Edge-Unfolding Orthogonal Polyhedra is Strongly ... - Semantic Scholar
Edge-Unfolding Orthogonal Polyhedra is Strongly NP-Complete Zachary Abel1 1 MIT
Erik D. Demaine2
Department of Mathematics 2 MIT
CSAIL
CCCG 2011
Zachary Abel, Erik D. Demaine (MIT)
Edge-Unfolding is NP-Complete
CCCG 2011
1/8
Background on Edge-Unfolding
(General) Unfolding: cut and fold flat without overlap Edge-Unfolding: cut only on edges
Zachary Abel, Erik D. Demaine (MIT)
Edge-Unfolding is NP-Complete
CCCG 2011
2/8
the boundary of a hat is an equilateral triangle, we can take four hats an Background Edge-Unfolding ainst the faces of aon regular tetrahedron (and then remove the guiding tetrah lt is a closed polyhedron with no edge unfolding, which we call a spiked tetra serve the following property of unfolded hats:
(General) Unfolding: cut and fold flat without overlap Edge-Unfolding: cut only on edges Edge-unfolding does not always exist [BDEKMS, CCCG ‘99] Figure 7: Spiked tetrahedra for both types of hats.
Zachary Abel, Erik D. Demaine (MIT)
Edge-Unfolding is NP-Complete
CCCG 2011
2/8
Background on Edge-Unfolding
(General) Unfolding: cut and fold flat without overlap Edge-Unfolding: cut only on edges Edge-unfolding does not always exist [BDDLOORW, CCCG ‘98]
Zachary Abel, Erik D. Demaine (MIT)
Edge-Unfolding is NP-Complete
CCCG 2011
2/8
Complexity of Edge-Unfolding
Decision question In NP? Not known! I
Needs high numerical precision (radicals)
Orthogonal edge-unfolding is in NP
Zachary Abel, Erik D. Demaine (MIT)
Edge-Unfolding is NP-Complete
CCCG 2011
3/8
Complexity of Edge-Unfolding
Decision question In NP? Not known! I
Needs high numerical precision (radicals)
Orthogonal edge-unfolding is in NP Our result: Precision is not the only barrier
Theorem Edge-unfolding is strongly NP hard even if the polyhedron must be orthogonal, topologically convex, and of genus zero.
Zachary Abel, Erik D. Demaine (MIT)
Edge-Unfolding is NP-Complete
CCCG 2011
3/8
A New Ununfoldable Polyhedron Orthogonal and Topologically Convex
drain
floor
cage
Zachary Abel, Erik D. Demaine (MIT)
Edge-Unfolding is NP-Complete
CCCG 2011
4/8
A New Ununfoldable Polyhedron Orthogonal and Topologically Convex
pipe1
pipe2
tower
Zachary Abel, Erik D. Demaine (MIT)
Edge-Unfolding is NP-Complete
CCCG 2011
4/8
A New Ununfoldable Polyhedron Orthogonal and Topologically Convex
pipe1
U-shaped polygons must be removed from floor and pipes.
Zachary Abel, Erik D. Demaine (MIT)
Edge-Unfolding is NP-Complete
CCCG 2011
4/8
A New Ununfoldable Polyhedron Orthogonal and Topologically Convex
Zachary Abel, Erik D. Demaine (MIT)
Edge-Unfolding is NP-Complete
CCCG 2011
4/8
A New Ununfoldable Polyhedron Orthogonal and Topologically Convex
Too many big faces on tower to fit in cage
Zachary Abel, Erik D. Demaine (MIT)
Edge-Unfolding is NP-Complete
CCCG 2011
4/8
Reduce from Square Packing Square Packing Problem Can n squares of (integer) side-lengths a1 , . . . , an be orthogonally packed into a square of side-length d? Cover the tower with large bricks b1 , . . . , bn and tiny filler material so the bricks must unfold inside the cage.
bn
b2 b1
Zachary Abel, Erik D. Demaine (MIT)
Edge-Unfolding is NP-Complete
CCCG 2011
5/8
Making a Square Packing Sufficent Need connectivity Need room for unused filler
Zachary Abel, Erik D. Demaine (MIT)
Edge-Unfolding is NP-Complete
CCCG 2011
6/8
Making a Square Packing Sufficent Need connectivity Need room for unused filler
Zachary Abel, Erik D. Demaine (MIT)
Edge-Unfolding is NP-Complete
CCCG 2011
6/8
Making a Square Packing Sufficent Need connectivity Need room for unused filler
Zachary Abel, Erik D. Demaine (MIT)
Edge-Unfolding is NP-Complete
CCCG 2011
6/8
Universal Filler Material Atom: this polyhedral surface Flatom: a 27 × 27 square 9
Theorem Any path of n edge-connected atoms can be edge-unfolded inside any path of n edge-connected flatoms, connecting to the initial and final edges at the midpoint (or one unit away depending on parity).
Zachary Abel, Erik D. Demaine (MIT)
Edge-Unfolding is NP-Complete
CCCG 2011
7/8
Universal Filler Material (Ctd.)
Zachary Abel, Erik D. Demaine (MIT)
Edge-Unfolding is NP-Complete
CCCG 2011
8/8
Erik D. Demaine2
Department of Mathematics 2 MIT
CSAIL
CCCG 2011
Zachary Abel, Erik D. Demaine (MIT)
Edge-Unfolding is NP-Complete
CCCG 2011
1/8
Background on Edge-Unfolding
(General) Unfolding: cut and fold flat without overlap Edge-Unfolding: cut only on edges
Zachary Abel, Erik D. Demaine (MIT)
Edge-Unfolding is NP-Complete
CCCG 2011
2/8
the boundary of a hat is an equilateral triangle, we can take four hats an Background Edge-Unfolding ainst the faces of aon regular tetrahedron (and then remove the guiding tetrah lt is a closed polyhedron with no edge unfolding, which we call a spiked tetra serve the following property of unfolded hats:
(General) Unfolding: cut and fold flat without overlap Edge-Unfolding: cut only on edges Edge-unfolding does not always exist [BDEKMS, CCCG ‘99] Figure 7: Spiked tetrahedra for both types of hats.
Zachary Abel, Erik D. Demaine (MIT)
Edge-Unfolding is NP-Complete
CCCG 2011
2/8
Background on Edge-Unfolding
(General) Unfolding: cut and fold flat without overlap Edge-Unfolding: cut only on edges Edge-unfolding does not always exist [BDDLOORW, CCCG ‘98]
Zachary Abel, Erik D. Demaine (MIT)
Edge-Unfolding is NP-Complete
CCCG 2011
2/8
Complexity of Edge-Unfolding
Decision question In NP? Not known! I
Needs high numerical precision (radicals)
Orthogonal edge-unfolding is in NP
Zachary Abel, Erik D. Demaine (MIT)
Edge-Unfolding is NP-Complete
CCCG 2011
3/8
Complexity of Edge-Unfolding
Decision question In NP? Not known! I
Needs high numerical precision (radicals)
Orthogonal edge-unfolding is in NP Our result: Precision is not the only barrier
Theorem Edge-unfolding is strongly NP hard even if the polyhedron must be orthogonal, topologically convex, and of genus zero.
Zachary Abel, Erik D. Demaine (MIT)
Edge-Unfolding is NP-Complete
CCCG 2011
3/8
A New Ununfoldable Polyhedron Orthogonal and Topologically Convex
drain
floor
cage
Zachary Abel, Erik D. Demaine (MIT)
Edge-Unfolding is NP-Complete
CCCG 2011
4/8
A New Ununfoldable Polyhedron Orthogonal and Topologically Convex
pipe1
pipe2
tower
Zachary Abel, Erik D. Demaine (MIT)
Edge-Unfolding is NP-Complete
CCCG 2011
4/8
A New Ununfoldable Polyhedron Orthogonal and Topologically Convex
pipe1
U-shaped polygons must be removed from floor and pipes.
Zachary Abel, Erik D. Demaine (MIT)
Edge-Unfolding is NP-Complete
CCCG 2011
4/8
A New Ununfoldable Polyhedron Orthogonal and Topologically Convex
Zachary Abel, Erik D. Demaine (MIT)
Edge-Unfolding is NP-Complete
CCCG 2011
4/8
A New Ununfoldable Polyhedron Orthogonal and Topologically Convex
Too many big faces on tower to fit in cage
Zachary Abel, Erik D. Demaine (MIT)
Edge-Unfolding is NP-Complete
CCCG 2011
4/8
Reduce from Square Packing Square Packing Problem Can n squares of (integer) side-lengths a1 , . . . , an be orthogonally packed into a square of side-length d? Cover the tower with large bricks b1 , . . . , bn and tiny filler material so the bricks must unfold inside the cage.
bn
b2 b1
Zachary Abel, Erik D. Demaine (MIT)
Edge-Unfolding is NP-Complete
CCCG 2011
5/8
Making a Square Packing Sufficent Need connectivity Need room for unused filler
Zachary Abel, Erik D. Demaine (MIT)
Edge-Unfolding is NP-Complete
CCCG 2011
6/8
Making a Square Packing Sufficent Need connectivity Need room for unused filler
Zachary Abel, Erik D. Demaine (MIT)
Edge-Unfolding is NP-Complete
CCCG 2011
6/8
Making a Square Packing Sufficent Need connectivity Need room for unused filler
Zachary Abel, Erik D. Demaine (MIT)
Edge-Unfolding is NP-Complete
CCCG 2011
6/8
Universal Filler Material Atom: this polyhedral surface Flatom: a 27 × 27 square 9
Theorem Any path of n edge-connected atoms can be edge-unfolded inside any path of n edge-connected flatoms, connecting to the initial and final edges at the midpoint (or one unit away depending on parity).
Zachary Abel, Erik D. Demaine (MIT)
Edge-Unfolding is NP-Complete
CCCG 2011
7/8
Universal Filler Material (Ctd.)
Zachary Abel, Erik D. Demaine (MIT)
Edge-Unfolding is NP-Complete
CCCG 2011
8/8
