Computational Complexity and Integer Programming Formulation of the Oredango Puzzle

TL;DR

This work analyzes the computational complexity of Oredango, a pencil puzzle on grids with circled positions and skewers. It establishes $NP$- and ASP-completeness by a polynomial-time, one-to-one reduction from $1$-in-$3$SAT, using three gadgets (G1, G2, G3) to enforce literal consistency and clause satisfaction, and shows this holds even when skewers and numbers are restricted to at most $1$. The paper also provides a complete $0$-$1$ integer programming formulation, with linear constraints for skewers, rows, and columns, and demonstrates practical solvability by solving $36$ puzzles with the Gurobi optimizer in under a second each. Collectively, the results place Oredango among hard combinatorial problems while also confirming that modern MILP/SOLVER approaches can efficiently solve many instances in practice, contributing to the puzzle complexity literature and providing a solid computational toolkit for this puzzle family.

Abstract

Oredango puzzle, one of the pencil puzzles, was originally created by Kanaiboshi and published in the popular puzzle magazine Nikoli. In this paper, we show NP- and ASP-completeness of Oredango by constructing a reduction from the 1-in-3SAT problem. Next, we formulate Oredango as an 0-1 integer-programming problem, and present numerical results obtained by solving Oredango puzzles from Nikoli and PuzzleSquare JP using a 0-1 optimization solver.

Figures (14)

• Figure 1: An example of a $4 \times 4$ Oredango (left) and its solution (right)
• Figure 2: Examples of wrong answers to the input of Figure \ref{['fig:ex']}
• Figure 3: Arrangement of literal circles for $C_1 = \{x_1, x_2, x_3\}$ in Oredango
• Figure 4: Arrangement of the rows corresponding to $C_1, C_2,$ and $C_3$
• Figure 5: Idea of constructing the board of Oredango from 1-in-3SAT

Theorems & Definitions (3)

• Theorem 1.1
• Example 2.1
• Example 2.2