Robust Algorithms for Recovering Planted $r$-Colorable Graphs

Anand Louis, Rameesh Paul, Prasad Raghavendra
Proceedings of Thirty Eighth Conference on Learning Theory, PMLR 291:3766-3794, 2025.

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 $V \setminus S$.

Cite this Paper

BibTeX
@InProceedings{pmlr-v291-louis25a, title = {Robust Algorithms for Recovering Planted $r$-Colorable Graphs}, author = {Louis, Anand and Paul, Rameesh and Raghavendra, Prasad}, booktitle = {Proceedings of Thirty Eighth Conference on Learning Theory}, pages = {3766--3794}, year = {2025}, editor = {Haghtalab, Nika and Moitra, Ankur}, volume = {291}, series = {Proceedings of Machine Learning Research}, month = {30 Jun--04 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v291/main/assets/louis25a/louis25a.pdf}, url = {https://proceedings.mlr.press/v291/louis25a.html}, 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 $V \setminus S$.} }
Endnote
%0 Conference Paper %T Robust Algorithms for Recovering Planted $r$-Colorable Graphs %A Anand Louis %A Rameesh Paul %A Prasad Raghavendra %B Proceedings of Thirty Eighth Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2025 %E Nika Haghtalab %E Ankur Moitra %F pmlr-v291-louis25a %I PMLR %P 3766--3794 %U https://proceedings.mlr.press/v291/louis25a.html %V 291 %X 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 $V \setminus S$.
APA
Louis, A., Paul, R. & Raghavendra, P.. (2025). Robust Algorithms for Recovering Planted $r$-Colorable Graphs. Proceedings of Thirty Eighth Conference on Learning Theory, in Proceedings of Machine Learning Research 291:3766-3794 Available from https://proceedings.mlr.press/v291/louis25a.html.

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