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A note on the integrality gap of the configuration LP for restricted Santa Claus [article]

Klaus Jansen, Lars Rohwedder
2018 arXiv   pre-print
In the original proof, a local search was used to obtain a bound of 4 for the ratio of the fractional to the integral optimum of the configuration LP (integrality gap).  ...  In the restricted Santa Claus problem we are given resources R and players P. Every resource j∈ R has a value v_j and every player i desires a set R(i) of resources.  ...  For the restricted Santa Claus problem there exists a strong LP relaxation, the configuration LP. The proof that this has a small integrality gap (see [2] ) is not trivial.  ... 
arXiv:1807.03626v1 fatcat:ymbla2q3tzdvlgkqscs7lmqhou

A polynomial time 12-approximation algorithm for restricted Santa Claus problem [article]

S Anil Kumar, N S Narayanaswamy
2020 arXiv   pre-print
Our algorithm starts by solving the configuration LP and uses the optimum value to get a 12-gap instance. This is then followed by the well-known clustering technique of Bansal and Sviridenko.  ...  In this paper, we consider the restricted case of the problem and improve the current best approximation ratio by presenting a polynomial time 12-approximation algorithm using linear programming and semi-definite  ...  This work is motivated by the existence of large gap between the best theoretical integrality gap of 3 for the restricted Santa Claus problem as evident from result due to Asadpour et al.  ... 
arXiv:2007.09849v2 fatcat:2iei7dt6zbc7xfs5rufzlfiwwe

Santa claus meets hypergraph matchings

Arash Asadpour, Uriel Feige, Amin Saberi
2012 ACM Transactions on Algorithms  
As a result, we prove that the integrality gap of the configuration LP is no worse than 1 4 .  ...  It was previously shown via a nonconstructive proof that uses the Lovász local lemma that the integrality gap of a certain configuration LP for the problem is no worse than some (unspecified) constant.  ...  Acknowledgements Part of this work was performed at Microsoft Research, Redmond, Washington.  ... 
doi:10.1145/2229163.2229168 fatcat:juenfplt6nahljvxesvpsairja

Improved Integrality Gap in Max-Min Allocation: or Topology at the North Pole [article]

Penny Haxell, Tibor Szabó
2022 arXiv   pre-print
We study the restricted variant, also known as the Santa Claus problem, where each resource has an intrinsic positive value, and each player covets a subset of the resources.  ...  The principal approach for obtaining approximation algorithms has been via the Configuration LP (CLP) of Bansal and Sviridenko.  ...  Acknowledgements: The authors are grateful to Lothar Narins for helpful discussions in the early stages of this work, and to Olaf Parczyk for computing the data in Section 7.  ... 
arXiv:2202.01143v1 fatcat:6q3h5ywhwvf3xjrsx5o3lhmeaq

Quasi-polynomial Local Search for Restricted Max-Min Fair Allocation [chapter]

Lukas Polacek, Ola Svensson
2012 Lecture Notes in Computer Science  
The restricted max-min fair allocation problem (also known as the restricted Santa Claus problem) is one of few problems that enjoys the intriguing status of having a better estimation algorithm than approximation  ...  Indeed, Asadpour et al. [1] proved that a certain configuration LP can be used to estimate the optimal value within a factor 1/(4 + ), for any > 0, but at the same time it is not known how to efficiently  ...  This case is known as the restricted max-min fair allocation or the restricted Santa Claus problem and is the focus of our paper.  ... 
doi:10.1007/978-3-642-31594-7_61 fatcat:rgeaa3asmje5zg4a6o2xorqqgy

Santa Claus Schedules Jobs on Unrelated Machines [article]

Ola Svensson
2011 arXiv   pre-print
The result is obtained by upper bounding the integrality gap of a certain strong linear program, known as configuration LP, that was previously successfully used for the related Santa Claus problem.  ...  One of the classic results in scheduling theory is the 2-approximation algorithm by Lenstra, Shmoys, and Tardos for the problem of scheduling jobs to minimize makespan on unrelated machines, i.e., job  ...  Acknowledgements I am grateful to Johan Håstad for many useful insights and comments. I also wish to thank Tobias Mömke and Lukáš Poláček for useful comments on the exposition.  ... 
arXiv:1011.1168v2 fatcat:gugzksvphfhiji37ao7rxfek6m

Quasi-Polynomial Local Search for Restricted Max-Min Fair Allocation [article]

Lukas Polacek, Ola Svensson
2014 arXiv   pre-print
The restricted max-min fair allocation problem (also known as the restricted Santa Claus problem) is one of few problems that enjoys the intriguing status of having a better estimation algorithm than approximation  ...  Indeed, Asadpour et al. proved that a certain configuration LP can be used to estimate the optimal value within a factor 1/(4+ϵ), for any ϵ>0, but at the same time it is not known how to efficiently find  ...  This case is known as the restricted max-min fair allocation or the restricted Santa Claus problem and is the focus of our paper.  ... 
arXiv:1205.1373v2 fatcat:ske4ykzfrnhonkjaqpo45ycyzq

Quasi-Polynomial Local Search for Restricted Max-Min Fair Allocation

Lukáš Poláček, Ola Svensson
2015 ACM Transactions on Algorithms  
The restricted max-min fair allocation problem (also known as the restricted Santa Claus problem) is one of few problems that enjoys the intriguing status of having a better estimation algorithm than approximation  ...  Indeed, Asadpour et al. [1] proved that a certain configuration LP can be used to estimate the optimal value within a factor 1/(4 + ), for any > 0, but at the same time it is not known how to efficiently  ...  This case is known as the restricted max-min fair allocation or the restricted Santa Claus problem and is the focus of our paper.  ... 
doi:10.1145/2818695 fatcat:5tnd4rqunfhkdn7fci6e2dfmyy

Santa Claus schedules jobs on unrelated machines

Ola Svensson
2011 Proceedings of the 43rd annual ACM symposium on Theory of computing - STOC '11  
The result is obtained by upper bounding the integrality gap of a certain strong linear program, known as configuration LP, that was previously successfully used for the related Santa Claus problem.  ...  One of the classic results in scheduling theory is the 2-approximation algorithm by Lenstra, Shmoys, and Tardos for the problem of scheduling jobs to minimize makespan on unrelated machines, i.e., job  ...  Acknowledgements I am grateful to Johan Håstad for many useful insights and comments. I also wish to thank Tobias Mömke and Lukáš Poláček for useful comments on the exposition.  ... 
doi:10.1145/1993636.1993718 dblp:conf/stoc/Svensson11 fatcat:mqmnmz55gnd5tcezipv2vv6bde

Santa Claus Schedules Jobs on Unrelated Machines

Ola Svensson
2012 SIAM journal on computing (Print)  
The result is obtained by upper bounding the integrality gap of a certain strong linear program, known as configuration LP, that was previously successfully used for the related Santa Claus problem.  ...  One of the classic results in scheduling theory is the 2-approximation algorithm by Lenstra, Shmoys, and Tardos for the problem of scheduling jobs to minimize makespan on unrelated machines, i.e., job  ...  Acknowledgements I am grateful to Johan Håstad for many useful insights and comments. I also wish to thank Tobias Mömke and Lukáš Poláček for useful comments on the exposition.  ... 
doi:10.1137/110851201 fatcat:52unyfqr6fcbbdb3w3mybau6ty

A Tale of Santa Claus, Hypergraphs and Matroids [article]

Sami Davies, Thomas Rothvoss, Yihao Zhang
2019 arXiv   pre-print
Our result can then be used as a blackbox to obtain a (4+ε)-approximation for Santa Claus. This factor also compares against a natural, compact LP for Santa Claus.  ...  In this paper, we introduce a matroid version of the Santa Claus problem.  ...  Interestingly, the integrality gap of the configuration LP is at least Ω( n) [BS06] .  ... 
arXiv:1807.07189v2 fatcat:i4nepjnh4ra6lb56av2zoc5aky

The Combinatorial Santa Claus Problem or: How to Find Good Matchings in Non-Uniform Hypergraphs [article]

Etienne Bamas, Paritosh Garg, Lars Rohwedder
2020 arXiv   pre-print
The non-trivial observation that any uniform regular hypergraph admits a relaxed matching for α = O(1) was a major step in obtaining a constant approximation rate for a special case of the Santa Claus  ...  We dub this problem the Combinatorial Santa Claus problem, since we show in this paper that this problem and the Santa Claus problem are almost equivalent in terms of their approximability.  ...  Namely, it implies an O(α) integrality gap for an LP relaxation of the restricted Santa Claus problem, which is the special case where for all valuations v ij ∈ {0, v j } (v j being a player independent  ... 
arXiv:2007.09116v2 fatcat:fz45a4smiza5fbqfh3jg3c2eh4

The Submodular Santa Claus Problem in the Restricted Assignment Case [article]

Etienne Bamas, Paritosh Garg, Lars Rohwedder
2020 arXiv   pre-print
The submodular Santa Claus problem was introduced in a seminal work by Goemans, Harvey, Iwata, and Mirrokni (SODA'09) as an application of their structural result.  ...  In particular, a line of research has shown that there is a polynomial time constant approximation algorithm for linear valuation functions in the restricted assignment case.  ...  A note on the integrality gap of the configuration lp for restricted santa claus. Information Processing Letters, 164:106025, 2020. [LLN06] Benny Lehmann, Daniel Lehmann, and Noam Nisan.  ... 
arXiv:2011.06939v1 fatcat:qh4ysuy2ufcg3cwvqowiy6mkwa

New Constructive Aspects of the Lovasz Local Lemma

Bernhard Haeupler, Barna Saha, Aravind Srin
2010 2010 IEEE 51st Annual Symposium on Foundations of Computer Science  
We demonstrate this idea on several applications. We give the first constant-factor approximation algorithm for the Santa Claus problem by making an LLL-based proof of Feige constructive.  ...  We show how a known bound on the probabilities of events in this distribution can be used for further probabilistic analysis and give new constructive and non-constructive results.  ...  The first author thanks Michel Goemans and Ankur Moitra for discussing the Santa Claus problem with him in an early stage of this work.  ... 
doi:10.1109/focs.2010.45 dblp:conf/focs/HaeuplerSS10 fatcat:uaon4s3s5fdhjj5rhdddzlvnum

New Constructive Aspects of the Lovász Local Lemma

Bernhard Haeupler, Barna Saha, Aravind Srinivasan
2011 Journal of the ACM  
We demonstrate this idea on several applications. We give the first constant-factor approximation algorithm for the Santa Claus problem by making an LLL-based proof of Feige constructive.  ...  We show how a known bound on the probabilities of events in this distribution can be used for further probabilistic analysis and give new constructive and non-constructive results.  ...  The first author thanks Michel Goemans and Ankur Moitra for discussing the Santa Claus problem with him in an early stage of this work.  ... 
doi:10.1145/2049697.2049702 fatcat:yshjonds4bgkpcwrhhss3tdmhi
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