Solving the Max-Cut SDP || @ CMU || Lecture 19c of CS Theory Toolkit

Value of the Information & Strength of the Argument

The lecture provides a clear and rigorous explanation of the algorithmic aspects of solving SDPs, specifically for Max-Cut. The argumentation is solid, building from the definition of PSD matrices to the implementation of the separation oracle via Cholesky decomposition. The instructor addresses potential pitfalls, such as numerical issues with eigenvalue computation, and justifies the use of the ellipsoid method. The value lies in its precise treatment of a fundamental technique in theoretical computer science, making it a valuable resource for advanced students and researchers.

Scientific Rigor, Source Quality, Title Accuracy

The lecture is scientifically rigorous, based on well-established results in convex optimization and linear algebra. The instructor cites two key references: the book by Grötschel, Lovász, and Schrijver on geometric algorithms and combinatorial optimization, and a paper by Delorme and Poljak on Laplacian eigenvalues and the maximum cut problem. These are appropriate and authoritative sources. The title accurately reflects the content, which is a focused lecture on solving the Max-Cut SDP. No public comments were provided for analysis.

The title accurately reflects the content: the lecture focuses on solving the SDP relaxation for Max-Cut using the ellipsoid method and PSD matrices.

8/10

Lecture by a recognized expert in theoretical computer science, based on established mathematical results (ellipsoid method, Cholesky decomposition). The content is rigorous and technically accurate, though it is a lecture without formal peer review.

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

• 00:00Introduction to the lecture and the problem of solving the SDP relaxation for Max-Cut.
• 00:30Definition of positive semidefinite (PSD) matrices and their characterization via infinitely many inequalities.
• 02:00Introduction of the ellipsoid algorithm and the need for a separation oracle.
• 04:00Explanation of the Cholesky decomposition as a symmetric Gaussian elimination to test PSD.
• 06:00Discussion of alternative methods like eigenvalue decomposition and their numerical issues.
• 08:00Use of the Cholesky-based separation oracle in the ellipsoid algorithm to solve SDPs.
• 10:00Technical note on the limitations of solving SDPs exactly and the need for approximate solutions.
• 12:00Proof that PSD matrices can be written as U^T U and the equivalence to vector dot products.
• 14:00Conclusion and preview of the next lecture on rounding the SDP solution for Max-Cut.

• Ryan O'Donnell's homepage — Instructor's academic page
• Course homepage on Diderot — Course materials and resources
• Rebecca Kiger Photography — Thumbnail photo credit

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