Robust Algorithms for Recovering Planted $r$-Colorable Graphs

Abstract

The planted clique problem is a fundamental problem in the study of algorithms and has been extensively studied in various random and semirandom models. It is known that a clique planted in a random graph can be efficiently recovered if the size of the clique is above the conjectured computational threshold of $\Omega_p(\sqrt{n})$. A natural question that arises then is: what other planted structures can be efficiently recovered? In this work, we investigate this question by considering random planted and semirandom models for the $r$-coloring problem. In our model, a subset $S \subseteq V$ of size $k$ is chosen, and an arbitrary $r$-colorable graph is planted on the subgraph induced by $S$. Edges between pairs in $V \setminus S$ are added independently with probability $p$, and an adversary may add arbitrary edges between $S$ and $V \setminus S$. Our main result is a polynomial-time algorithm that recovers most of the vertices of the planted $r$-colorable graph when $k \geq c r \sqrt{n/p}$, for some constant $c$. The key technical contribution is a novel semidefinite programming (SDP) relaxation and a rounding algorithm. Our algorithm is also robust to the presence of a monotone adversary that can insert edges within $V \setminus S$.