The Maximum Exposure Problem Neeraj Kumar, Stavros Sintos, Subhash - - PowerPoint PPT Presentation

The Maximum Exposure Problem

Neeraj Kumar, Stavros Sintos, Subhash Suri University of California, Santa Barbara Duke University

Problem Description

Set of points P in the plane,

Problem Description

integer parameter k Set of points P in the plane, set of rectangular ranges R covering them,

Problem Description

integer parameter k find k ranges to delete so as to ‘expose’ a maximum number of points Set of points P in the plane, set of rectangular ranges R covering them,

Problem Description

integer parameter k

k = 1

find k ranges to delete so as to ‘expose’ a maximum number of points Set of points P in the plane, set of rectangular ranges R covering them,

⇒

Problem Description

integer parameter k

k = 2

find k ranges to delete so as to ‘expose’ a maximum number of points Set of points P in the plane, set of rectangular ranges R covering them,

⇒

Problem Description

integer parameter k

k = 3

find k ranges to delete so as to ‘expose’ a maximum number of points Set of points P in the plane, set of rectangular ranges R covering them,

⇒

Motivation

Reliability of coverage: points correspond to clients, ranges correspond

to coverage of facilities

Motivation

Which k facilities to disable so as to affect maximum number of clients?

Reliability of coverage: points correspond to clients, ranges correspond

to coverage of facilities

Motivation

Which k facilities to disable so as to affect maximum number of clients?

Reliability of coverage: points correspond to clients, ranges correspond Geometric constraint removal: ranges correspond to constraints,

Maximize rewards by removing at most k constraints points correspond to rewards to coverage of facilities

Hardness of Max Exposure

Geometric counterpart of the densest k-subhypergraph problem

– studied recently in (APPROX’16, SODA’17), conditionally hard to approximate within ∣V ∣1−ϵ

Hardness of Max Exposure

Geometric counterpart of the densest k-subhypergraph problem

– ranges R correspond to vertices of the hypergraph, points P correspond to edges (defined by containment relation) – studied recently in (APPROX’16, SODA’17), conditionally hard to approximate within ∣V ∣1−ϵ

Hardness of Max Exposure

Geometric counterpart of the densest k-subhypergraph problem

– ranges R correspond to vertices of the hypergraph, points P correspond to edges (defined by containment relation)

With convex polygons, max-exposure is as hard as densest k-subhypergraph

– Hypergraph H = (X, E) can be transformed into max-exposure of convex ranges R and points P – studied recently in (APPROX’16, SODA’17), conditionally hard to approximate within ∣V ∣1−ϵ

Hardness of Max Exposure

Geometric counterpart of the densest k-subhypergraph problem

– ranges R correspond to vertices of the hypergraph, points P correspond to edges (defined by containment relation)

With convex polygons, max-exposure is as hard as densest k-subhypergraph

– Hypergraph H = (X, E) can be transformed into max-exposure of convex ranges R and points P

What about rectangle ranges?

– studied recently in (APPROX’16, SODA’17), conditionally hard to approximate within ∣V ∣1−ϵ

Hardness of Max Exposure

Geometric counterpart of the densest k-subhypergraph problem

– ranges R correspond to vertices of the hypergraph, points P correspond to edges (defined by containment relation)

With convex polygons, max-exposure is as hard as densest k-subhypergraph

– Hypergraph H = (X, E) can be transformed into max-exposure of convex ranges R and points P

What about rectangle ranges? when rectangles in R are translates of two fixed rectangles

NP-hard and also ‘conditionally’ hard to approximate within O(n1/4) even

n = ∣R∣

– studied recently in (APPROX’16, SODA’17), conditionally hard to approximate within ∣V ∣1−ϵ

Hardness of Max Exposure

Geometric counterpart of the densest k-subhypergraph problem

– ranges R correspond to vertices of the hypergraph, points P correspond to edges (defined by containment relation)

With convex polygons, max-exposure is as hard as densest k-subhypergraph

– Hypergraph H = (X, E) can be transformed into max-exposure of convex ranges R and points P

What about rectangle ranges? when rectangles in R are translates of two fixed rectangles

NP-hard and also ‘conditionally’ hard to approximate within O(n1/4) even

Simple reduction from densest k-subgraph on bipartite graphs (bipartite-DkS)

1 2 3 a b c 1 2 3 a b c

n = ∣R∣

– studied recently in (APPROX’16, SODA’17), conditionally hard to approximate within ∣V ∣1−ϵ

Hardness of Max Exposure

Geometric counterpart of the densest k-subhypergraph problem

– ranges R correspond to vertices of the hypergraph, points P correspond to edges (defined by containment relation)

With convex polygons, max-exposure is as hard as densest k-subhypergraph

– Hypergraph H = (X, E) can be transformed into max-exposure of convex ranges R and points P

What about rectangle ranges? when rectangles in R are translates of two fixed rectangles

NP-hard and also ‘conditionally’ hard to approximate within O(n1/4) even

Simple reduction from densest k-subgraph on bipartite graphs (bipartite-DkS) – Assuming Dense Vs Random conjecture, bipartite-DkS is hard to approximate within O(∣V ∣1/4)

1 2 3 a b c 1 2 3 a b c

n = ∣R∣

– studied recently in (APPROX’16, SODA’17), conditionally hard to approximate within ∣V ∣1−ϵ

Approximation Algorithms

What happens if we only allow translates of a single rectangle? Can we do somewhat better for arbitrary rectangles?

Approximation Algorithms

A bicriteria O(k)-approximation for arbitrary rectangles

– Expose at least Ω(1/k) of optimal points by removing k2 rectangles – Approximation factor improves to O(

√ k) if rectangles have bounded aspect ratio

What happens if we only allow translates of a single rectangle? Can we do somewhat better for arbitrary rectangles?

Approximation Algorithms

There exists a PTAS when R consists of translates of a single rectangle A bicriteria O(k)-approximation for arbitrary rectangles

– Expose at least Ω(1/k) of optimal points by removing k2 rectangles – Builds upon a polynomial time 2-approximation using shifting techniques – Approximation factor improves to O(

√ k) if rectangles have bounded aspect ratio

What happens if we only allow translates of a single rectangle? Can we do somewhat better for arbitrary rectangles?

Approximation Algorithms

There exists a PTAS when R consists of translates of a single rectangle A bicriteria O(k)-approximation for arbitrary rectangles

– Expose at least Ω(1/k) of optimal points by removing k2 rectangles – Builds upon a polynomial time 2-approximation using shifting techniques

rest of this talk

– Approximation factor improves to O(

√ k) if rectangles have bounded aspect ratio

– Gives a constant approximation if ratio of smallest and longest sidelengths is bounded

What happens if we only allow translates of a single rectangle? Can we do somewhat better for arbitrary rectangles?

A Simple Bicriteria Approximation

The algorithm is essentially greedy: R(p) = set of ranges that contain point p

A Simple Bicriteria Approximation

The algorithm is essentially greedy: R(p) = set of ranges that contain point p

Discard all points for which ∣R(p)∣ > k

A Simple Bicriteria Approximation

The algorithm is essentially greedy: R(p) = set of ranges that contain point p

Discard all points for which ∣R(p)∣ > k Partition P into a set G of groups:

each group is an equivalence class of points with same R(p)

A Simple Bicriteria Approximation

The algorithm is essentially greedy: R(p) = set of ranges that contain point p

Discard all points for which ∣R(p)∣ > k Partition P into a set G of groups:

each group is an equivalence class of points with same R(p)

Sort groups in G by decreasing size and return points in first k groups

A Simple Bicriteria Approximation

The algorithm is essentially greedy: R(p) = set of ranges that contain point p

Discard all points for which ∣R(p)∣ > k Partition P into a set G of groups:

each group is an equivalence class of points with same R(p)

Sort groups in G by decreasing size and return points in first k groups

Total deleted ranges is at most k ⋅ max ∣R(p)∣ = k2

A Simple Bicriteria Approximation

The algorithm is essentially greedy: R(p) = set of ranges that contain point p

Discard all points for which ∣R(p)∣ > k Partition P into a set G of groups:

each group is an equivalence class of points with same R(p)

Sort groups in G by decreasing size and return points in first k groups

Total deleted ranges is at most k ⋅ max ∣R(p)∣ = k2 # of groups G∗ in optimal ≤ # of cells in arrangement of k rectangles ≤ c ⋅ k2

A Simple Bicriteria Approximation

The algorithm is essentially greedy: R(p) = set of ranges that contain point p

Discard all points for which ∣R(p)∣ > k Partition P into a set G of groups:

each group is an equivalence class of points with same R(p)

Sort groups in G by decreasing size and return points in first k groups

Total deleted ranges is at most k ⋅ max ∣R(p)∣ = k2 # of groups G∗ in optimal ≤ # of cells in arrangement of k rectangles ≤ c ⋅ k2 Holds for any polygon with O(1) complexity

Translates of a Single Rectangle

Translates of a Single Rectangle

First, scale the rectangles so that they become squares Goal now is to compute max-exposure of unit square ranges ⇒

Does not change any point-rectangle containment

Translates of a Single Rectangle

First, scale the rectangles so that they become squares Goal now is to compute max-exposure of unit square ranges ⇒ Consider an even simpler problem: all points lie inside a unit square

Does not change any point-rectangle containment

Translates of a Single Rectangle

First, scale the rectangles so that they become squares Goal now is to compute max-exposure of unit square ranges ⇒ Consider an even simpler problem: all points lie inside a unit square Roadmap Within a unit square → Within a horizontal strip of unit width → PTAS (polytime) (polytime) (shifting techniques)

Does not change any point-rectangle containment

⇒ 4-approximation ⇒ 2-approximation

Translates of a Single Rectangle

First, scale the rectangles so that they become squares Goal now is to compute max-exposure of unit square ranges ⇒ Consider an even simpler problem: all points lie inside a unit square Roadmap Within a unit square → Within a horizontal strip of unit width → PTAS (polytime) (polytime) (shifting techniques)

Does not change any point-rectangle containment

⇒ 4-approximation ⇒ 2-approximation

Max-Exposure Within a Unit Square

Consider the dynamic programming formulation : DP-template-0

Max-Exposure Within a Unit Square

Consider the dynamic programming formulation : DP-template-0 – Process points in P by increasing x-coordinates Active ranges : ranges that have at least one corner to the right of x = xi x = xi

Max-Exposure Within a Unit Square

Consider the dynamic programming formulation : DP-template-0 – Process points in P by increasing x-coordinates Active ranges : ranges that have at least one corner to the right of x = xi Expose pi ⇔ delete all ranges in R(pi) x = xi

Max-Exposure Within a Unit Square

Consider the dynamic programming formulation : DP-template-0 – Process points in P by increasing x-coordinates Active ranges : ranges that have at least one corner to the right of x = xi S(i, k′, Rd) = max ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ S(i + 1, k′, Rd) do not expose pi S(i + 1, k′ − ki, Rd ∪ R(pi)) + 1 expose pi Expose pi ⇔ delete all ranges in R(pi) x = xi

Max-Exposure Within a Unit Square

Consider the dynamic programming formulation : DP-template-0 – Process points in P by increasing x-coordinates Active ranges : ranges that have at least one corner to the right of x = xi S(i, k′, Rd) = max ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ S(i + 1, k′, Rd) do not expose pi S(i + 1, k′ − ki, Rd ∪ R(pi)) + 1 expose pi # of ranges that can be deleted to right of x = xi (0 ≤ k′ ≤ k) Expose pi ⇔ delete all ranges in R(pi) x = xi

Max-Exposure Within a Unit Square

Consider the dynamic programming formulation : DP-template-0 – Process points in P by increasing x-coordinates Active ranges : ranges that have at least one corner to the right of x = xi S(i, k′, Rd) = max ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ S(i + 1, k′, Rd) do not expose pi S(i + 1, k′ − ki, Rd ∪ R(pi)) + 1 expose pi # of ranges that can be deleted to right of x = xi (0 ≤ k′ ≤ k) Set of active ranges that were already deleted Expose pi ⇔ delete all ranges in R(pi) x = xi

Max-Exposure Within a Unit Square

Consider the dynamic programming formulation : DP-template-0 – Process points in P by increasing x-coordinates Active ranges : ranges that have at least one corner to the right of x = xi S(i, k′, Rd) = max ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ S(i + 1, k′, Rd) do not expose pi S(i + 1, k′ − ki, Rd ∪ R(pi)) + 1 expose pi # of ranges that can be deleted to right of x = xi (0 ≤ k′ ≤ k) Set of active ranges that were already deleted Expose pi ⇔ delete all ranges in R(pi) Optimal solution : S(0, k, ∅) x = xi

Max-Exposure Within a Unit Square

Consider the dynamic programming formulation : DP-template-0 – Process points in P by increasing x-coordinates Active ranges : ranges that have at least one corner to the right of x = xi S(i, k′, Rd) = max ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ S(i + 1, k′, Rd) do not expose pi S(i + 1, k′ − ki, Rd ∪ R(pi)) + 1 expose pi # of ranges that can be deleted to right of x = xi (0 ≤ k′ ≤ k) Set of active ranges that were already deleted Expose pi ⇔ delete all ranges in R(pi) Optimal solution : S(0, k, ∅) x = xi

Max-Exposure Within a Unit Square

Consider the dynamic programming formulation : DP-template-0 – Process points in P by increasing x-coordinates Active ranges : ranges that have at least one corner to the right of x = xi S(i, k′, Rd) = max ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ S(i + 1, k′, Rd) do not expose pi S(i + 1, k′ − ki, Rd ∪ R(pi)) + 1 expose pi # of ranges that can be deleted to right of x = xi (0 ≤ k′ ≤ k) Set of active ranges that were already deleted Expose pi ⇔ delete all ranges in R(pi) Optimal solution : S(0, k, ∅) x = xi

Max-Exposure Within a Unit Square

Consider the dynamic programming formulation : DP-template-0 – Process points in P by increasing x-coordinates Active ranges : ranges that have at least one corner to the right of x = xi S(i, k′, Rd) = max ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ S(i + 1, k′, Rd) do not expose pi S(i + 1, k′ − ki, Rd ∪ R(pi)) + 1 expose pi # of ranges that can be deleted to right of x = xi (0 ≤ k′ ≤ k) Set of active ranges that were already deleted Expose pi ⇔ delete all ranges in R(pi) Optimal solution : S(0, k, ∅) x = xi

Max-Exposure Within a Unit Square

Consider the dynamic programming formulation : DP-template-0 – Process points in P by increasing x-coordinates Active ranges : ranges that have at least one corner to the right of x = xi S(i, k′, Rd) = max ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ S(i + 1, k′, Rd) do not expose pi S(i + 1, k′ − ki, Rd ∪ R(pi)) + 1 expose pi # of ranges that can be deleted to right of x = xi (0 ≤ k′ ≤ k) Set of active ranges that were already deleted Expose pi ⇔ delete all ranges in R(pi) ki = ∣R(pi) \ Rd∣ Optimal solution : S(0, k, ∅) x = xi

Max-Exposure Within a Unit Square

How do we keep track of deleted range set Rd using polynomial space?

(0, 0)

Max-Exposure Within a Unit Square

Type-0: Unit square ranges that intersect x = 0 How do we keep track of deleted range set Rd using polynomial space?

(0, 0)

Max-Exposure Within a Unit Square

Type-0: Unit square ranges that intersect x = 0 Type-1: Unit square ranges that intersect x = 1 How do we keep track of deleted range set Rd using polynomial space?

(0, 0)

Max-Exposure Within a Unit Square

Type-0: Unit square ranges that intersect x = 0 Type-1: Unit square ranges that intersect x = 1 How do we keep track of deleted range set Rd using polynomial space? Suppose we only had Type-0 ranges:

(0, 0)

ℓ0 ℓ1 x = xi

R3

R3 is ‘anchored’ to ℓ0

Max-Exposure Within a Unit Square

Type-0: Unit square ranges that intersect x = 0 Type-1: Unit square ranges that intersect x = 1 How do we keep track of deleted range set Rd using polynomial space? Suppose we only had Type-0 ranges: q0 = Exposed point to left of x = xi closest to ℓ0

(0, 0)

ℓ0 ℓ1

already deleted

x = xi

R3

⇒ must contain q0 R3 is ‘anchored’ to ℓ0

Max-Exposure Within a Unit Square

Type-0: Unit square ranges that intersect x = 0 Type-1: Unit square ranges that intersect x = 1 How do we keep track of deleted range set Rd using polynomial space? Suppose we only had Type-0 ranges: q0 = Exposed point to left of x = xi closest to ℓ0

(0, 0)

q1 = Exposed point to left of x = xi closest to ℓ1 ℓ0 ℓ1

already deleted

x = xi

R3

⇒ must contain q0 R3 is ‘anchored’ to ℓ0

Max-Exposure Within a Unit Square

Type-0: Unit square ranges that intersect x = 0 Type-1: Unit square ranges that intersect x = 1 How do we keep track of deleted range set Rd using polynomial space? Suppose we only had Type-0 ranges: Rd = R(q0) ∪ R(q1) q0 = Exposed point to left of x = xi closest to ℓ0

(0, 0)

q1 = Exposed point to left of x = xi closest to ℓ1 ℓ0 ℓ1

already deleted

x = xi

R3

⇒ must contain q0 R3 is ‘anchored’ to ℓ0

Max-Exposure Within a Unit Square

Type-0: Unit square ranges that intersect x = 0 Type-1: Unit square ranges that intersect x = 1 How do we keep track of deleted range set Rd using polynomial space? Suppose we only had Type-0 ranges: Rd = R(q0) ∪ R(q1) q0 = Exposed point to left of x = xi closest to ℓ0

(0, 0)

q1 = Exposed point to left of x = xi closest to ℓ1 ℓ0 ℓ1

already deleted

x = xi

R3

⇒ must contain q0 Can keep track of Type-0 deleted ranges by remembering q0,q1 R3 is ‘anchored’ to ℓ0

Handling Type-1 Ranges

Need an alternative dynamic programming formulation : DP-template-1

Handling Type-1 Ranges

Need an alternative dynamic programming formulation : DP-template-1 – Process ‘events’ in P by increasing x-coordinates xi

begin-range events point events

Handling Type-1 Ranges

Need an alternative dynamic programming formulation : DP-template-1 – Process ‘events’ in P by increasing x-coordinates xi

begin-range events point events

Active Points : with x-coordinates ≥ xi

Handling Type-1 Ranges

Need an alternative dynamic programming formulation : DP-template-1 – Process ‘events’ in P by increasing x-coordinates xi

begin-range events point events

S(i, k′, Pf ) = max ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ S(i + 1, k′ − 1, Pf ) delete range Ri S(i + 1, k′, Pf ∪ P(Ri)) do not delete Ri = max ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ S(i + 1, k′, Pf ) if pi ∈ Pf , cannot expose pi S(i + 1, k′, Pf ) + 1

• therwise, expose pi

Maintain set of forbidden points Pf

active points that lie in a range that was not deleted

Optimal solution : S(0, k, ∅) Active Points : with x-coordinates ≥ xi

Handling Type-1 Ranges

Need an alternative dynamic programming formulation : DP-template-1 – Process ‘events’ in P by increasing x-coordinates xi S(i, k′, Pf ) = max ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ S(i + 1, k′ − 1, Pf ) delete range Ri S(i + 1, k′, Pf ∪ P(Ri)) do not delete Ri = max ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ S(i + 1, k′, Pf ) if pi ∈ Pf , cannot expose pi S(i + 1, k′, Pf ) + 1

• therwise, expose pi

begin-range Ri

Maintain set of forbidden points Pf

active points that lie in a range that was not deleted

Optimal solution : S(0, k, ∅) Active Points : with x-coordinates ≥ xi x = xi

Handling Type-1 Ranges

Need an alternative dynamic programming formulation : DP-template-1 – Process ‘events’ in P by increasing x-coordinates xi S(i, k′, Pf ) = max ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ S(i + 1, k′ − 1, Pf ) delete range Ri S(i + 1, k′, Pf ∪ P(Ri)) do not delete Ri = max ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ S(i + 1, k′, Pf ) if pi ∈ Pf , cannot expose pi S(i + 1, k′, Pf ) + 1

• therwise, expose pi

begin-range Ri

Maintain set of forbidden points Pf

active points that lie in a range that was not deleted

Optimal solution : S(0, k, ∅)

All points contained in Ri

Active Points : with x-coordinates ≥ xi x = xi

Handling Type-1 Ranges

Need an alternative dynamic programming formulation : DP-template-1 – Process ‘events’ in P by increasing x-coordinates xi S(i, k′, Pf ) = max ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ S(i + 1, k′ − 1, Pf ) delete range Ri S(i + 1, k′, Pf ∪ P(Ri)) do not delete Ri = max ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ S(i + 1, k′, Pf ) if pi ∈ Pf , cannot expose pi S(i + 1, k′, Pf ) + 1

• therwise, expose pi

Maintain set of forbidden points Pf

active points that lie in a range that was not deleted Point pi

Optimal solution : S(0, k, ∅) Active Points : with x-coordinates ≥ xi x = xi

Handling Type-1 Ranges

How do we keep track of forbidden points Pf using polynomial space? ℓ0 ℓ1 Q0 = Undeleted range to left of x = xi farthest from ℓ0 x = xi

p

Handling Type-1 Ranges

How do we keep track of forbidden points Pf using polynomial space? ℓ0 ℓ1 Q0 = Undeleted range to left of x = xi farthest from ℓ0 Q1 = Undeleted range to left of x = xi farthest from ℓ1 Pf = P(Q0) ∪ P(Q1) x = xi

p

if p ∈ Pf , then p must lie in either Q0 or Q1

Handling Type-1 Ranges

How do we keep track of forbidden points Pf using polynomial space? ℓ0 ℓ1 Q0 = Undeleted range to left of x = xi farthest from ℓ0 Q1 = Undeleted range to left of x = xi farthest from ℓ1 Pf = P(Q0) ∪ P(Q1) Can keep track of forbidden points by remembering Q0,Q1 x = xi

p

if p ∈ Pf , then p must lie in either Q0 or Q1

Handling Type-1 Ranges

How do we keep track of forbidden points Pf using polynomial space? ℓ0 ℓ1 Q0 = Undeleted range to left of x = xi farthest from ℓ0 Q1 = Undeleted range to left of x = xi farthest from ℓ1 Pf = P(Q0) ∪ P(Q1) Can keep track of forbidden points by remembering Q0,Q1 x = xi Combine DP-template-0 and DP-template-1 to solve wihin a unit square: Subproblems defined as : S(i, k′, q0, q1, Q0, Q1)

updated appropriately at begin-range and point events

p

if p ∈ Pf , then p must lie in either Q0 or Q1

In Summary:

In Summary:

Max-exposure : to expose maximum points by deleting k ranges

In Summary:

Max-exposure : to expose maximum points by deleting k ranges Hard to approximate – even with restricted rectangular ranges

In Summary:

Max-exposure : to expose maximum points by deleting k ranges Hard to approximate – even with restricted rectangular ranges Exhibits a PTAS for unit-square ranges

– Gives a constant approximation for rectangles if ratio of smallest and longest sidelengths is bounded

In Summary:

Max-exposure : to expose maximum points by deleting k ranges Hard to approximate – even with restricted rectangular ranges Exhibits a PTAS for unit-square ranges

– Gives a constant approximation for rectangles if ratio of smallest and longest sidelengths is bounded

Bi-criteria O(k)-approximation algorithm for rectangles, O(

√ k) for squares

In Summary:

Max-exposure : to expose maximum points by deleting k ranges Hard to approximate – even with restricted rectangular ranges Exhibits a PTAS for unit-square ranges

– Gives a constant approximation for rectangles if ratio of smallest and longest sidelengths is bounded

Bi-criteria O(k)-approximation algorithm for rectangles, O(

√ k) for squares

Does there exist a constant approximation for arbitrary squares?

In Summary:

Max-exposure : to expose maximum points by deleting k ranges Hard to approximate – even with restricted rectangular ranges Exhibits a PTAS for unit-square ranges

– Gives a constant approximation for rectangles if ratio of smallest and longest sidelengths is bounded

Bi-criteria O(k)-approximation algorithm for rectangles, O(

√ k) for squares

Does there exist a constant approximation for arbitrary squares?

Thanks!

Backup: Combined DP

= max ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ S(i + 1, k′, q0, q1, Q0, Q1)

if pi ∈ Pf , cannot expose pi

S(i + 1, k′, q0, q1, Q0, Q1)

choose to not expose pi

S(i + 1, k′ − ki, closer(pi, q0), closer(pi, q1), Q0, Q1) + 1

• therwise, expose pi

= max ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ S(i + 1, k′ − 1, q0, q1, Q0, Q1)

delete Type-1 range Ri

S(i + 1, k′, q0, q1, farther(Ri, Q0), Q1) Ri is not deleted and anchored to ℓ0 S(i + 1, k′, q0, q1, Q0, farther(Ri, Q1)) Ri is not deleted and anchored to ℓ1 S(i, k′, q0, q1, Q0, Q1)

begin-range Ri Point pi

Backup: Combined DP

= max ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ S(i + 1, k′, q0, q1, Q0, Q1)

if pi ∈ Pf , cannot expose pi

S(i + 1, k′, q0, q1, Q0, Q1)

choose to not expose pi

S(i + 1, k′ − ki, closer(pi, q0), closer(pi, q1), Q0, Q1) + 1

• therwise, expose pi

= max ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ S(i + 1, k′ − 1, q0, q1, Q0, Q1)

delete Type-1 range Ri

S(i + 1, k′, q0, q1, farther(Ri, Q0), Q1) Ri is not deleted and anchored to ℓ0 S(i + 1, k′, q0, q1, Q0, farther(Ri, Q1)) Ri is not deleted and anchored to ℓ1 S(i, k′, q0, q1, Q0, Q1)

begin-range Ri Point pi

Backup: Combined DP

= max ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ S(i + 1, k′, q0, q1, Q0, Q1)

if pi ∈ Pf , cannot expose pi

S(i + 1, k′, q0, q1, Q0, Q1)

choose to not expose pi

S(i + 1, k′ − ki, closer(pi, q0), closer(pi, q1), Q0, Q1) + 1

• therwise, expose pi

= max ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ S(i + 1, k′ − 1, q0, q1, Q0, Q1)

delete Type-1 range Ri

S(i + 1, k′, q0, q1, farther(Ri, Q0), Q1) Ri is not deleted and anchored to ℓ0 S(i + 1, k′, q0, q1, Q0, farther(Ri, Q1)) Ri is not deleted and anchored to ℓ1 S(i, k′, q0, q1, Q0, Q1)

begin-range Ri Point pi