How Bad is the Freedom to Flood-It?

TL;DR

This paper compares Fixed-Flood-It and Free-Flood-It puzzles, revealing that allowing vertex choice increases complexity in some cases and analyzing the relationship between their solution lengths.

Contribution

It demonstrates how free vertex choice impacts problem complexity and provides bounds relating optimal solutions of the two variants.

Findings

Some Fixed-Flood-It cases become intractable in Free-Flood-It.

Tractable Fixed-Flood-It cases remain tractable in Free-Flood-It but require more complex proofs.

The optimal solution length for Fixed-Flood-It is at most twice that of Free-Flood-It, and this bound is tight.

Abstract

Fixed-Flood-It and Free-Flood-It are combinatorial problems on graphs that generalize a very popular puzzle called Flood-It. Both problems consist of recoloring moves whose goal is to produce a monochromatic ("flooded") graph as quickly as possible. Their difference is that in Free-Flood-It the player has the additional freedom of choosing the vertex to play in each move. In this paper, we investigate how this freedom affects the complexity of the problem. It turns out that the freedom is bad in some sense. We show that some cases trivially solvable for Fixed-Flood-It become intractable for Free-Flood-It. We also show that some tractable cases for Fixed-Flood-It are still tractable for Free-Flood-It but need considerably more involved arguments. We finally present some combinatorial properties connecting or separating the two problems. In particular, we show that the length of an optimal solution for Fixed-Flood-It is always at most twice that of Free-Flood-It, and this is tight.