# the lasserre hierarchy in approximation algorithms

. Lectures notes. algorithms. The limitations of the Lasserre hierarchy have also been studied. Approximate Algorithms Introduction: An Approximate Algorithm is a way of approach NP-COMPLETENESS for the optimization problem. When can local constraints help in approximating a global property (eg. . There has been a fair bit of recent interest in Lasserre hierarchy based approximation algorithms [CS08,KMN10,GS11,BRS11,RT12,AG11,GS12a]. 2009. The Lasserre Hierarchy Start with a 0/1 integer quadratic program. . . . In light of the above, the power and limitations of the Lasserre hierarchy merit further inves-tigation. We ﬁrst provide integrality gaps for dispersers in the Lasse rre hierarchy. The Lasserre hierarchy of semidefinite programming approximations to convex polynomial optimization problems is known to converge finitely under some assumptions. This technique does not guarantee the best solution. These problems are notorious for the existence of huge gaps between the known algorith-mic results and NP-hardness results. ity is via designing approximation algorithms to efﬁciently approximate the optimal solutions with provable guarantees. Local Constraints in Approximation Algorithms LP or SDP based approximation algorithms impose constraints onfew variablesat a time. Slides. In this work, we study the power of Lasserre/Sum-of-Squares SDP Hierar-chy. We prove the integrality gaps in the Lasserre hierarchy, which is a strong algo-rithmic tool in approximation algorithm design such that most currently known semideﬁnite programming based algorithms can be derived by a constant number of levels in this hierarchy. MAPSP Tutorial: The Lasserre Hierarchy in Approximation Algorithms. In many interesting cases, for small constant ‘, the ‘th level of the Lasserre hierarchy provides the best known polynomial-time computable approximation. . Our algorithm is based on rounding semidefinite pro-grams from the Lasserre hierarchy, and the analysis uses bounds for low-rank approximations of a matrix in Frobenius norm using columns of the matrix. First part of this work focuses on using Lasserre/Sum-of-Squares SDP Hi-erarchy to achieve better approximation ratio for certain CSPs with global cardi-nality constraints. Vertex Cover, Chromatic Number)? the Lasserre hierarchy, for several of these problems, and a broader class of quadratic integer pro-gramming problems with linear constraints (more details are in Section1.1below). laxations would be able to achieve better approximation ratio for Max CSPs and their variants. . (in the Lasserre Hierarchy) Madhur Tulsiani UC Berkeley. On the other hand, given an NP-hard optimization problem, we are also interested in the best possible approx- ... 5.2.3 The Lasserre hierarchy for DENSEkSUBGRAPH. Think “big" variables Z S = Q i2S z i. The goal of an approximation algorithm is to come as close as possible to the optimum value in a reasonable amount of time which is at the most polynomial time. .69 Our algorithm is based on rounding semidefinite programs from the Lasserre hierarchy, and the analysis uses bounds for low-rank approximations of a matrix in Frobenius norm using columns of the matrix. Our algorithms deliver a good approximation ratio if the eigenvalues of the Laplacian of … Lecture for PhD students held at EPFL in Lausanne, Switzerland in Fall 2009 (2 hours/week for one semester) Most of the known lower bounds for the hierarchy originated in the works of Grigoriev [17, 18] (also independently rediscovered later by Schoenebeck [36]). For a more detailed overview on the use of hierarchies in approximation algorithms, see the surveys [11, 25, 26]. Approximation Algorithms. These problems are notorious for the existence of huge gaps between the known algorithmic results and NP-hardness results. . For instance, our work [GS11]

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