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Sum-Of-Squares To Approximate Knapsack

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arxiv 2502.13292 v1 pith:OZVULYXS submitted 2025-02-18 cs.DS math.OC

classification cs.DSmath.OC
keywords sum-of-squaresknapsacknotesadvancedalgorithmalgorithmsapproximateapproximation
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These notes give a self-contained exposition of Karlin, Mathieu and Nguyen's tight estimate of the integrality gap of the sum-of-squares semidefinite program for solving the knapsack problem. They are based on a sequence of three lectures in CMU course on Advanced Approximation Algorithms in Fall'21 that used the KMN result to introduce the Sum-of-Squares method for algorithm design. The treatment in these notes uses the pseudo-distribution view of solutions to the sum-of-squares SDPs and only rely on a few basic, reusable results about pseudo-distributions.

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