On the Hardest Problem Formulations for the $$0/1$$ Lasserre Hierarchy

Kurpisz, Adam; Leppänen, Samuli; Mastrolilli, Monaldo

doi:10.1007/978-3-662-47672-7_71

Cited by 7 publications

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References 33 publications

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“…In this paper we disprove Laurent's conjecture, and show that indeed the SoS rank of K is bounded between Ω( √ n) and n − Ω(n 1/3 ). Interestingly, Au [1] and the authors of this paper [12] independently considered the rank of a variation of the set K where on the right hand side of the inequalities there is an exponentially small constant instead of 1 2 . Both works show that the rank of the modified K is exactly n.…”

Section: Our Resultsmentioning

confidence: 99%

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Tight Sum-of-squares Lower Bounds for Binary Polynomial Optimization Problems

Kurpisz,

Leppänen,

Mastrolilli

2024

ACM Trans. Comput. Theory

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For binary polynomial optimization problems of degree 2 d with n variables Sakaue, Takeda, Kim and Ito [SIAM J. Optim., 2017] proved that the $\lceil \frac{n+2d-1}{2}\rceil $ th semidefinite (SDP) relaxation in the SoS/Lasserre hierarchy of SDP relaxations provides the exact optimal value. When n is an odd number, we show that their analysis is tight, i.e. we prove that $\frac{n+2d-1}{2} $ levels of the SoS/Lasserre hierarchy are also necessary. Laurent [Math. Oper. Res., 2003] showed that the Sherali-Adams hierarchy requires n levels to detect the empty integer hull of a linear representation of a set with no integral points. She conjectured that the SoS/Lasserre rank for the same problem is n − 1. In this paper we disprove this conjecture and derive lower and upper bounds for the rank.

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Section: Our Resultsmentioning

confidence: 99%

“…Note that polynomial G h (k) in ( 13) is nonnegative in a real interval, and in (12) it is zero over a set of integers. Moreover, constraints ( 14) are trivially satisfied for h > n/2 .…”

Section: T} That Satisfy the Following Conditionsmentioning

confidence: 99%

Tight Sum-of-squares Lower Bounds for Binary Polynomial Optimization Problems

Kurpisz,

Leppänen,

Mastrolilli

2024

ACM Trans. Comput. Theory

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Sum-of-Squares Hierarchy Lower Bounds for Symmetric Formulations

Kurpisz

Leppänen

Mastrolilli

2016

Integer Programming and Combinatorial Optimization

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We introduce a method for proving Sum-of-Squares (SoS)/ Lasserre hierarchy lower bounds when the initial problem formulation exhibits a high degree of symmetry. Our main technical theorem allows us to reduce the study of the positive semidefiniteness to the analysis of "well-behaved" univariate polynomial inequalities.We illustrate the technique on two problems, one unconstrained and the other with constraints. More precisely, we give a short elementary proof of Grigoriev/Laurent lower bound for finding the integer cut polytope of the complete graph. We also show that the SoS hierarchy requires a non-constant number of rounds to improve the initial integrality gap of 2 for the Min-Knapsack linear program strengthened with cover inequalities.

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A Lasserre Lower Bound for the Min-Sum Single Machine Scheduling Problem

Kurpisz

Leppänen

Mastrolilli

2015

Algorithms - ESA 2015

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The Min-sum single machine scheduling problem (denoted 1|| f j ) generalizes a large number of sequencing problems. The first constant approximation guarantees have been obtained only recently and are based on natural time-indexed LP relaxations strengthened with the so called Knapsack-Cover inequalities (see Bansal and Pruhs, Cheung and Shmoys and the recent (4 + ǫ)-approximation by Mestre and Verschae). These relaxations have an integrality gap of 2, since the Min-knapsack problem is a special case. No APX-hardness result is known and it is still conceivable that there exists a PTAS. Interestingly, the Lasserre hierarchy relaxation, when the objective function is incorporated as a constraint, reduces the integrality gap for the Min-knapsack problem to 1 + ǫ.In this paper we study the complexity of the Min-sum single machine scheduling problem under algorithms from the Lasserre hierarchy. We prove the first lower bound for this model by showing that the integrality gap is unbounded at level Ω( √ n) even for a variant of the problem that is solvable in O(n log n) time by the Moore-Hodgson algorithm, namely Min-number of tardy jobs. We consider a natural formulation that incorporates the objective function as a constraint and prove the result by partially diagonalizing the matrix associated with the relaxation and exploiting this characterization. * Supported by the Swiss National Science Foundation project 200020-144491/1 "Approximation Algorithms for Machine Scheduling Through Theory and Experiments". † A preliminary version of this paper appeared in 23rd European Symposium on Algorithms -ESA 2015.been recently improved to 4 + ǫ: Cheung and Shmoys [5] gave a primal-dual algorithm and claimed that is a (2 + ǫ)-approximation; recently, Mestre and Verschae [16] showed that the analysis in [5] cannot yield an approximation better than 4 and provided a proof that the algorithm in [5] has an approximation ratio of 4 + ǫ.A particular difficulty in approximating this problem lies in the fact that the ratio (integrality gap) between the optimal IP solution to the optimal solution of "natural" LPs can be arbitrarily large, since the Min-knapsack LP is a common special case. Thus, in [1,5] the authors strengthen natural time-indexed LP relaxations by adding (exponentially many) Knapsack-Cover (KC) inequalities introduced by Wolsey [24] (see also [4]) that have proved to be a useful tool to address capacitated covering problems.One source of improvements could be the use of semidefinite relaxations such as the powerful Lasserre/Sum-of-Squares hierarchy [13,19,22] (we defer the definition and related results to Section 2). Indeed, it is known [10] that for Min-knapsack the Lasserre hierarchy relaxation, when the objective function is incorporated as a constraint in the natural LP, reduces the gap to (1 + ε) at level O(1/ε), for any ε > 0. 1 In light of this observation, it is therefore tempting to understand whether the Lasserre hierarchy relaxation can replace the use of exponentially many KC inequalities to get a better ap...

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On the Hardest Problem Formulations for the $$0/1$$ Lasserre Hierarchy

Cited by 7 publications

References 33 publications

Tight Sum-of-squares Lower Bounds for Binary Polynomial Optimization Problems

Tight Sum-of-squares Lower Bounds for Binary Polynomial Optimization Problems

Sum-of-Squares Hierarchy Lower Bounds for Symmetric Formulations

A Lasserre Lower Bound for the Min-Sum Single Machine Scheduling Problem

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