Tromino Tilings with Pegs via Flow Networks

TL;DR

This paper introduces a linear-time algorithm for deciding and counting tromino tilings with pegs using flow network reductions, advancing tiling problem solutions with efficient computational methods.

Contribution

It characterizes peg-based tromino tilings via flow networks and provides linear-time algorithms for tiling decision and counting, improving computational efficiency.

Findings

Existence of a linear-time reduction to maximum-flow problem.

Counting tilings can be performed in linear time.

Algorithms enable efficient tiling decision in O(n) time.

Abstract

A tromino tiling problem is a packing puzzle where we are given a region of connected lattice squares and we want to decide whether there exists a tiling of the region using trominoes with the shape of an L. In this work we study a slight variation of the tromino tiling problem where some positions of the region have pegs and each tromino comes with a hole that can only be placed on top of the pegs. We present a characterization of this tiling problem with pegs using flow networks and show that (i) there exists a linear-time parsimonious reduction to the maximum-flow problem, and (ii) counting the number of such tilings can be done in linear-time. The proofs of both results contain algorithms that can then be used to decide the tiling of a region with pegs in $O(n)$ time.