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A polyomino is a polygonal region with axis parallel edges and corners of integral coordinates, which may have holes. In this paper, we consider planar tiling and packing problems with polyomino pieces and a polyomino container $P$. We give two polynomial time algorithms, one for deciding if $P$ can be tiled with $ktimes k$ squares for any fixed $k$ which can be part of the input (that is, deciding if $P$ is the union of a set of non-overlapping $ktimes k$ squares) and one for packing $P$ with a maximum number of non-overlapping and axis-parallel $2times 1$ dominos, allowing rotations by $90^circ$. As packing is more general than tiling, the latter algorithm can also be used to decide if $P$ can be tiled by $2times 1$ dominos. These are classical problems with important applications in VLSI design, and the related problem of finding a maximum packing of $2times 2$ squares is known to be NP-Hard [J. Algorithms 1990]. For our three problems there are known pseudo-polynomial time algorithms, that is, algorithms with running times polynomial in the area of $P$. However, the standard, compact way to represent a polygon is by listing the coordinates of the corners in binary. We use this representation, and thus present the first polynomial time algorithms for the problems. Concretely, we give a simple $O(nlog n)$ algorithm for tiling with squares, and a more involved $O(n^3,text{polylog}, n)$ algorithm for packing and tiling with dominos, where $n$ is the number of corners of $P$.
In this article, we consider a collection of geometric problems involving points colored by two colors (red and blue), referred to as bichromatic problems. The motivation behind studying these problems is two fold; (i) these problems appear naturally
We provide a tight result for a fundamental problem arising from packing squares into a circular container: The critical density of packing squares into a disk is $delta=frac{8}{5pi}approx 0.509$. This implies that any set of (not necessarily equal)
The Split Packing algorithm cite{splitpacking_ws, splitpackingsoda, splitpacking} is an offline algorithm that packs a set of circles into triangles and squares up to critical density. In this paper, we develop an online alternative to Split Packing
We give an asymptotic approximation scheme (APTAS) for the problem of packing a set of circles into a minimum number of unit square bins. To obtain rational solutions, we use augmented bins of height $1+gamma$, for some arbitrarily small number $gamm
Finding cliques in random graphs and the closely related planted clique variant, where a clique of size t is planted in a random G(n,1/2) graph, have been the focus of substantial study in algorithm design. Despite much effort, the best known polynom