ON THE COMPLEXITY OF BARRIER RESILIENCE FOR FAT REGIONS Matias - PowerPoint PPT Presentation

IDEA: Cut open the domain along π . Now the problem looks almost like computing the resilience of a strip! . . . but, the optimal solution may still cross π . . . LEMMA: There are only 2 O ( r ) possible crossings patterns. W s t

IDEA: Cut open the domain along π . Now the problem looks almost like computing the resilience of a strip! . . . but, the optimal solution may still cross π . . . LEMMA: There are only 2 O ( r ) possible crossings patterns. So. . . we just try them all. W s t

IDEA: Cut open the domain along π . Now the problem looks almost like computing the resilience of a strip! . . . but, the optimal solution may still cross π . . . LEMMA: There are only 2 O ( r ) possible crossings patterns. So. . . we just try them all. W s t

Now we have a nice, clean problem. W s t

Now we have a nice, clean problem. Given a simply connected region with pairs of points on the boundary, remove the smallest number of disks such that all pairs are connected. W s t

Now we have a nice, clean problem. Given a simply connected region with pairs of points on the boundary, remove the smallest number of disks such that all pairs are connected. OBSERVATION: The geometry no longer matters. W s t

Now we have a nice, clean problem. Given a simply connected region with pairs of points on the boundary, remove the smallest number of disks such that all pairs are connected. OBSERVATION: The geometry no longer matters. W t s

Consider the intersection graph. W t s

Consider the intersection graph. t s

Consider the intersection graph. Add special terminal vertices for pieces of boundary (at most O ( r ) ). t s

Consider the intersection graph. Add special terminal vertices for pieces of boundary (at most O ( r ) ). t s

Consider the intersection graph. Add special terminal vertices for pieces of boundary (at most O ( r ) ). We want to cut this graph “along the dotted lines”. t s

Consider the intersection graph. Add special terminal vertices for pieces of boundary (at most O ( r ) ). We want to cut this graph “along the dotted lines”. OBSERVATION: This is the same as cutting between all pairs of terminal vertices that are separated by at least one dotted line. t s

Consider the intersection graph. Add special terminal vertices for pieces of boundary (at most O ( r ) ). We want to cut this graph “along the dotted lines”. OBSERVATION: This is the same as cutting between all pairs of terminal vertices that are separated by at least one dotted line. t s

Consider the intersection graph. Add special terminal vertices for pieces of boundary (at most O ( r ) ). We want to cut this graph “along the dotted lines”. OBSERVATION: This is the same as cutting between all pairs of terminal vertices that are separated by at least one dotted line. t s

Consider the intersection graph. Add special terminal vertices for pieces of boundary (at most O ( r ) ). We want to cut this graph “along the dotted lines”. OBSERVATION: This is the same as cutting between all pairs of terminal vertices that are separated by at least one dotted line. t s

This is known as vertex multicut .

This is known as vertex multicut . PROBLEM: Given a graph G = ( V, E ) and a set of forbidden pairs of vertices L , compute the smallest subset D ⊂ V such that all pairs in L are disconnected in G \ D .

This is known as vertex multicut . PROBLEM: Given a graph G = ( V, E ) and a set of forbidden pairs of vertices L , compute the smallest subset D ⊂ V such that all pairs in L are disconnected in G \ D . An optimal vertex multicut can be computed in O (2 f ( | D | , | L | ) n 3 ) time. [Xiao, 2010]

This is known as vertex multicut . PROBLEM: Given a graph G = ( V, E ) and a set of forbidden pairs of vertices L , compute the smallest subset D ⊂ V such that all pairs in L are disconnected in G \ D . An optimal vertex multicut can be computed in O (2 f ( | D | , | L | ) n 3 ) time. [Xiao, 2010]

This is known as vertex multicut . PROBLEM: Given a graph G = ( V, E ) and a set of forbidden pairs of vertices L , compute the smallest subset D ⊂ V such that all pairs in L are disconnected in G \ D . An optimal vertex multicut can be computed in O (2 f ( | D | , | L | ) n 3 ) time. [Xiao, 2010]

This is known as vertex multicut . PROBLEM: Given a graph G = ( V, E ) and a set of forbidden pairs of vertices L , compute the smallest subset D ⊂ V such that all pairs in L are disconnected in G \ D . An optimal vertex multicut can be computed in O (2 f ( | D | , | L | ) n 3 ) time. [Xiao, 2010] t s

This is known as vertex multicut . PROBLEM: Given a graph G = ( V, E ) and a set of forbidden pairs of vertices L , compute the smallest subset D ⊂ V such that all pairs in L are disconnected in G \ D . An optimal vertex multicut can be computed in O (2 f ( | D | , | L | ) n 3 ) time. [Xiao, 2010] s

This is known as vertex multicut . PROBLEM: Given a graph G = ( V, E ) and a set of forbidden pairs of vertices L , compute the smallest subset D ⊂ V such that all pairs in L are disconnected in G \ D . An optimal vertex multicut can be computed in O (2 f ( | D | , | L | ) n 3 ) time. [Xiao, 2010] s

This is known as vertex multicut . PROBLEM: Given a graph G = ( V, E ) and a set of forbidden pairs of vertices L , compute the smallest subset D ⊂ V such that all pairs in L are disconnected in G \ D . An optimal vertex multicut can be computed in O (2 f ( | D | , | L | ) n 3 ) time. [Xiao, 2010] W s t

This is known as vertex multicut . PROBLEM: Given a graph G = ( V, E ) and a set of forbidden pairs of vertices L , compute the smallest subset D ⊂ V such that all pairs in L are disconnected in G \ D . An optimal vertex multicut can be computed in O (2 f ( | D | , | L | ) n 3 ) time. [Xiao, 2010] s t

This is known as vertex multicut . PROBLEM: Given a graph G = ( V, E ) and a set of forbidden pairs of vertices L , compute the smallest subset D ⊂ V such that all pairs in L are disconnected in G \ D . An optimal vertex multicut can be computed in O (2 f ( | D | , | L | ) n 3 ) time. [Xiao, 2010] s t

This is known as vertex multicut . PROBLEM: Given a graph G = ( V, E ) and a set of forbidden pairs of vertices L , compute the smallest subset D ⊂ V such that all pairs in L are disconnected in G \ D . An optimal vertex multicut can be computed in O (2 f ( | D | , | L | ) n 3 ) time. [Xiao, 2010] s t

ε -APPROXIMATION ALGORITHM

The ply δ of a point is the total number of regions that contain it.

The ply δ of a point is the total number of regions that contain it.

The ply δ of a point is the total number of regions that contain it. δ = 1 δ = 3

The ply δ of a point is the total number of regions that contain it. The ply ∆ of R is the maximum ply over all points in the plane. δ = 1 δ = 3

The ply δ of a point is the total number of regions that contain it. The ply ∆ of R is the maximum ply over all points in the plane. ∆ = 4 δ = 1 δ = 3

Let k = ⌈ 4∆ /ε 2 ⌉ .

Let k = ⌈ 4∆ /ε 2 ⌉ . Compute all pairs of resilience at most k exactly, using the FPT algorithm.

Let k = ⌈ 4∆ /ε 2 ⌉ . Compute all pairs of resilience at most k exactly, using the FPT algorithm.

Let k = ⌈ 4∆ /ε 2 ⌉ . Compute all pairs of resilience at most k exactly, using the FPT algorithm. s t