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The Box Mover Problem

The Box Mover Problem (BMP) addresses the complexities of robotic interactions in a warehouse environment where an agent (robot) must manipulate boxes. The robot is constrained by rules such as the inability to step on boxes and specific limits on the number of boxes it can push, pull, or lift. Different versions of the problem exist, including scenarios with fixed walls, various movement capabilities, and obstacles. This paper explores problem feasibility, optimized robot operations, and the implications of NP-hardness within the context of box manipulation tasks.

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The Box Mover Problem

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  1. The Box Mover Problem Valentin Polishchuk Dept. of Applied Math. & Statistics Stony Brook University

  2. A Puzzle

  3. A Puzzle • Pixeled domain

  4. A Puzzle • Pixeled domain • Items • 1 per pixel

  5. A Puzzle • Pixeled domain • Items • 1 per pixel • Agent (Robot) travels moves the items www.cs.utah.edu/~mflatt/

  6. A Puzzle • Pixeled domain • Items • 1 per pixel • Agent (Robot) travels moves the items • initial position

  7. A Puzzle • Pixeled domain • Items • 1 per pixel • Agent (Robot) travels moves the items • initial position • Rules/Constraints • What happens after a move • What is forbidden

  8. A Puzzle (contd.) • Target configuration • Sink

  9. A Puzzle (contd.) • Target configuration • Sink • Goal: • items • initial → target

  10. A Puzzle (contd.) • Target configuration • Sink • Goal: • items • initial → target • robot • back

  11. The Box Mover Problem

  12. The Box Mover Problem • Domain – warehouse • Items – boxes

  13. The Box Mover Problem • Domain – warehouse • Items – boxes • Agent (robot) – warehouse-keeper, pushing and/or pulling and/or lifting boxes

  14. The Box Mover Problem • Domain – warehouse • Items – boxes • Agent (robot) – warehouse-keeper, pushing and/or pulling and/or lifting boxes • Rules • don’t step on boxes

  15. The Box Mover Problem • Domain – warehouse • Items – boxes • Agent (robot) – warehouse-keeper, pushing and/or pulling and/or lifting boxes • Rules • don’t step on boxes • ≤k pushed • ≤p pulled • ≤l lifted BMP(k,p,l)

  16. Versions of the Problem

  17. Versions of the Problem • All boxes movable?

  18. Versions of the Problem • All boxes movable? • Some fixed (rigid walls) BMP(k,p,l)-F • Robot’s path • No self-intersections BMP(k,p,l)-X • Boxes’ IDs #-BMP(k,p,l) E.g.: SOKOBAN = BMP(1,0,0)-F Push-k = BMP(k,0,0), Push-* = BMP(∞,0,0) BMP(∞, ∞, ∞) – Omnipotent Robot Problem

  19. Versions of the Problem • All boxes movable? • Some fixed (rigid walls) BMP(k,p,l)-F • Robot’s path • No self-intersections BMP(k,p,l)-X • Boxes’ IDs #-BMP(k,p,l) E.g.: SOKOBAN = BMP(1,0,0)-F Push-k = BMP(k,0,0), Push-* = BMP(∞,0,0) BMP(∞, ∞, ∞) – Omnipotent Robot Problem

  20. Versions of the Problem • All boxes movable? • Some fixed (rigid walls) BMP(k,p,l)-F • Robot’s path • No self-intersections BMP(k,p,l)-X • Boxes’ IDs #-BMP(k,p,l) E.g.: SOKOBAN = BMP(1,0,0)-F Push-k = BMP(k,0,0), Push-* = BMP(∞,0,0) BMP(∞, ∞, ∞) – Omnipotent Robot Problem

  21. Versions of the Problem • All boxes movable? • Some fixed (rigid walls) BMP(k,p,l)-F • Robot’s path • No self-intersections BMP(k,p,l)-X • Boxes’ IDs #-BMP(k,p,l) E.g.: SOKOBAN = BMP(1,0,0)-F Push-k = BMP(k,0,0), Push-* = BMP(∞,0,0) BMP(∞, ∞, ∞) – Omnipotent Robot Problem

  22. Previous work: Feasibility

  23. Previous work: Feasibility • P? • monotone [Dhagat and O’Rourke,’92] • NP? • only 1 version (-F-X) [Demaine and Hoffmann,’01] • NP-hard, PSPACE-complete [Dhagat and O’Rourke,’92], [Dor and Zwick,’99], [Culberson,’98], [Demaine, Demaine and O'Rourke,’00], [Demaine, Demaine, Hoffmann and O'Rourke,’01], [Demaine, Hearn and Hoffmann,’02], [Hearn and Demaine,’02]

  24. Previous work: Feasibility • P? • monotone [Dhagat and O’Rourke,’92] • NP? • only 1 version (-F-X) [Demaine and Hoffmann,’01] • NP-hard, PSPACE-complete [Dhagat and O’Rourke,’92], [Dor and Zwick,’99], [Culberson,’98], [Demaine, Demaine and O'Rourke,’00], [Demaine, Demaine, Hoffmann and O'Rourke,’01], [Demaine, Hearn and Hoffmann,’02], [Hearn and Demaine,’02]

  25. “Reality”: Optimization

  26. “Reality”: Optimization • “Infeasible!”– not acceptable • Upgrade robot, warehouse • Feasibility insured • Original SOKOBAN problem

  27. “Reality”: Optimization • “Infeasible!”– not acceptable • Upgrade robot, warehouse • Feasibility insured • Original SOKOBAN problem • Optimization • cost = workload (pushes + pulls + lifts) • unloaded motion – no cost • [Sabey,’96]

  28. Proving Hardness • All boxes movable, no walls, infinite plane • leakage is a problem • l = 0 • kp = 0 – a wall of thickness max(k, p)+1 is rigid • p = 0 – k+1-by-k+1 square is unmovable • k = 0 – p+1-by-p+1 square is unmovable Palliative, what if kp > 0 or l > 0 ? • Our hardness proof – wall thickness is constant

  29. Making Puzzle “more tractable” • k ↑, p ↑, l ↑ ? • [Demaine, Demaine and O'Rourke,’00]: introducing powerful robot “makes it relatively easy to construct intractable puzzles” • constrain robot’s motion: heavier constructions • l>0 – none? • Mosaic Rearranging Problem (“flying” robot) is in P – solved by assignment • Limiting the robot’s capabilities • most problems (even the “more tractable” ones): • exact complexity unknown, some PSPACE-complete

  30. Our Results • NP-hard for any (k, p, l) • including infinite values • Open BMP instances • in NP

  31. Another Problem – Crossovers • [Demaine and Hoffman,‘01], [Demaine, Demaine, Hoffmann and O'Rourke,’01] contrasted their work to “all previous approaches of building circuits based on graphs, which seem to inherently require [problematic] crossings” • Our construction • no crossings • reduction from HC problem for planar graphs

  32. [Culberson,’98]

  33. The Reduction • HC in planar directed graphs • each node v: out(v) + in(v) = 3 [Plesnik,‘79]

  34. BMP(1,0,0)-F Edges – corridors of width 1Nodes – T-intersections

  35. Node gadget Edge gadget The Gadgets Checked box – initial position Shaded box – target position Robot initially – somewhere

  36. Node gadget Edge gadget The Gadgets Checked box – initial position Shaded box – target position Robot initially – somewhere

  37. Node gadget Edge gadget The Gadgets Checked box – initial position Shaded box – target position Robot initially – somewhere

  38. Node gadget Edge gadget The Gadgets Checked box – initial position Shaded box – target position Robot initially – somewhere

  39. Node gadget Edge gadget The Gadgets Checked box – initial position Shaded box – target position Robot initially – somewhere

  40. Node gadget Edge gadget The Gadgets Checked box – initial position Shaded box – target position Robot initially – somewhere

  41. Node gadget Edge gadget The Gadgets Checked box – initial position Shaded box – target position Robot initially – somewhere

  42. Node gadget Edge gadget The Gadgets Checked box – initial position Shaded box – target position Robot initially – somewhere

  43. Node gadget Edge gadget The Gadgets Checked box – initial position Shaded box – target position Robot initially – somewhere

  44. Node gadget Edge gadget The Gadgets Checked box – initial position Shaded box – target position Robot initially – somewhere

  45. Node gadget Edge gadget The Gadgets Checked box – initial position Shaded box – target position Robot initially – somewhere

  46. Node gadget Edge gadget The Gadgets Checked box – initial position Shaded box – target position Robot initially – somewhere

  47. Node gadget Edge gadget The Gadgets Checked box – initial position Shaded box – target position Robot initially – somewhere

  48. Node gadget Edge gadget The Gadgets Checked box – initial position Shaded box – target position Robot initially – somewhere

  49. Node gadget Edge gadget The Gadgets Checked box – initial position Shaded box – target position Robot initially – somewhere

  50. Node gadget Edge gadget The Gadgets Checked box – initial position Shaded box – target position Robot initially – somewhere

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