1. Shang Yang Stony Brook Univ. 08/17/2011 1.CONVEX 2.CLOSED 3.STABS EVERY ELEMENT

2. 1.CONVEX 3.STABS EVERYONE 2.CLOSED Esther M.Arkin Christian Knauer Claudia Dieckmann Lena Schlipf Shang Yang Joseph S.B. Mitchell Valentin Polishchuk

3. Computing a convex transversal 1987: original problem proposed Arik Tamir (NYU CG Day 3/13) * Comp. Vision, Graphics & Image Processing 49(2):152170 1990: 2d parallel segments solved Goodrich & Snoeyink*

4. Curve Reconstruction Line Simplification Motion Planning Surface Reconstruction Motivation

5. 2d segments, squares, 3d balls proved NP-hard from 3-SAT Our contributions Disjoint segments, pseudodisks Polytime (DP) If vertices are from a given set

6. NP-Completeness Proof: Stabbing Arbitrary Segments 6 From 3-SAT

7. NP-Completeness Proof: Stabbing Arbitrary Segments 7 From 3-SAT : a b c d e A Variable Gadget: a b c A Clause Gadget: Variable Gadgets Clause Gadgets

8. 8 From 3-SAT : a b c d e A Variable Gadget: a b c A Clause Gadget: Variable Gadgets Clause Gadgets

9. 9 From 3-SAT : a b c d e A Variable Gadget: Variable Gadgets Clause Gadgets inner

10. 10 From 3-SAT : a b c d e A Variable Gadget: Variable Gadgets Clause Gadgets inner

11. 11 From 3-SAT : a b c d e A Variable Gadget: Variable Gadgets Clause Gadgets inner

12. 12 From 3-SAT : a b c d e A Variable Gadget: Variable Gadgets Clause Gadgets inner e f

13. 13 From 3-SAT : a b c d e A Variable Gadget: Variable Gadgets Clause Gadgets inner e f

14. 14 From 3-SAT : a b c d e A Variable Gadget: Variable Gadgets Clause Gadgets inner x i is T x i is F x i is T e f

15. 15 From 3-SAT : a b c d e A Variable Gadget: Variable Gadgets Clause Gadgets inner x i is T x i is F x i is T e f

16. 16 From 3-SAT : Variable Gadgets Clause Gadgets a b c A Clause Gadget: inner

17. 17 From 3-SAT : Variable Gadgets Clause Gadgets a b c A Clause Gadget: df inner e

18. 18 From 3-SAT : Variable Gadgets Clause Gadgets a b c A Clause Gadget: inner d e f C i = x 1 v ⌐x 2 v x 3 x 3 is F x 2 is T x 1 is F In a Variable Gadget, x i FALSE: the blue segments are not stabbed x i TRUE : the green segments are not stabbed

19. 19 a b c a b c a b c a b c A Clause Gadget : Can stab any 2 (but not all 3) of segs d, e, f d e f d e f d e f d e f

20. 20 Variable Gadgets Clause Gadgets F = C 1 ∩ C 2 x1x1 x4x4 x3x3 x2x2 C1C1 C2C2 C 1 = x 1 U ⌐x 2 U x 3 C 2 = ⌐ x 1 U ⌐x 3 U x 4

21. Additional Hardness Results 21 Finding a convex transversal for: 1. a set of unit-length segments in 2D 2. a set of pairwise-disjoint segments in 3D 3. a set of disjoint balls in 3D 4. a set of disjoint unit balls in 3D Conjectured is NP-Complete.

22. Polytime Algorithm: First Step 3-link chain: Bridge 3-link chain: Bridge Segment chord

23. Chords Are NEVER INTERSECTED By Any Input Segments

24. Chords & Bridges

25. A Bridge Partitions the problem into 2 halves p q t r q t B p r B Stab(qp, tr, B)=1 Stab(pq, rt, B)=1 Succeed!!

26. DP Current State (pq, rt, qabt) p q t r a b b a c

27. p q t r a b DP Recursion( abc Case 1) Stab(pq, rt, qabt) (c) = Stab(pq, rt,qbt) p q t r b (c) a

28. DP Recursion( abc Case 2) Stab(pq, rt, qabt) p q t r a b z c p q t r b =Stab(pz, rt,zcbt) z c a

29. DP Recursion( abc Case 3) Stab(pq, rt, qabt) p q t r a b z c p q t r b = Stab(pz, rt,zbt) z c a

30. DP Recursion( abc Case 4) u z c a q p v r t b z c Stab(pq, uz, qacz) & Stab(rt, vz, tbcz)

31. Base Case(1 & 2) p q t r a b z c Stab(pq, rt, qabt) = FALSE

32. p q t r a b Base Case(3) Stab(pq, rt, qabt) = TRUE

33. q t B p r B Review

34. Symmetry Detection Stabbing with Regular Polygon Polynomial Time Algorithm

35. Optimization Versions of the Problem Maximize the number of objects stabbed by the convex transversal (DP) Minimize the length of the stabber: – TSP with Neighborhoods (TSPN) – Require convexity: Shortest convex stabber – Example: TSPN for lines Minimize movement of objects to make them have a convex stabber: optimal “Convexification” 35

36. Convexification Fast 2-Approximation & PTAS

37. Let Q’ be the convex hull of Q, and let D be the maximum distance from a point, q, in Q to the boundary of Q’. Convexification: 2-Approx 37 Q’ D q

38. Convexification: 2-Approx 38 D OPT

39. Convexification: 2-Approx 39 D OPT

40. Summary Settle the open problem in 2D: – Deciding existence of a convex transversal is NP- complete, in general – If objects S are disjoint, or form set of pseudodisks, then poly-time algorithm to decide, and to max # objects stabbed 3D: NP-complete, even for disjoint disks 40 Hard even for terrain stabbers! Assumes candidate set P of corners of stabber is given.

41. 3 Open Problems Candidate Points NOT Given? Allowing < k reflex vertices? Fast 2d Unit Disk Case Solver?

42. Esther M.Arkin Christian Knauer Claudia Dieckmann Lena Schlipf Joseph S.B. Mitchell Valentin Polishchuk Thank you! Questions Please ! Shang Yang