Tiling Problem: Complexity & Reductions

• Tiling problems are defined as determining whether a given region can be exactly covered by a finite set of prototiles with no gaps or overlaps under predefined local constraints.
• Research highlights efficient algorithmic methods, including height function techniques, alongside NP-completeness proofs via reductions from SAT variants.
• Extensions involve counting tilings, converting symbolic Wang tiles to geometric models, and framing tiling issues as graph connectivity problems.

A tiling problem asks, given a finite set of prototiles and a target region (often $\mathbb{Z}^2$ or the Euclidean plane), whether the region can be exactly covered (without gaps or overlaps) by translates or isometries of the prototiles, subject to local adjacency constraints. The problem admits extensive study in both theoretical computer science (complexity, undecidability, algorithms) and combinatorial and geometric tiling theory.

1. Formal Definitions and Fundamental Variants

Given a finite set $R = \{(a_1,b_1),\dots,(a_k,b_k)\}$ of axis-aligned rectangle types, with $(a_i,b_i)$ denoting an $a_i \times b_i$ rectangle in $\mathbb{Z}^{2}$, the TILE$(R)$ problem is to decide, for a simply connected region $\Gamma$ (a finite union of grid squares, interior connected, no holes), whether $\Gamma$ can be exactly tiled by nonoverlapping translates of rectangles in $R$. The counting variant $\#\mathrm{TILE}(R)$ asks for the number of distinct tilings of $\Gamma$ by elements of $R$ (Pak et al., 2013).

For symbolic tiling, the session focuses on Wang tiles: unit squares with colored edges. A set $T \subset \Sigma^4$ of such 4-tuples determines legal placements by requiring matching of adjacent edge colors. TILE$(T)$ is the respective tileability problem for a finite region or the entire $\mathbb{Z}^2$ grid (Yang et al., 2024, Schwarzentruber, 2019).

Variants across the literature include:

• Tiling with arbitrary polyominoes rather than rectangles (Yang et al., 24 Jun 2025).
• Tiling the whole plane (total or partial, infinite or finite regions).
• Constraints: translation-only, with/without rotations/reflections, or imposing local adjacency rules.

2. Algorithmic and Complexity-Theoretic Milestones

Efficiently Solvable Cases

For two fixed rectangles $R = \{A,B\}$, Rémila established that tileability of a simply connected region of area $N$ is decidable in $O(N^2)$ time. The approach employs height function methods: each square receives an integer height, with legal placements corresponding to local height differences. There always exists a minimal height function capturing tileability. If such a function can be computed via graph-theoretic shortest-path methods, and local "flip" moves connect all tilings (when they exist), failure implies non-tileability (Pak et al., 2013).

For certain highly structured sets of Wang tiles (e.g., Brick Wang tiles), linear-time algorithms exist based on structural characterizations (for instance, the presence of at least one cycle in the region's adjacency graph) (Derouet-Jourdan et al., 2016).

NP-Completeness and Intractability

For $|R| \ge 3$ rectangles, or for large (but fixed) sets of other shapes or Wang tiles, the complexity landscape shifts sharply:

• NP-Completeness: There exists a finite set $R_0$ of at most $10^6$ rectangles such that TILE$(R_0)$ on simply connected regions is NP-complete. Membership in NP is immediate, as a tiling is a polynomial-size certificate. The proof reduces from Cubic Monotone 1-in-3 SAT (an NP-complete problem) to tiling with 23 Wang tiles (Yang et al., 2024) and then simulates Wang-tiling into tiling by rectangles (Pak et al., 2013).
• Counting complexity: The generalized counting problem $\#\mathrm{TILE}(R_0)$ is $\#\textrm{P}$-complete for this hard set $R_0$. The reduction is parsimonious—solution count is preserved through each step (Pak et al., 2013).
• Symbolic tiling (Wang tiles): For the general tiling decision problem $TILING_{n,n}$ (tiling $n \times n$ square), the problem is NP-complete, and for $TILING_{2^n,n}$ (height exponential in $n$), it is PSPACE-complete (Schwarzentruber, 2019).

3. Methods of Reduction and Equivalence

The hard instances universally arise by encoding computational problems into tiling constraints. The high-level strategy:

• Reduce from SAT variants (notably Cubic Monotone 1-in-3 SAT) to tiling—formulate Boolean constraints as boundary and adjacency requirements in a tiling instance.
• Simulate Turing machine configuration graphs by spatial encodings, often via Wang tile gadgets enforcing local stepwise transitions.
• Convert symbolic tiling (Wang tiles) to geometric tiling (rectangles, polyominoes) via gadget constructions: expand each Wang tile into an intricate rectangle, encode edge colors as shifts/modifications, ensure only allowed adjacency patterns correspond to legal Wang transitions.

A representative reduction sequence:

Step	Source Problem	Target Tiling	Reference
1	Cubic Monotone 1-in-3 SAT	23 Wang tiles	(Yang et al., 2024)
2	Wang tiles (finite set)	Rectangles	(Pak et al., 2013)

Key to these reductions is simulating logical constraints through local geometric constraints (adjacent colors, allowed placements), essentially encoding computation into the combinatorial structure of the tiling problem.

4. Table of Key Complexity Results

Problem Type	Class or Bound	Reference
TILE$(\{A,B\})$ (2 rectangles, simply connected)	P (quadratic)	(Pak et al., 2013)
TILE$(R_0)$ ($|R_0| \leq 10^6$, simply connected)	NP-complete	(Pak et al., 2013)
TILE$(T_0)$ ($|T_0|=23$ Wang tiles) (simply connected)	NP-complete	(Yang et al., 2024)
$\#\mathrm{TILE}(R_0)$ (counting, $|R_0| \leq 10^6$)	$\#$P-complete	(Pak et al., 2013)
TILING$_{n,n}$ (square region, general tiles)	NP-complete	(Schwarzentruber, 2019)
TILING$_{2^n,n}$, TILING$_{*,n}$ (exponential height)	PSPACE-complete	(Schwarzentruber, 2019)

5. Equivalence with Other Tiling Models

The following decision problems are all linear-time equivalent (reducible one to another with only overhead linear in the instance size):

1. Tiling simply connected regions with a fixed set of axis-aligned rectangles.
2. With a fixed finite set of (general) polyominoes.
3. With a fixed set of generalized Wang tiles (edge-labeled).
4. With a fixed set of relational Wang tiles (arbitrary local relations) (Pak et al., 2013).

Thus, hardness and tractability results port freely between these models. This equivalence is facilitated by the detailed gadget constructions that simulate the constraints of one type of tile by another.

6. Proof Structure and Reduction Gadgets

Central reduction tools:

• Wang-to-rectangles expansions: Each Wang tile is converted to a unique axis-aligned rectangle, possibly via a base expansion (e.g., perturbing by powers of 5) and careful control of shifts so that only the intended Wang tile adjacencies are permitted. Small rectangles and "perturbations" enforce the differentiation (see $(M,e)$-expansion gadgets).
• Graph-theoretic interpretation: Tileability is recast as a graph connectivity or shortest-path problem, enabling algorithmic solution or proof of hardness by simulating logical flows.
• Parsimonious reductions: The number of solutions is preserved through the reduction, supporting counting complexity results ($\#\mathrm{P}$-completeness).

7. Open Problems and Current Frontiers

• Reducing the size of the hard rectangle set $R_0$ below the order of $10^6$ or the corresponding hard Wang tile set below $23$ remains open. There is a conjecture that as few as $3$ rectangles might suffice for NP-completeness (Yang et al., 2024).
• For non-simply connected or more general regions, the gap between tractable and intractable classes for tile sets of small cardinality persists.
• Exact complexity for tiling by three or more rectangles in non-simply connected contexts (with holes) is unresolved.
• Further developments in symbolic-to-geometric reductions could further clarify the lower bounds for necessary rectangle or Wang tile set sizes.
• The precise dividing line between tractable (polynomial, even quadratic) and intractable (NP-hard) tiling problems under geometric constraints is an active research direction.

• Rémila, C. "Tiling a polygon with two kinds of rectangles", Discrete & Computational Geometry 34 (2005), 313–330 (Pak et al., 2013).
• Pak, I. and Yang, J. "Tiling simply connected regions with rectangles", JCTA (2012) (Pak et al., 2013).
• Yang, J. and Zhang, J. "NP-completeness of Tiling Finite Simply Connected Regions with a Fixed Set of Wang Tiles" (Yang et al., 2024).
• van Emde Boas, P. "The Convenience of Tilings" (1997) (Schwarzentruber, 2019).