Existence of Simple Tours through Imprecise Points Maarten L - PowerPoint PPT Presentation

Existence of Simple Tours through Imprecise Points Maarten L¨ offler Center for Geometry, Imaging and Virtual Environments Utrecht University 1-1

Overview • Imprecise points • Imprecise simple polygons • NP-hardness proof • Planar 3-SAT • Variables as scissors • Clauses • Further details • Other results and consequences 2-1 Existence of Simple Tours through Imprecise Points Maarten L¨ offler, March 7, 2007

Imprecise Points • Unknown location • Known region of possible locations • Regions are simple geometric objects • Disc • Square • Rectangle • Convex polygon • Line segment 3-1 Existence of Simple Tours through Imprecise Points Maarten L¨ offler, March 7, 2007

Imprecise Points • Unknown location • Known region of possible locations • Regions are simple geometric objects • Disc • Square • Rectangle • Convex polygon • Line segment 3-2 Existence of Simple Tours through Imprecise Points Maarten L¨ offler, March 7, 2007

Imprecise Points • Unknown location • Known region of possible locations • Regions are simple geometric objects • Disc • Square • Rectangle • Convex polygon • Line segment 3-3 Existence of Simple Tours through Imprecise Points Maarten L¨ offler, March 7, 2007

Imprecise Points • Unknown location • Known region of possible locations • Regions are simple geometric objects • Disc • Square • Rectangle • Convex polygon • Line segment 3-4 Existence of Simple Tours through Imprecise Points Maarten L¨ offler, March 7, 2007

Imprecise Points • Unknown location • Known region of possible locations • Regions are simple geometric objects • Disc • Square • Rectangle • Convex polygon • Line segment 3-5 Existence of Simple Tours through Imprecise Points Maarten L¨ offler, March 7, 2007

Imprecise Simple Polygons • Sequence of imprecise 1 points 2 • Place one vertex in 4 each region 10 • The result should be 5 a simple polygon 3 9 • This problem is NP-hard 8 7 6 4-1 Existence of Simple Tours through Imprecise Points Maarten L¨ offler, March 7, 2007

Imprecise Simple Polygons • Sequence of imprecise 1 points 2 • Place one vertex in 4 each region 10 • The result should be 5 a simple polygon 3 9 • This problem is NP-hard 8 7 6 4-2 Existence of Simple Tours through Imprecise Points Maarten L¨ offler, March 7, 2007

Imprecise Simple Polygons • Sequence of imprecise 1 points 2 • Place one vertex in 4 each region 10 • The result should be 5 a simple polygon 3 9 • This problem is NP-hard 8 7 6 4-3 Existence of Simple Tours through Imprecise Points Maarten L¨ offler, March 7, 2007

Planar 3-SAT • Boolean satisfiability a ∨ b ∨ ¬ c problem b • At most three a variables per clause b ∨ ¬ c ∨ e • Variable-clause graph c must be planar e • Known to be NP-hard [Lichtenstein 1982] d ¬ a ∨ c ∨ ¬ d 5-1 Existence of Simple Tours through Imprecise Points Maarten L¨ offler, March 7, 2007

Planar 3-SAT • Boolean satisfiability a ∨ b ∨ ¬ c problem b • At most three a variables per clause b ∨ ¬ c ∨ e • Variable-clause graph c must be planar e • Known to be NP-hard [Lichtenstein 1982] d ¬ a ∨ c ∨ ¬ d 5-2 Existence of Simple Tours through Imprecise Points Maarten L¨ offler, March 7, 2007

Scissors Gadget • Configuration of four imprecise points 1 4 • Two distinct possible solutions • Each variable will be represented by a 3 2 number of scissors • Positive slope: true • Negative slope: false 6-1 Existence of Simple Tours through Imprecise Points Maarten L¨ offler, March 7, 2007

Scissors Gadget • Configuration of four imprecise points 1 4 • Two distinct possible solutions • Each variable will be represented by a 3 2 number of scissors • Positive slope: true • Negative slope: false 6-2 Existence of Simple Tours through Imprecise Points Maarten L¨ offler, March 7, 2007

Scissors Gadget • Configuration of four imprecise points 1 4 • Two distinct possible solutions • Each variable will be represented by a 3 2 number of scissors • Positive slope: true • Negative slope: false 6-3 Existence of Simple Tours through Imprecise Points Maarten L¨ offler, March 7, 2007

Scissors Chain 1 4 1 4 3 3 2 3 2 2 1 4 7-1 Existence of Simple Tours through Imprecise Points Maarten L¨ offler, March 7, 2007

Scissors Chain 1 4 1 4 3 3 2 3 2 2 1 4 7-2 Existence of Simple Tours through Imprecise Points Maarten L¨ offler, March 7, 2007

Scissors Chain 1 4 1 4 3 3 2 3 2 2 1 4 7-3 Existence of Simple Tours through Imprecise Points Maarten L¨ offler, March 7, 2007

Scissors Chain 1 4 1 4 3 3 2 3 2 2 1 4 7-4 Existence of Simple Tours through Imprecise Points Maarten L¨ offler, March 7, 2007

Scissors Chain 1 4 1 4 3 3 2 3 2 2 1 4 7-5 Existence of Simple Tours through Imprecise Points Maarten L¨ offler, March 7, 2007

Scissors Chain 1 4 1 4 3 3 2 3 2 2 1 4 7-6 Existence of Simple Tours through Imprecise Points Maarten L¨ offler, March 7, 2007

Scissors Chain 1 4 1 4 3 3 2 3 2 2 1 4 7-7 Existence of Simple Tours through Imprecise Points Maarten L¨ offler, March 7, 2007

Clause Gadget • Configuration of three 1 imprecise points • Two fixed parts of the tour • Three distinct 2 possible solutions • Each solution will be connected to one of the clause’s variables 3 8-1 Existence of Simple Tours through Imprecise Points Maarten L¨ offler, March 7, 2007

Clause Gadget • Configuration of three 1 imprecise points • Two fixed parts of the tour • Three distinct 2 possible solutions • Each solution will be connected to one of the clause’s variables 3 8-2 Existence of Simple Tours through Imprecise Points Maarten L¨ offler, March 7, 2007

Clause Gadget • Configuration of three 1 imprecise points • Two fixed parts of the tour • Three distinct 2 possible solutions • Each solution will be connected to one of the clause’s variables 3 8-3 Existence of Simple Tours through Imprecise Points Maarten L¨ offler, March 7, 2007

Clause Gadget • Configuration of three 1 imprecise points • Two fixed parts of the tour • Three distinct 2 possible solutions • Each solution will be connected to one of the clause’s variables 3 8-4 Existence of Simple Tours through Imprecise Points Maarten L¨ offler, March 7, 2007

Variables and Clauses 9-1 Existence of Simple Tours through Imprecise Points Maarten L¨ offler, March 7, 2007

Variables and Clauses 9-2 Existence of Simple Tours through Imprecise Points Maarten L¨ offler, March 7, 2007

Variables and Clauses 9-3 Existence of Simple Tours through Imprecise Points Maarten L¨ offler, March 7, 2007

Variables and Clauses 9-4 Existence of Simple Tours through Imprecise Points Maarten L¨ offler, March 7, 2007

Variables and Clauses 9-5 Existence of Simple Tours through Imprecise Points Maarten L¨ offler, March 7, 2007

Variables and Clauses 9-6 Existence of Simple Tours through Imprecise Points Maarten L¨ offler, March 7, 2007

Variables and Clauses 9-7 Existence of Simple Tours through Imprecise Points Maarten L¨ offler, March 7, 2007

Variables and Clauses 9-8 Existence of Simple Tours through Imprecise Points Maarten L¨ offler, March 7, 2007

Variables and Clauses 9-9 Existence of Simple Tours through Imprecise Points Maarten L¨ offler, March 7, 2007

Variables and Clauses 9-10 Existence of Simple Tours through Imprecise Points Maarten L¨ offler, March 7, 2007

Further Details • Splitting variables • Variables will occur in many clauses • Chains need to move vertically • Connecting tour parts • Many small pieces of tour need to become one big tour • Bridges to cross chains 10-1 Existence of Simple Tours through Imprecise Points Maarten L¨ offler, March 7, 2007

Further Details • Splitting variables • Variables will occur in many clauses • Chains need to move vertically • Connecting tour parts • Many small pieces of tour need to become one big tour • Bridges to cross chains 10-2 Existence of Simple Tours through Imprecise Points Maarten L¨ offler, March 7, 2007

Further Details • Splitting variables • Variables will occur in many clauses • Chains need to move vertically • Connecting tour parts • Many small pieces of tour need to become one big tour • Bridges to cross chains 10-3 Existence of Simple Tours through Imprecise Points Maarten L¨ offler, March 7, 2007

Further Details • Splitting variables • Variables will occur in many clauses • Chains need to move vertically • Connecting tour parts • Many small pieces of tour need to become one big tour • Bridges to cross chains 10-4 Existence of Simple Tours through Imprecise Points Maarten L¨ offler, March 7, 2007

Conclusions • Finding a simple polygon with imprecise vertices as vertical line segments is NP-hard • Other results • Imprecision model: any other connected region • Finding the shortest simple tour through n regions 11-1 Existence of Simple Tours through Imprecise Points Maarten L¨ offler, March 7, 2007