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On the Hardest Problem Formulations for the 0/1 Lasserre Hierarchy [article]

Adam Kurpisz, Samuli Leppänen, Monaldo Mastrolilli
2015 arXiv   pre-print
These problems are the hardest for the Lasserre hierarchy in this sense.  ...  It is known that the hierarchy converges to the 0/1 polytope in n levels and captures the convex relaxations used in the best available approximation algorithms for a wide variety of optimization problems  ...  We thank Ola Svensson and the anonymous reviewers for helpful comments.  ... 
arXiv:1510.01891v1 fatcat:6rxxi3mh2rev3hjimbmaa5lb4y

Strengthening Convex Relaxations of 0/1-Sets Using Boolean Formulas [article]

Samuel Fiorini, Tony Huynh, Stefan Weltge
2018 arXiv   pre-print
As one result, interpreting an iterated application of our procedure as a hierarchy, our findings simplify, improve, and extend previous results by Bienstock and Zuckerberg on covering problems.  ...  On the other hand, various methods have been designed for obtaining strengthened relaxations for very specific sets that arise in combinatorial optimization.  ...  The hierarchies of Sherali & Adams [25] , Lovász & Schrijver [19] , and Lasserre [18] , which are tailored to 0/1-sets S ⊆ {0, 1} n , are methods of that type.  ... 
arXiv:1711.01358v3 fatcat:p3yvosmvdjef3cob6juvpyb4oy

Strengthening convex relaxations of 0/1-sets using Boolean formulas

Samuel Fiorini, Tony Huynh, Stefan Weltge
2020 Mathematical programming  
As one application, interpreting an iterated application of our procedure as a hierarchy, our findings simplify, improve, and extend previous results by Bienstock and Zuckerberg on covering problems.  ...  Our procedure strengthens any convex set containing a set S ⊆ { 0 , 1 } n by exploiting certain additional information about S.  ...  We also thank all the anonymous referees for their constructive comments on the paper.  ... 
doi:10.1007/s10107-020-01542-w pmid:34776534 pmcid:PMC8550646 fatcat:i2qtr3izxzhuta5ahhsobak7am

Constant Factor Lasserre Integrality Gaps for Graph Partitioning Problems [article]

Venkatesan Guruswami, Ali Kemal Sinop, Yuan Zhou
2014 arXiv   pre-print
This complements recent algorithmic results in Guruswami and Sinop (2011) which used the Lasserre hierarchy to give an approximation scheme for these problems (with runtime depending on the spectrum of  ...  For this problem, and variants such as the Uniform Sparsest Cut problem where the goal is to minimize the fraction of pairs on opposite sides of the cut that are connected by an edge, there are large gaps  ...  b i ∈ {0, 1}.  ... 
arXiv:1202.6071v3 fatcat:ankt6qg7yvgmlonzx4eptheivi

Constant Factor Lasserre Integrality Gaps for Graph Partitioning Problems

Venkatesan Guruswami, Ali Kemal Sinop, Yuan Zhou
2014 SIAM Journal on Optimization  
This complements recent algorithmic results in Guruswami and Sinop (2011) which used the Lasserre hierarchy to give an approximation scheme for these problems (with runtime depending on the spectrum of  ...  For this problem, and variants such as the Uniform Sparsest Cut problem where the goal is to minimize the fraction of pairs on opposite sides of the cut that are connected by an edge, there are large gaps  ...  b i ∈ {0, 1}.  ... 
doi:10.1137/13093025x fatcat:mihce365nfem7jpk44cgl476re

High Degree Sum of Squares Proofs, Bienstock-Zuckerberg hierarchy and Chvatal-Gomory cuts [article]

Monaldo Mastrolilli
2019 arXiv   pre-print
In this paper we present a novel polynomial time SOS hierarchy for 0/1 problems with a custom subspace of high degree polynomials (not the standard subspace of low-degree polynomials).  ...  Moreover, for a class of polytopes (e.g. set covering and packing problems), the resulting SOS hierarchy optimizes in polynomial time over the polytope resulting from any constant rounds of CG-cuts, up  ...  These problems are the "hardest" for the SoS hierarchy in this sense.  ... 
arXiv:1709.07966v7 fatcat:ns2h3t4qwzaavc6sgdfpcvyuyu

A Semidefinite Programming approach for minimizing ordered weighted averages of rational functions [article]

V. Blanco, S. El-Haj Ben-Ali, J. Puerto
2011 arXiv   pre-print
We illustrate this methodology with some extensive computational results on location problems in the plane and the 3-dimension space.  ...  This reformulation admits a hierarchy of SDP relaxations that approximates, up to any degree of accuracy, the optimal value of those problems.  ...  ¿From our tables we conclude that Weber problem is the simplest one whereas the trimmed-mean problem is the hardest one, as expected.  ... 
arXiv:1106.2407v2 fatcat:uqn6ri3dtvco5bm67tnmcupaa4

Affine reductions for LPs and SDPs [article]

Gábor Braun, Sebastian Pokutta, Daniel Zink
2016 arXiv   pre-print
In the case of SDPs, we obtain inapproximability results for these problems relative to the SDP-inapproximability of MaxCUT.  ...  We define a reduction mechanism for LP and SDP formulations that degrades approximation factors in a controlled fashion.  ...  We are indebted to Siu On Chan for some of the PCP inapproximability bounds as well as Santosh Vempala for the helpful discussions.  ... 
arXiv:1410.8816v5 fatcat:tqklld6tnbbmvokxxag2l6sjem

Revisiting several problems and algorithms in continuous location with ℓ_p norms [article]

Víctor Blanco, Justo Puerto, Safae El Haj Ben Ali
2013 arXiv   pre-print
for any p> 1).  ...  The ultimate goal is to provide a common approach to solve the family of continuous ℓ_p ordered median location problems in dimension d (including of course the ℓ_p minisum or Fermat-Weber location problem  ...  Acknowledgements The authors were partially supported by the project FQM-5849 (Junta de Andalucía\FEDER).  ... 
arXiv:1312.7473v1 fatcat:6kftlishjbedxngz6vm6tjr7xa

Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs [chapter]

Ryan O'Donnell, John Wright, Chenggang Wu, Yuan Zhou
2013 Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms  
Building on work of Cai, Fürer, and Immerman [CFI92], we show two hardness results for the Graph Isomorphism problem.  ...  , this holds for the hardest known instances of UNIQUEGAMES [BBH + 12].  ...  SOS/Lasserre hierarchy One way to formulate the SOS/Lasserre hierarchy is via the pseudo-expectation view. We briefly recall the formulation as follows.  ... 
doi:10.1137/1.9781611973402.120 dblp:conf/soda/ODonnellWWZ14 fatcat:fvq6dhxkxvaetecklp74d64hpi

A Bundle Approach for SDPs with Exact Subgraph Constraints [article]

Elisabeth Gaar, Franz Rendl
2019 arXiv   pre-print
Computational experiments on the Max-Cut, stable set and coloring problem show the efficiency of this approach.  ...  We introduce a computational framework for these relaxations designed to cope with these difficulties.  ...  Among the most prominent hierarchies are the polyhedral ones from Boros, Crama and Hammer [3] as well as the ones from Sherali and Adams [20] , Lovász and Schrijver [15] and Lasserre [13] which  ... 
arXiv:1902.05345v1 fatcat:ra4ckvzbdvaklpulpc5hzxhfdm

Computational complexity of the quantum separability problem [article]

Lawrence M. Ioannou
2007 arXiv   pre-print
Finally, I survey all the proposed (deterministic) algorithms for the quantum separability problem, including the bounded search for symmetric extensions (via semidefinite programming), based on the recent  ...  First, I review the one-sided tests for separability, paying particular attention to the semidefinite programming methods.  ...  Let B = {X i : i = 0, 1, . . . , M 2 N 2 − 1} be an orthonormal, Hermitian basis for H M,N , where X 01 √ MN I.  ... 
arXiv:quant-ph/0603199v7 fatcat:px4365cqurda3lvhmrub3bb4rm

Rounding sum-of-squares relaxations

Boaz Barak, Jonathan A. Kelner, David Steurer
2014 Proceedings of the 46th Annual ACM Symposium on Theory of Computing - STOC '14  
We present a general approach to rounding semidefinite programming relaxations obtained by the Sum-of-Squares method (Lasserre hierarchy).  ...  This is a natural analytical relaxation of the problem of finding the sparsest element in a subspace, and is also motivated by a connection to the Small Set Expansion problem shown by Barak et al.  ...  We thank Aram Harrow, Alex Wein, and an anonymous STOC referee for pointing out many typos and errors in a previous version of this paper.  ... 
doi:10.1145/2591796.2591886 dblp:conf/stoc/BarakKS14 fatcat:krbxr5m5mzcnlkzohy73mfqfiy

A computational study of exact subgraph based SDP bounds for Max-Cut, stable set and coloring

Elisabeth Gaar, Franz Rendl
2020 Mathematical programming  
Finally computational experiments on the Max-Cut, stable set and coloring problem show the excellent quality of the bounds obtained with this approach.  ...  We introduce a computational framework for these relaxations designed to cope with these difficulties.  ...  from the copyright holder.  ... 
doi:10.1007/s10107-020-01512-2 pmid:32863433 pmcid:PMC7441529 fatcat:ciy6gf6nybemzbhal7xbd2nlzq

Verifying the output of quantum optimizers with ground-state energy lower bounds

Flavio Baccari, Christian Gogolin, Peter Wittek, Antonio Acín
2020 Physical Review Research  
We consider the use of relaxations to the ground-state problem as a benchmark for the output of quantum optimizers.  ...  This yields certified solutions for many of the problems that are currently addressed by heuristic optimization algorithms more efficiently and for larger system sizes.  ...  With the restriction J i, j ∈ {−1, 0, 1} this even holds for toroidal graphs (grids on a torus, i.e., systems with periodic boundary conditions) [25] .  ... 
doi:10.1103/physrevresearch.2.043163 fatcat:czyyqjyfrnhkho2oj2gzofj4gq
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