Hard Tiling Problems with Simple Tiles

Abstract

Abstract. It is well-known that the question of whether a given finite region can be tiled with a given set of tiles is NP-complete. We show that the same is true for the right tromino and square tetromino on the square lattice, or for the right tromino alone. In the process, we show that Monotone 1-in-3 Satisfiability is NP-complete for planar cubic graphs. In higher dimensions, we show NP-completeness for the domino and straight tromino for general regions on the cubic lattice, and for simplyconnected regions on the four-dimensional hypercubic lattice.

Jigsaw Puzzles, Edge Matching, and Polyomino Packing: Connections and Complexity

“…Polyomino packing puzzles are NP-complete when the target shape is complicated (a polyomino with holes) and the pieces are all identical (either 2 × 2 squares, 1 × 3 rectangles, or 2 × 2 L shapes) [28]. In contrast, the polyomino packing puzzles we consider have a simple target shape (a larger square) and the problem is all about using the different tiles given.…”

Variations of the maximum leaf spanning tree problem for bipartite graphs

“…For each x ∈ X, let min x denote the least-cost edge between x and any vertex in Y it is adjacent to. min x is welldefined by condition (2). Note that computing the value of min x can be done in polynomial time.…”

“…2 Problem 2.3 remains NP-complete if we assume the bipartite graph is planar. This is due to the following result from [2]. It can easily be seen that an instance of the restricted 1-in-3 satisfiability problem, given by the conditions in Theorem 2.5, can be reduced to an instance of the X3C problem (X, B) by associating X with the clauses and B with the variables of the satisfiability problem instance.…”

Graph coloring with cardinality constraints on the neighborhoods

“…Proof. We use a transformation from the CUBIC PLANAR MONOTONE 1-in-3SAT problem which is known to be NP-complete (see [20]). In this problem we are given a set X of variables and a set C of clauses of the form (a ∨ b ∨ c) where a, b and c are distinct variables without negation such that the underlying bipartite graph G = (X ∪ C , E) = (X ∪ C , {[x i ,ĉ]|x i occurring in clauseĉ ∈ C }) is 3-regular and planar.…”