Drainability and Fillability of Polyominoes in Diverse Models of Global Control

TL;DR

The paper studies drainability and fillability of polyomino boards under global tilt controls, comparing full-tilt and single-step models and their extensions. It develops general tools to relate different models, establishes a key duality between drainability in FT and fillability in S1, and shows that relaxing global control does not materially increase drainability power. It proves NP-hardness for obstacle-placement to guarantee drainability and provides a constant-factor approximation for scaled boards (4-approx for $k=3$, 6-approx for $k>3$) via minimum-weight arborescences on extended large tilt graphs. Through generalized model analysis and duality, it shows fillability and drainability align across several models, and identifies open questions on optimal filling sequences and the vacancy/occupancy problems.

Abstract

Tilt models offer intuitive and clean definitions of complex systems in which particles are influenced by global control commands. Despite a wide range of applications, there has been almost no theoretical investigation into the associated issues of filling and draining geometric environments. This is partly because a globally controlled system (i.e., passive matter) exhibits highly complex behavior that cannot be locally restricted. Thus, there is a strong need for theoretical studies that investigate these models both (1) in terms of relative power to each other, and (2) from a complexity theory perspective. In this work, we provide (1) general tools for comparing and contrasting different models of global control, and (2) both complexity and algorithmic results on filling and draining.

Figures (15)

• Figure 1: Movement of a single bubble in the single step model (left) is identical to the movement of a single particle in the full tilt model (right).
• Figure 2: (a) A board with a pixel $p$ at the intersection of its row and column segments (shaded area), along with the boundary pixels of these segments. Note that $p=p^u$. (b) A configuration of a board $B$ that is minimal with respect to a sink $s$ in the full tilt model. Particles are depicted as gray squares. (c) A drainable sub-board of $B$.
• Figure 3: Draining three particles to a sink $s$ using full tilt moves.
• Figure 4: (a) The small tilt graph, $G_S(B)$, of a board $B$. (b) The vertices of the large tilt graph, $G_L(B, \{s\})$, of $B$ and a sink $s$. The lightly-shaded pixels are added due to the reflex corner $C$ (though some are also part of the graph as corner pixels). Vertices marked with a cross are added as intersections. We only depict the edges incident on vertex $v$ to avoid clutter.
• Figure 5: Overview of the -hardness reduction for the full tilt variant. The depicted instance is due to the 3Sat-3 formula $\varphi = (x_1 \lor \neg x_2 \lor x_3) \land (\neg x_1 \lor x_4 \lor \neg x_5) \land (x_2 \lor x_3 \lor \neg x_4) \land (x_1 \lor x_4) \land (\neg x_3 \lor x_5)$.

Theorems & Definitions (38)

• Theorem 1
• Lemma 2
• Corollary 2
• Lemma 2
• Theorem 3
• Theorem 4
• Lemma 5
• Corollary 6
• Lemma 7
• Theorem 8