On the effect of randomness on planted 3-coloring models

TL;DR

This paper introduces a unified framework for studying planted 3-coloring problems, analyzing how randomness and adversarial choices affect the complexity and solvability of graph coloring.

Contribution

It presents a new framework that generalizes previous models, allowing for both adversarial and random components, and adapts existing algorithms to these new settings.

Findings

The Alon-Kahale 3-coloring algorithm extends to spectral expander graphs with adversarial planted colorings.

Finding 3-colorings is NP-hard in certain regimes with adversarial planted colorings.

The framework clarifies the role of randomness in the success of coloring algorithms.

Abstract

We present the hosted coloring framework for studying algorithmic and hardness results for the $k$-coloring problem. There is a class ${\cal H}$ of host graphs. One selects a graph $H\in{\cal H}$ and plants in it a balanced $k$-coloring (by partitioning the vertex set into $k$ roughly equal parts, and removing all edges within each part). The resulting graph $G$ is given as input to a polynomial time algorithm that needs to $k$-color $G$ (any legal $k$-coloring would do -- the algorithm is not required to recover the planted $k$-coloring). Earlier planted models correspond to the case that ${\cal H}$ is the class of all $n$-vertex $d$-regular graphs, a member $H\in{\cal H}$ is chosen at random, and then a balanced $k$-coloring is planted at random. Blum and Spencer [1995] designed algorithms for this model when $d=n^{\delta}$ (for $0<\delta\le1$), and Alon and Kahale [1997] managed to do so even when $d$ is a sufficiently large constant. The new aspect in our framework is that it need not involve randomness. In one model within the framework (with $k=3$) $H$ is a $d$ regular spectral expander (meaning that except for the largest eigenvalue of its adjacency matrix, every other eigenvalue has absolute value much smaller than $d$) chosen by an adversary, and the planted 3-coloring is random. We show that the 3-coloring algorithm of Alon and Kahale [1997] can be modified to apply to this case. In another model $H$ is a random $d$-regular graph but the planted balanced $3$-coloring is chosen by an adversary, after seeing $H$. We show that for a certain range of average degrees somewhat below $\sqrt{n}$, finding a 3-coloring is NP-hard. Together these results (and other results that we have) help clarify which aspects of randomness in the planted coloring model are the key to successful 3-coloring algorithms.