[Paper Review] 2-FREE-FLOOD-IT is polynomial

Aurélie Lagoutte|arXiv (Cornell University)|Aug 18, 2010

Constraint Satisfaction and Optimization1 references3 citations

TL;DR

This paper proves that the 2-Free-Flood-It problem is solvable in polynomial time by showing that the minimum number of flooding operations needed to monochromatize a 2-colored graph is exactly equal to the radius of its reduced graph. The algorithm computes the radius of the reduced graph in O(nm) time, providing an optimal and efficient solution for both general graphs and the N×N square grid case.

ABSTRACT

We study a discrete diffusion process introduced in some combinatorial games called FLOODIT and MADVIRUS that can be played online and whose computational complexity has been recently studied by Arthur et al (FUN'2010). The flooding dynamics used in those games can be defined for any colored graph. It has been shown in a first report (in french, hal-00509488 on HAL archive) that studying this dynamics directly on general graph is a valuable approach to understand its specificities and extract uncluttered key patterns or algorithms that can be applied with success to particular cases like the square grid of FLOODIT or the hexagonal grid of MADVIRUS, and many other classes of graphs. This report is the translation from french to english of the section in the french report showing that the variant of the problem called 2-FREE-FLOOD-IT can be solved with a polynomial algorithm, answering a question raised in the previous study of FLOODIT by Arthur et al.

Motivation & Objective

• To resolve the open computational complexity question of 2-Free-Flood-It, which had been unresolved despite NP-hardness for 3 or more colors.
• To establish that 2-Free-Flood-It admits a polynomial-time algorithm, answering a question posed in prior works [3, 4].
• To provide a general and efficient algorithm applicable to all connected graphs with a 2-coloration, including the square grid used in FLOODIT.
• To formalize the use of reduced graphs as a key abstraction, showing equivalence between original and reduced graph dynamics.

Proposed method

• The problem is transformed into a graph reduction process by constructing the reduced graph, where each vertex represents a monochromatic connected zone in the original graph.
• The reduced graph is bipartite due to the proper 2-coloration, and each flooding operation corresponds to a neighborhood contraction around a vertex.
• The key insight is that flooding a vertex x in the reduced graph corresponds to contracting x and its neighbors into a single vertex, forming G/x.
• The radius R(G) of the reduced graph G is used as the optimal number of steps, with each step reducing the radius by exactly one when the center vertex is chosen.
• The algorithm computes the radius of the reduced graph in O(nm) time using BFS from each vertex to determine eccentricities.
• For the N×N grid, the time complexity is O(N⁴), derived from n = N² and m = 4N² − 4N.

Experimental results

Research questions

• RQ1 Is 2-Free-Flood-It solvable in polynomial time, despite the NP-hardness of the 3-color variant?
• RQ2 Can the minimum number of flooding operations be characterized precisely using graph-theoretic invariants?
• RQ3 Does the radius of the reduced graph serve as an exact lower and upper bound for the number of steps needed to monochromatize the graph?
• RQ4 Can neighborhood contractions in the reduced graph preserve the necessary structural properties to ensure optimality?

Key findings

• The minimum number of flooding operations required to monochromatize a 2-colored graph is exactly equal to the radius R(G) of its reduced graph.
• The radius of the reduced graph decreases by exactly one in each optimal flooding step when the center vertex is selected, proving R(G) is both necessary and sufficient.
• A polynomial-time algorithm exists for 2-Free-Flood-It, running in O(nm) time on the reduced graph, where n and m are the number of vertices and edges.
• For the N×N square grid, the algorithm runs in O(N⁴) time, which is efficient and optimal for this class.
• The reduced graph construction is linear in time, and the radius computation is the dominant cost, solvable via BFS from each vertex.
• The neighborhood contraction operation G/x preserves the essential structure for radius analysis, and R(G/x) ≥ R(G) − 1 holds universally.