All Paths Lead to Rome

TL;DR

This paper analyzes the computational complexity of the puzzle game Roma, proving various complexity results including NP-completeness and providing algorithms based on Catalan structures.

Contribution

It establishes the complexity classifications of Roma puzzle problems and introduces a dynamic programming algorithm for solving them efficiently.

Findings

Completing Roma is NP-complete.

Counting solutions is #P-time complete.

Determining minimal preset arrows for unique solutions is ^P-complete.

Abstract

All roads lead to Rome is the core idea of the puzzle game Roma. It is played on an $n \times n$ grid consisting of quadratic cells. Those cells are grouped into boxes of at most four neighboring cells and are either filled, or to be filled, with arrows pointing in cardinal directions. The goal of the game is to fill the empty cells with arrows such that each box contains at most one arrow of each direction and regardless where we start, if we follow the arrows in the cells, we will always end up in the special Roma-cell. In this work, we study the computational complexity of the puzzle game Roma and show that completing a Roma board according to the rules is an \NP-complete task, counting the number of valid completions is #Ptime-complete, and determining the number of preset arrows needed to make the instance \emph{uniquely} solvable is $\Sigma_2^P$-complete. We further show that the problem of completing a given Roma instance on an $n\times n$ board cannot be solved in time $\mathcal{O}\left(2^{{o}(n)}\right)$ under ETH and give a matching dynamic programming algorithm based on the idea of Catalan structures.