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Breaking symmetries to rescue Sum of Squares in the case of makespan scheduling [article]

Victor Verdugo and José Verschae and Andreas Wiese
2020 arXiv   pre-print
In this paper, we study this for the case of the minimum makespan problem on identical machines.  ...  Then, we consider the weaker assignment linear program and add a well chosen set of symmetry breaking inequalities that removes a subset of the machine permutation symmetries.  ...  to yield a (1 + ε) integrality gap (and hence also the weaker assignment linear program below).  ... 
arXiv:1811.08539v3 fatcat:gctu2ebycjfutgo6es55vrde3q

Convex programming for scheduling unrelated parallel machines

Yossi Azar, Amir Epstein
2005 Proceedings of the thirty-seventh annual ACM symposium on Theory of computing - STOC '05  
To the best of our knowledge, this is the first time that general convex programming techniques (apart from SDPs and CQPs) are used in the area of scheduling.  ...  We consider the classical problem of scheduling parallel unrelated machines. Each job is to be processed by exactly one machine. Processing job j on machine i requires time pij .  ...  There are several applications of linear programming relaxations for machine scheduling problems, see, e.g., [14, 16, 10, 4, 5] .  ... 
doi:10.1145/1060590.1060639 dblp:conf/stoc/AzarE05 fatcat:jksogjheznfvfgghnpvplmsviq

Convex quadratic and semidefinite programming relaxations in scheduling

Martin Skutella
2001 Journal of the ACM  
Finally, for the special case of two machines we introduce a more sophisticated semidefinite programming relaxation and apply the random hyperplane technique introduced by Goemans and Williamson for the  ...  We also consider preemptive scheduling problems and derive approximation algorithms and results on the power of preemption which improve upon the best previously known results for these settings.  ...  The author is grateful to Michel Goemans, Andreas Schulz, and an anonymous referee for helpful comments.  ... 
doi:10.1145/375827.375840 fatcat:5dbcodih3vdkdddei236hzi2l4

Semidefinite Programming in Timetabling and Mutual-Exclusion Scheduling [article]

Jakub Marecek, Andrew J. Parkes
2019 arXiv   pre-print
This corresponds to the vertex colouring problem in graph theory, for which there are well-known semidefinite programming (SDP) relaxations.  ...  We present SDP relaxations for a variety of mutual-exclusion scheduling and timetabling problems, starting from a bound on the number of tasks executed within each period, which corresponds to graph colouring  ...  Notation and Related Work Semidefinite Programming Firstly, let us revisit the definition of semidefinite programming, which is a popular extension of linear programming.  ... 
arXiv:1904.03539v1 fatcat:csewztv5draqhp7zdx5foqtsbq

A strong conic quadratic reformulation for machine-job assignment with controllable processing times

M. Seli̇m Aktürk, Alper Atamtürk, Si̇nan Gürel
2009 Operations Research Letters  
We describe a polynomial-size conic quadratic reformulation for a machine-job assignment problem with separable convex cost.  ...  Gürel was a visiting scholar at the University of California-Berkeley with support from the Scientific and Technological Research Council of Turkey under Grant 2214. A.  ...  In Table 1 we also observe that with tighter machining capacity, the integrality gap is higher for all problems sizes.  ... 
doi:10.1016/j.orl.2008.12.009 fatcat:jyoj5nypyffnrkrahsunx4ym64

Sum-of-squares hierarchy lower bounds for symmetric formulations [article]

Adam Kurpisz, Samuli Leppänen, Monaldo Mastrolilli
2016 arXiv   pre-print
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.  ...  We illustrate the technique on two problems, one unconstrained and the other with constraints.  ...  The authors would like to express their gratitude to Ola Svensson for helpful discussions and ideas regarding this paper.  ... 
arXiv:1407.1746v2 fatcat:wxu7jef6gvc2nk3iq4kh3pukhq

Human-machine interaction for real-time linear optimization

Simon Hamel, Jonathan Gaudreault, Claude-Guy Quimper, Mathieu Bouchard, Philippe Marier
2012 2012 IEEE International Conference on Systems, Man, and Cybernetics (SMC)  
Mixed-Initiative-Systems (MIS) are hybrid decisionmaking systems in which human and machine collaborate in order to produce a solution.  ...  However, this delay is unacceptable for iterative and interactive decision-making contexts where users need to provide their input.  ...  The two first constraints of this linear program force the variable g to be at least as large as j j x x ′ − . The unknown of this problem are the vector α and the gap variable g.  ... 
doi:10.1109/icsmc.2012.6377804 dblp:conf/smc/HamelGQBM12 fatcat:arzkzixr6zcpbf4cnkytgiluwi

Approximability of average completion time scheduling on unrelated machines

René Sitters
2016 Mathematical programming  
This provides one of the last missing pieces in the complexity classification of machine scheduling with (weighted) sum of completion times objective.  ...  The proof is based on a mixed integer linear program. This means that verification of the reduction is partly done by an ILPsolver. This gives a concise proof which is easy to verify.  ...  reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.  ... 
doi:10.1007/s10107-016-1004-8 fatcat:6u5cw6zic5awlfaaomymfwwiwy

Lift-and-Round to Improve Weighted Completion Time on Unrelated Machines [article]

Nikhil Bansal, Aravind Srinivasan, Ola Svensson
2015 arXiv   pre-print
To do this, we first introduce a new lift-and-project based SDP relaxation for the problem. This is necessary as the previous convex programming relaxations have an integrality gap of 3/2.  ...  We consider the problem of scheduling jobs on unrelated machines so as to minimize the sum of weighted completion times.  ...  We thank Nick Harvey and Bruce Shepherd for organizing the Bellairs Workshop on Combinatorial Optimization 2015, which was the starting point for this work.  ... 
arXiv:1511.07826v2 fatcat:2qsw2asj7zdxjpuh2a6oiw6o54

Lift-and-round to improve weighted completion time on unrelated machines

Nikhil Bansal, Aravind Srinivasan, Ola Svensson
2016 Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing - STOC 2016  
To do this, we first introduce a new lift-andproject based SDP relaxation for the problem. This is necessary as the previous convex programming relaxations have an integrality gap of 3/2.  ...  We consider the problem of scheduling jobs on unrelated machines so as to minimize the sum of weighted completion times.  ...  We thank Nick Harvey and Bruce Shepherd for organizing the Bellairs Workshop on Combinatorial Optimization 2015, which was the starting point for this work.  ... 
doi:10.1145/2897518.2897572 dblp:conf/stoc/BansalSS16 fatcat:anurs75s6ra2ve5bmr3mzyt6sm

Unrelated Machine Scheduling of Jobs with Uniform Smith Ratios

Christos Kalaitzis, Ola Svensson, Jakub Tarnawski
2017 Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms  
In this work, we take a different approach and propose to use the same elegant rounding scheme for the weighted completion time objective as devised by Shmoys and Tardos for optimizing a linear function  ...  We consider the classic problem of scheduling jobs on unrelated machines so as to minimize the weighted sum of completion times.  ...  The integrality gap of the Configuration-LP for R|| p j C j is at least 1.08.  ... 
doi:10.1137/1.9781611974782.175 dblp:conf/soda/KalaitzisST17 fatcat:234eas2xvjantfqay6hsopna3e

A branch-and-cut algorithm based on semidefinite programming for the minimum k-partition problem

Bissan Ghaddar, Miguel F. Anjos, Frauke Liers
2008 Annals of Operations Research  
The main contribution of this paper is the design and implementation of a branch-and-cut algorithm based on semidefinite programming (SBC) for the MkP problem.  ...  The two key ingredients for this algorithm are: the combination of semidefinite programming (SDP) with polyhedral results; and the iterative clustering heuristic (ICH) that finds feasible solutions for  ...  Usually, (linear programming) relaxations are used and strengthened during the run of the algorithm.  ... 
doi:10.1007/s10479-008-0481-4 fatcat:vxq6dt22tfc4xikzxxqwhscxd4

Progress in Linear Programming-Based Algorithms for Integer Programming: An Exposition

Ellis L. Johnson, George L. Nemhauser, Martin W.P. Savelsbergh
2000 INFORMS journal on computing  
We show how these techniques are useful in important application areas such as network design and crew scheduling.  ...  Finally, we discuss the relatively new research areas of parallel integer programming and stochastic integer programming.  ...  Consider the GAP in which all machines are identical.  ... 
doi:10.1287/ijoc. fatcat:zticaywmezc4rl7uji44l7u6tu

Fifty years of scheduling: a survey of milestones

C N Potts, V A Strusevich
2009 Journal of the Operational Research Society  
A discussion of the main topics of scheduling research for the past five decades is provided, highlighting the key contributions that helped shape the subject.  ...  The main topics covered in the respective decades are combinatorial analysis, branch and bound, computational complexity and classification, approximate solution algorithms, and enhanced scheduling models  ...  challenges remain for new and established researchers in this exciting and rewarding area.  ... 
doi:10.1057/jors.2009.2 fatcat:trzjs3iaqbdara3waf7u2fagam

Guaranteed Accuracy for Conic Programming Problems in Vector Lattices [article]

Christian Jansson
2007 arXiv   pre-print
Numerical results, applications and software for linear and semidefinite programming problems are described.  ...  In the case of linear programming, second order cone programming, and semidefinite programming specialized formulas are deduced yielding guaranteed accuracy.  ...  A detailed description of this algorithm can be found in the case of linear programming in [14] , for smooth convex programming problems in [13] , and for semidefinite programming problems and linear  ... 
arXiv:0707.4366v1 fatcat:wdcd5f2bvne47khgvntofwwvi4
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