131 |
Dimer statistics and phase transitions.
- Kasteleyn
- 1963
(Show Context)
Citation Context ... l and m are both at least 2. In the case that l = m = 2 this problem can be formulated in terms of counting perfect matchings of planar graphs which Kasteleyn showed could be done in polynomial time =-=[3]-=-. Thus we look at the case either l or m is at least 3; in particular we will show: Theorem 1.1. Counting the number of ways to tile a planar figure with 1× l and m × 1 rectangles is #P-complete if at... |

35 | Tiling a polygon with rectangles,
- Kenyon, Kenyon
- 1992
(Show Context)
Citation Context ...ct this problem to simply connected regions. As mentioned above, in the general case determining the existence of a tiling is NP-complete, but in the simply-connected case Kenyon and Kenyon showed in =-=[4]-=-, that determining the existence of a tiling can be done in linear time. Thus there is a fundamental difference between these two cases. Additionally, Pak and Yang showed in [5] that there is a large ... |

8 | Tiling simply connected regions with rectangles
- Pak, Yang
(Show Context)
Citation Context ...n and Kenyon showed in [4], that determining the existence of a tiling can be done in linear time. Thus there is a fundamental difference between these two cases. Additionally, Pak and Yang showed in =-=[5]-=- that there is a large set of rectangles for which tiling simply connected regions is NP-complete and #P-complete, so it would be interesting to see if it is possible to reduce the size of that set to... |

1 |
Eric Remila, and Mike Robson, Tiling figures of the plane with two bars
- Beauquier, Nivat
- 1995
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Citation Context ...hat the problem of determining the number of ways of tiling a planar figure with a horizontal and a vertical bar is #P-complete. We build off of the results of Beauquier, Nivat, Remila, and Robson in =-=[1]-=- in which they showed that the problem determining existence of such tilings was shown to be NP-complete. 1. Introduction For this problem we consider the plane as a grid of unit squares, and we defin... |

1 |
The complexity of planar counting problems, CoRR cs.CC/9809017
- Marathe, Radhakrishnan, et al.
- 1998
(Show Context)
Citation Context ...ues to each variable such that all clauses in the expression are true? Counting solutions of planar 1-Ex3MonoSat expressions was shown to be #Pcomplete by Hunt, Marathe, Radhakrishnan, and Stearns in =-=[2]-=-. 2. Proof of the case l = 2, m = 3 We will first prove our claim for the case l = 2,m = 3, and then extend these results to l ≥ 2, m ≥ 3. In order to do this we introduce variable figures, clause 1 2... |

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