The Inapproximability of Illuminating Polygons by α-floodlights

Transcription

1 The Inapproximability of Illuminating Polygons by α-floodlights A. Abdelkader 1 A. Saeed 2 K. Harras 3 A. Mohamed 4 1 Department of Computer Science University of Maryland at College Park 2 Department of Computer Science Georgia Institute of Technology 3 Department of Computer Science Carnegie Mellon University 4 Department of Computer Science and Engineering Qatar University CCCG, 2015 Abdelkader, Saeed, Harras, Mohamed Illuminating Polygons by α-floodlights CCCG, / 15

2 Guarding & Illumination Motivation Guarding Illumination < α < 360 Abdelkader, Saeed, Harras, Mohamed Illuminating Polygons by α-floodlights CCCG, / 15

3 Guarding & Illumination Motivation Guarding Illumination < α < 360 Abdelkader, Saeed, Harras, Mohamed Illuminating Polygons by α-floodlights CCCG, / 15

4 Guarding & Illumination Motivation Guarding Illumination < α < 360 Abdelkader, Saeed, Harras, Mohamed Illuminating Polygons by α-floodlights CCCG, / 15

5 Guarding & Illumination Motivation Guarding Illumination < α < 360 Abdelkader, Saeed, Harras, Mohamed Illuminating Polygons by α-floodlights CCCG, / 15

6 Guarding & Illumination Motivation Guarding Illumination < α < 360 Abdelkader, Saeed, Harras, Mohamed Illuminating Polygons by α-floodlights CCCG, / 15

7 Guarding & Illumination Motivation Guarding Illumination < α < 360 Abdelkader, Saeed, Harras, Mohamed Illuminating Polygons by α-floodlights CCCG, / 15

8 Guarding & Illumination Motivation Guarding Illumination < α < 360 Abdelkader, Saeed, Harras, Mohamed Illuminating Polygons by α-floodlights CCCG, / 15

9 Guarding & Illumination Motivation Guarding Illumination < α < 360 Abdelkader, Saeed, Harras, Mohamed Illuminating Polygons by α-floodlights CCCG, / 15

10 Preliminaries Definition (α-floodlight) An α-floodlight at point p, with orientation θ, is the infinite wedge W (p, α, θ) bounded between the two rays v l and v r starting at p with angles θ ± α 2. In a polygon P, a point q belongs to the α-floodlight if pq lies entirely in both P and W (p, α, θ). Abdelkader, Saeed, Harras, Mohamed Illuminating Polygons by α-floodlights CCCG, / 15

11 Preliminaries Definition (α-floodlight) An α-floodlight at point p, with orientation θ, is the infinite wedge W (p, α, θ) bounded between the two rays v l and v r starting at p with angles θ ± α 2. In a polygon P, a point q belongs to the α-floodlight if pq lies entirely in both P and W (p, α, θ). Problem (Polygon Illumination by α-floodlights (PFIP)) Given a simple polygon P with n sides, a positive integer m and an angular aperture α, determine if P can be illuminated by at most m α-floodlights placed in its interior. Abdelkader, Saeed, Harras, Mohamed Illuminating Polygons by α-floodlights CCCG, / 15

12 Brief History Problem (Art Gallery Problem) Let P be a simple polygon without holes. Find the minimum subset S of the vertices of P such that the interior of P is visible from S. 1 Lee, D.-T. and Lin, A. K. (1986). Computational complexity of art gallery problems. Information Theory, IEEE Transactions on, 32(2): Eidenbenz, S., Stamm, C., and Widmayer, P. (2001). Inapproximability results for guarding polygons and terrains. Algorithmica, 31(1): King, J. and Kirkpatrick, D. (2011). Improved approximation for guarding simple galleries from the perimeter. Discrete & Computational Geometry, 46(2): Abdelkader, Saeed, Harras, Mohamed Illuminating Polygons by α-floodlights CCCG, / 15

13 Brief History Problem (Art Gallery Problem) Let P be a simple polygon without holes. Find the minimum subset S of the vertices of P such that the interior of P is visible from S. Decision version is NP-hard 1. 1 Lee, D.-T. and Lin, A. K. (1986). Computational complexity of art gallery problems. Information Theory, IEEE Transactions on, 32(2): Eidenbenz, S., Stamm, C., and Widmayer, P. (2001). Inapproximability results for guarding polygons and terrains. Algorithmica, 31(1): King, J. and Kirkpatrick, D. (2011). Improved approximation for guarding simple galleries from the perimeter. Discrete & Computational Geometry, 46(2): Abdelkader, Saeed, Harras, Mohamed Illuminating Polygons by α-floodlights CCCG, / 15

14 Brief History Problem (Art Gallery Problem) Let P be a simple polygon without holes. Find the minimum subset S of the vertices of P such that the interior of P is visible from S. Decision version is NP-hard 1. APX-hard 2. 1 Lee, D.-T. and Lin, A. K. (1986). Computational complexity of art gallery problems. Information Theory, IEEE Transactions on, 32(2): Eidenbenz, S., Stamm, C., and Widmayer, P. (2001). Inapproximability results for guarding polygons and terrains. Algorithmica, 31(1): King, J. and Kirkpatrick, D. (2011). Improved approximation for guarding simple galleries from the perimeter. Discrete & Computational Geometry, 46(2): Abdelkader, Saeed, Harras, Mohamed Illuminating Polygons by α-floodlights CCCG, / 15

15 Brief History Problem (Art Gallery Problem) Let P be a simple polygon without holes. Find the minimum subset S of the vertices of P such that the interior of P is visible from S. Decision version is NP-hard 1. APX-hard 2. O(log log OPT )-approximation 3. 1 Lee, D.-T. and Lin, A. K. (1986). Computational complexity of art gallery problems. Information Theory, IEEE Transactions on, 32(2): Eidenbenz, S., Stamm, C., and Widmayer, P. (2001). Inapproximability results for guarding polygons and terrains. Algorithmica, 31(1): King, J. and Kirkpatrick, D. (2011). Improved approximation for guarding simple galleries from the perimeter. Discrete & Computational Geometry, 46(2): Abdelkader, Saeed, Harras, Mohamed Illuminating Polygons by α-floodlights CCCG, / 15

16 Reduction from 3SAT [Eidenbenz et al.] Big Picture [Fig. 3 in Eidenbenz, ISAAC 98] Abdelkader, Saeed, Harras, Mohamed Illuminating Polygons by α-floodlights CCCG, / 15

17 Reduction from 3SAT [Eidenbenz et al.] Clauses & Literals [Fig. 2 in Eidenbenz, ISAAC 98] Abdelkader, Saeed, Harras, Mohamed Illuminating Polygons by α-floodlights CCCG, / 15

18 Reduction from 3SAT [Eidenbenz et al.] Variables & Literals [Fig. 4 in Eidenbenz, ISAAC 98] Abdelkader, Saeed, Harras, Mohamed Illuminating Polygons by α-floodlights CCCG, / 15

19 Requirements on Guard Coverage? x 1 x 1 x 2 x 3 [Fig. 3 in Eidenbenz, ISAAC 98] Abdelkader, Saeed, Harras, Mohamed Illuminating Polygons by α-floodlights CCCG, / 15

20 Requirements on Guard Coverage? x 1 x 1 x 2 x 3 [Fig. 3 in Eidenbenz, ISAAC 98] Abdelkader, Saeed, Harras, Mohamed Illuminating Polygons by α-floodlights CCCG, / 15

21 Requirements on Guard Coverage? x 1 x 1 x 2 x 3 [Fig. 3 in Eidenbenz, ISAAC 98] Abdelkader, Saeed, Harras, Mohamed Illuminating Polygons by α-floodlights CCCG, / 15

22 Requirements on Guard Coverage? x 1 x 1 x 2 x 3 [Fig. 3 in Eidenbenz, ISAAC 98] Abdelkader, Saeed, Harras, Mohamed Illuminating Polygons by α-floodlights CCCG, / 15

23 Requirements on Guard Coverage? [Fig. 3 in Eidenbenz, ISAAC 98] Abdelkader, Saeed, Harras, Mohamed Illuminating Polygons by α-floodlights CCCG, / 15

24 Beam Machine Culberson and Reckhow, FOCS 88 & JAlg 94 [Fig. 7 in Eidenbenz and Widmayer, SICOMP 03] Abdelkader, Saeed, Harras, Mohamed Illuminating Polygons by α-floodlights CCCG, / 15

25 Beam Machine Culberson and Reckhow, FOCS 88 & JAlg 94 [Fig. 9 in Eidenbenz and Widmayer, SICOMP 03] Abdelkader, Saeed, Harras, Mohamed Illuminating Polygons by α-floodlights CCCG, / 15

26 Beam Machine Bagga, Gewali and Glasser, CCCG 96 [Fig. 1 in Bagga, Gewali and Glasser, CCCG 94] Abdelkader, Saeed, Harras, Mohamed Illuminating Polygons by α-floodlights CCCG, / 15

27 Beam Machine Bagga, Gewali and Glasser, CCCG 96 [Fig. 5 in Bagga, Gewali and Glasser, CCCG 94] Abdelkader, Saeed, Harras, Mohamed Illuminating Polygons by α-floodlights CCCG, / 15

28 Floodlight Gadget a c a b b e e d d c Abdelkader, Saeed, Harras, Mohamed Illuminating Polygons by α-floodlights CCCG, / 15

29 Floodlight Gadget Abdelkader, Saeed, Harras, Mohamed Illuminating Polygons by α-floodlights CCCG, / 15

30 New Beam Machine Configuration Floodlight possible locations Aux PFG 2 Aux PFG 1 Nozzle Abdelkader, Saeed, Harras, Mohamed Illuminating Polygons by α-floodlights CCCG, / 15

31 New Beam Machine Abdelkader, Saeed, Harras, Mohamed Illuminating Polygons by α-floodlights CCCG, / 15

32 New Beam Machine Z Z` B X B` A D D` A` C E E` C` Abdelkader, Saeed, Harras, Mohamed Illuminating Polygons by α-floodlights CCCG, / 15

33 New Beam Machine Abdelkader, Saeed, Harras, Mohamed Illuminating Polygons by α-floodlights CCCG, / 15

34 New Beam Machine Abdelkader, Saeed, Harras, Mohamed Illuminating Polygons by α-floodlights CCCG, / 15

35 Beam Coupling Abdelkader, Saeed, Harras, Mohamed Illuminating Polygons by α-floodlights CCCG, / 15

36 New Clause Gadget 1 2 Abdelkader, Saeed, Harras, Mohamed Illuminating Polygons by α-floodlights CCCG, / 15

37 New Variable Gadget Leftmost F literal Leftmost T literal a b a b Rightmost T spike Rightmost F spike [Fig. 1 in Eidenbenz, ISAAC 98] Abdelkader, Saeed, Harras, Mohamed Illuminating Polygons by α-floodlights CCCG, / 15

38 New Variable Gadget Abdelkader, Saeed, Harras, Mohamed Illuminating Polygons by α-floodlights CCCG, / 15

39 Hardness Results Abdelkader, Saeed, Harras, Mohamed Illuminating Polygons by α-floodlights CCCG, / 15

40 Hardness Results Theorem PFIP is NP-hard. Abdelkader, Saeed, Harras, Mohamed Illuminating Polygons by α-floodlights CCCG, / 15

41 Hardness Results Theorem PFIP is NP-hard. Theorem FIP is NP-hard. Abdelkader, Saeed, Harras, Mohamed Illuminating Polygons by α-floodlights CCCG, / 15

42 APX-hardness Gap Preserving Reductions (informal) I S Reduction Transformation I (Approximation) Algorithm Abdelkader, Saeed, Harras, Mohamed Illuminating Polygons by α-floodlights CCCG, / 15 S

43 APX-hardness An APX Representative Problem (5-OCC-MAX-3-SAT) Given a boolean formula Φ in conjunctive normal form, with m clauses and n variables, 3 literals at most per clause, and 5 literals at most per variable, find an assignment of the variables that satisfies as many clauses as possible. Abdelkader, Saeed, Harras, Mohamed Illuminating Polygons by α-floodlights CCCG, / 15

44 APX-hardness Transformation Process BM d 1 BM d 2 9 d 3 Abdelkader, Saeed, Harras, Mohamed Illuminating Polygons by α-floodlights CCCG, / 15

45 APX-hardness Definition (Flushing Condition) An α-floodlight is flush with the vertices of the polygon P if at least one of v l or v r passes through some vertex of P, different from p, such that θ is determined implicitly. Abdelkader, Saeed, Harras, Mohamed Illuminating Polygons by α-floodlights CCCG, / 15

46 APX-hardness Inapproximability Results Definition (Flushing Condition) An α-floodlight is flush with the vertices of the polygon P if at least one of v l or v r passes through some vertex of P, different from p, such that θ is determined implicitly. Theorem R-PFIP is APX-hard. Abdelkader, Saeed, Harras, Mohamed Illuminating Polygons by α-floodlights CCCG, / 15

47 APX-hardness Inapproximability Results Definition (Flushing Condition) An α-floodlight is flush with the vertices of the polygon P if at least one of v l or v r passes through some vertex of P, different from p, such that θ is determined implicitly. Theorem R-PFIP is APX-hard. Theorem R-FIP is APX-hard. Abdelkader, Saeed, Harras, Mohamed Illuminating Polygons by α-floodlights CCCG, / 15

48 Thank You Questions? Abdelkader, Saeed, Harras, Mohamed Illuminating Polygons by α-floodlights CCCG, / 15