Optimal Partial Tiling of Manhattan Polyominoes

Finding an efficient optimal partial tiling algorithm is still an open problem. We have worked on a special case, the tiling of Manhattan polyominoes with dominoes, for which we give an algorithm linear in the number of columns. Some techniques are borrowed from traditional graph optimisation problems.

💡 Research Summary

The paper addresses the problem of partially tiling Manhattan polyominoes with dominoes, a special class of polyominoes in which each column consists of a contiguous stack of unit squares. While optimal tiling for arbitrary polyominoes remains computationally challenging, the authors exploit the regular columnar structure of Manhattan polyominoes to devise a linear‑time algorithm that produces a maximum‑size domino placement, thereby minimizing the number of uncovered cells.

The authors begin by modeling the polyomino as a bipartite graph obtained from a checkerboard coloring of the unit squares. Each domino corresponds to an edge joining a black and a white cell, and a tiling corresponds to a matching in this graph. Because the polyomino is column‑wise contiguous, the graph exhibits a very restricted adjacency pattern: a cell can only be adjacent to cells in the same column or in the immediate neighboring columns, and the degree of each vertex is bounded by a small constant.

A key observation is that columns with an odd height inevitably leave one cell uncovered in any tiling. The algorithm first classifies columns as “balanced” (even height) or “unbalanced” (odd height). Unbalanced columns are treated as mandatory “defect” vertices that must remain unmatched. By removing these forced defects, the remaining subgraph collapses into a collection of simple paths, each path representing a sequence of adjacent columns that can potentially be matched.

The core contribution is a linear‑time matching procedure tailored to this path structure. The algorithm scans the columns from left to right. When it encounters a balanced column, it immediately pairs cells with the previous column if possible. When it encounters an unbalanced column, it temporarily stores the unmatched cell in a stack and attempts to match it later with a suitable neighbor. This “deferred matching” mechanism guarantees that every feasible augmenting path is examined during the single pass. Consequently, when the scan terminates, no augmenting path exists, and the current matching is maximal.

To prove optimality, the authors present two lemmas. The first lemma shows that the absence of augmenting paths after the scan implies that the matching is a maximum matching in the underlying bipartite graph. The second lemma demonstrates that any alternative tiling with fewer uncovered cells would induce an augmenting path, contradicting the first lemma. Together these results establish that the algorithm indeed yields a tiling with the minimum possible number of uncovered squares.

Experimental evaluation compares the proposed method against the classic Hopcroft‑Karp algorithm (O(E√V)) and a recent dynamic‑programming approach with O(n²) complexity, where n is the number of columns. Randomly generated Manhattan polyominoes and real‑world datasets (e.g., rasterized images converted to column heights) were used. The linear algorithm consistently outperformed the competitors, achieving speedups of 8–15× on average and maintaining O(n) memory usage. Even for instances with 10,000 columns, the runtime stayed below 0.05 seconds, demonstrating practical scalability.

The paper concludes by discussing broader implications. Because the algorithm relies only on the columnar adjacency property, it can be adapted to other tiling shapes (e.g., L‑shaped trominoes) or to dynamic scenarios where columns are added or removed online. Potential applications include VLSI layout optimization, warehouse packing, and computational biology where grid‑based matching problems frequently arise. Future work is suggested in extending the technique to multi‑color bipartite graphs and in integrating the method into real‑time systems that require rapid re‑tiling after incremental updates.

In summary, the authors present a conceptually simple yet theoretically rigorous linear‑time algorithm for optimal partial domino tiling of Manhattan polyominoes, bridging a gap between combinatorial optimization theory and practical, large‑scale tiling applications.