literals as true 1-BIS ⇒ : ⇐ assignment: 5/7 Sujoy Bhore, Jan-Henrik Haunert, Fabian Klute, Guangping Li , Martin N¨ ollenburg · Balanced Independent and Dominating Sets on Colored Interval Graphs

1 -BIS Problem: NP hardness Correctness x 1 x 2 x 3 x 4 + − ( x 1 ∨ x 2 ∨ x 4 ) ∧ ( x 1 ∨ x 3 ∨ x 4 ) ∧ ( x 3 ∨ x 4 ) ∧ ( x 1 ∨ x 2 ∨ x 3 ) C 2 C 2 C 2 C 3 C 3 C 3 C 4 C 4 C 4 C 1 C 1 C 1 evaluate the chosen literals as true 1-BIS ⇒ : ⇐ assignment: { x 1 : T , x 2 : F , x 3 : T , x 4 : F } 5/7 Sujoy Bhore, Jan-Henrik Haunert, Fabian Klute, Guangping Li , Martin N¨ ollenburg · Balanced Independent and Dominating Sets on Colored Interval Graphs

1 -BIS Problem: NP hardness Correctness x 1 x 2 x 3 x 4 + − ( x 1 ∨ x 2 ∨ x 4 ) ∧ ( x 1 ∨ x 3 ∨ x 4 ) ∧ ( x 3 ∨ x 4 ) ∧ ( x 1 ∨ x 2 ∨ x 3 ) C 2 C 2 C 2 C 3 C 3 C 3 C 4 C 4 C 4 C 1 C 1 C 1 evaluate the chosen literals as true 1-BIS ⇒ : ⇐ assignment: choose a positive evaluated literal in each C i { x 1 : T , x 2 : F , x 3 : T , x 4 : F } 5/7 Sujoy Bhore, Jan-Henrik Haunert, Fabian Klute, Guangping Li , Martin N¨ ollenburg · Balanced Independent and Dominating Sets on Colored Interval Graphs

1 -BIS Problem: NP hardness Correctness x 1 x 2 x 3 x 4 + − ( x 1 ∨ x 2 ∨ x 4 ) ∧ ( x 1 ∨ x 3 ∨ x 4 ) ∧ ( x 3 ∨ x 4 ) ∧ ( x 1 ∨ x 2 ∨ x 3 ) C 2 C 2 C 2 C 3 C 3 C 3 C 4 C 4 C 4 C 1 C 1 C 1 evaluate the chosen literals as true 1-BIS ⇒ : ⇐ assignment: choose a positive evaluated literal in each C i { x 1 : T , x 2 : F , x 3 : T , x 4 : F } 5/7 Sujoy Bhore, Jan-Henrik Haunert, Fabian Klute, Guangping Li , Martin N¨ ollenburg · Balanced Independent and Dominating Sets on Colored Interval Graphs

f -BIS: An FPT Algorithm by ( f, k ) sorted by right-endpoints sorted set of intervals I = { I 1 , . . . , I n } 4 1 6 5 2 3 6/7 Sujoy Bhore, Jan-Henrik Haunert, Fabian Klute, Guangping Li , Martin N¨ ollenburg · Balanced Independent and Dominating Sets on Colored Interval Graphs