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Wataridori is NP-Complete

Suthee Ruangwises

TL;DR

This work proves that the Wataridori puzzle is $NP$-complete by presenting a direct polynomial-time reduction from Numberlink. The core idea is a block-based encoding where each Numberlink cell is replaced by a $(4k+5) imes (4k+5)$ block, and endpoints are labeled with distinct numbers so that solutions correspond one-to-one. A key part of the method is selecting $k = ceil((p-1)/2)$ to guarantee at least $p$ distinct region-count values, encoding the original pairings via the region-visit counts. The reduction preserves solvability in both directions and fits within polynomial time, thereby establishing the hardness component. Overall, the paper contributes a direct cross-puzzle reduction technique to the literature on pencil-puzzle complexity and demonstrates the $NP$-completeness of Wataridori.

Abstract

Wataridori is a pencil puzzle involving drawing paths to connect all circles in a rectangular grid into pairs, in order to satisfy several constraints. In this paper, we prove that deciding solvability of a given Wataridori puzzle is NP-complete via reduction from Numberlink, another pencil puzzle that has already been proved to be NP-complete.

Wataridori is NP-Complete

TL;DR

This work proves that the Wataridori puzzle is -complete by presenting a direct polynomial-time reduction from Numberlink. The core idea is a block-based encoding where each Numberlink cell is replaced by a block, and endpoints are labeled with distinct numbers so that solutions correspond one-to-one. A key part of the method is selecting to guarantee at least distinct region-count values, encoding the original pairings via the region-visit counts. The reduction preserves solvability in both directions and fits within polynomial time, thereby establishing the hardness component. Overall, the paper contributes a direct cross-puzzle reduction technique to the literature on pencil-puzzle complexity and demonstrates the -completeness of Wataridori.

Abstract

Wataridori is a pencil puzzle involving drawing paths to connect all circles in a rectangular grid into pairs, in order to satisfy several constraints. In this paper, we prove that deciding solvability of a given Wataridori puzzle is NP-complete via reduction from Numberlink, another pencil puzzle that has already been proved to be NP-complete.
Paper Structure (10 sections, 1 theorem, 5 figures)

This paper contains 10 sections, 1 theorem, 5 figures.

Table of Contents

  1. Introduction
  2. Wataridori Puzzle
  3. Related Work
  4. Our Contribution
  5. Numberlink Puzzle
  6. Reduction from Numberlink to Wataridori
  7. Constructing a Block Representing Each Cell
  8. Idea of Reduction
  9. Proof of Correctness
  10. Future Work

Key Result

Theorem 1

$G$ has a solution if and only if $H$ has a solution.

Figures (5)

  • Figure 1: An example of a $6 \times 6$ Wataridori puzzle (left) and its solution (right)
  • Figure 2: An example of a $6 \times 6$ Numberlink puzzle (left) and its solution (right)
  • Figure 3: A $13 \times 13$ number block in Wataridori puzzle $H$ for $k=2$ (bottom) representing a cell with number $x$ in Numberlink puzzle $G$ (top), with the red lines showing one possible solution corresponding to the path going towards the right of the cell in $G$
  • Figure 4: A $13 \times 13$ empty block in Wataridori puzzle $H$ for $k=2$ (bottom) representing an empty cell in Numberlink puzzle $G$ (top), with the red lines showing one possible solution corresponding to the path coming from the left, turning left, and going towards the top of the cell in $G$
  • Figure 5: Three diffrent ways the path can exit the block on the right side, passing through six, eight, and ten different regions, respectively

Theorems & Definitions (2)

  • Theorem 1
  • proof