The SDP Relaxation for Max-Cut || @ CMU || Lecture 19b of CS Theory Toolkit

The SDP Relaxation for Max-Cut || @ CMU || Lecture 19b of CS Theory Toolkit

by Ryan O'Donnell

Formal & Physical Sciences Computing & Cybersecurity UYComputer scienceUYAMathematical theory of computation
🎙 Ryan O'Donnell 👥 14K 📅 June 11, 2020 ⏱ 33 min 👁 3K 📄 lecture 🧭 2026-08-17
Available in: English (current) Français

Keywords

Max-CutSemidefinite ProgrammingRelaxationEllipsoid AlgorithmApproximation

Summary

This lecture, part of the CS Theory Toolkit course at CMU, introduces semidefinite programming (SDP) as a generalization of linear programming, motivated by the Max-Cut problem. The instructor, Ryan O’Donnell, begins by explaining the Max-Cut problem and why the standard LP relaxation fails to provide a good approximation. He then presents the key idea of Delorme and Poljak: reformulate the problem using ±1 variables and introduce variables y_vw representing the product x_v x_w. The constraint that such x_v exist is called the moment constraint. Since this constraint is not linear, the lecture shows how to relax it to an SDP constraint, which consists of infinitely many linear inequalities (one for each vector c). The lecture explains how the ellipsoid algorithm can handle such infinite constraints via a separation oracle. The SDP relaxation is then shown to be solvable in polynomial time and yields the Goemans-Williamson approximation algorithm with a factor of 0.878. The lecture also mentions that this is optimal under the Unique Games Conjecture. The presentation includes detailed derivations and examples, making it a rigorous introduction to SDP for Max-Cut.

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Critical Evaluation

Value of the Information & Strength of the Argument

The lecture provides a thorough and rigorous introduction to the SDP relaxation for Max-Cut. It clearly explains the motivation, the mathematical formulation, and the reasoning behind each step. The argumentation is solid: the instructor carefully derives the SDP constraint from the moment constraint, proves that the SDP constraint is a relaxation, and explains how the ellipsoid algorithm can solve the resulting program. The value of the information is high for students and researchers in theoretical computer science, as it bridges linear programming and SDP and demonstrates a key technique in approximation algorithms.

Scientific Rigor, Source Quality, Title Accuracy

The lecture is scientifically rigorous, based on well-established results in combinatorial optimization. The instructor references the work of Delorme and Poljak and the book by Grötschel, Lovász, and Schrijver, which are authoritative sources. The title accurately reflects the content, and the lecture is well-structured. The presentation is clear and includes proofs and explanations, ensuring the reliability of the information.

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Title / Content Match

The title accurately reflects the content: the lecture focuses on the SDP relaxation for Max-Cut, as part of a CS Theory Toolkit course.

Quality & Reliability

9/10

Lecture from a graduate course at Carnegie Mellon University by a recognized expert in theoretical computer science. The content is rigorous, well-structured, and based on established results (Goemans-Williamson, Delorme-Poljak). The presentation is clear and includes proofs and explanations.

Key Moments

Markers derived by PSI from the transcript: the creator did not define chapters.

Cited Sources

  • Ryan O'Donnell's homepage — Instructor's academic page, providing background and additional resources.
  • Course homepage on Diderot — Course materials and lecture notes for CS Theory Toolkit.
  • Rebecca Kiger Photography — Photographer of the thumbnail image.

Concurring Sources

  • Geometric Algorithms and Combinatorial Optimization — Book by Grötschel, Lovász, and Schrijver, referenced as a resource for the lecture.
  • Laplacian eigenvalues and the maximum cut problem — Paper by Delorme and Poljak, referenced as the source of the key idea.

Contribution & Novelties

This lecture provides a clear and detailed exposition of the SDP relaxation for Max-Cut, making the connection between the moment constraint and the SDP constraint explicit. It explains how the ellipsoid algorithm can handle the infinite constraints via a separation oracle, and highlights the Goemans-Williamson approximation algorithm. The lecture is particularly valuable for its pedagogical approach, breaking down the derivation step by step.

Pour aller plus loin :

107 words

Radar Profile

The radar profile shows high scores across all dimensions, indicating a lecture that is both information-dense and technically rigorous. The balance between quantity and quality of information is strong, and the technical level is appropriate for a graduate-level audience. The overall reliability is high, reflecting the instructor's expertise and the use of established sources.

Reliability 9/10