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• W2748623618 abstract "Author(s): Weitz, Benjamin | Advisor(s): Raghavendra, Prasad | Abstract: Semidenite programming (SDP) relaxations have been a popular choice for approximationalgorithm design ever since Goemans and Williamson used one to improve the best approximationof Max-Cut in 1992. In the effort to construct stronger and stronger SDP relaxations,the Sum-of-Squares (SOS) or Lasserre hierarchy has emerged as the most promisingset of relaxations. However, since the SOS hierarchy is relatively new, we still do not knowthe answer to even very basic questions about its power. For example, we do not even knowwhen the SOS SDP is guaranteed to run correctly in polynomial time!In this dissertation, we study the SOS hierarchy and make positive progress inunderstanding the above question, among others. First, we give a sufficient, simple criteriawhich implies that an SOS SDP will run in polynomial time, as well as confirm that ourcriteria holds for a number of common applications of the SOS SDP. We also present anexample of a Boolean polynomial system which has SOS certificates that require exponential time to find, even though the certificates are degree two. This answers a conjecture of [54].Second, we study the power of the SOS hierarchy relative to other symmetric SDP relaxationsof comparable size. We show that in some situations, the SOS hierarchy achievesthe best possible approximation among every symmetric SDP relaxation. In particular, weshow that the SOS SDP is optimal for the Matching problem. Together with an SOS lowerbound due to Grigoriev [32], this implies that the Matching problem has no subexponentialsize symmetric SDP relaxation. This can be viewed as an SDP analogy of Yannakakis'original symmetric LP lower bound [72].As a key technical tool, our results make use of low-degree certificates of ideal membershipfor the polynomial ideal formed by polynomial constraints. Thus an important step in ourproofs is constructing certificates for arbitrary polynomials in the corresponding constraintideals. We develop a meta-strategy for exploiting symmetries of the underlying combinatorialproblem. We apply our strategy to get low-degree certificates for Matching, Balanced CSP, TSP, and others." @default.
• W2748623618 created "2017-08-31" @default.
• W2748623618 creator A5067283662 @default.
• W2748623618 date "2017-01-01" @default.
• W2748623618 modified "2023-09-24" @default.
• W2748623618 title "Polynomial Proof Systems, Effective Derivations, and their Applications in the Sum-of-Squares Hierarchy" @default.
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