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

We consider the problem of scheduling jobs on unrelated machines so as to minimize the sum of weighted completion times. Our main result is a (3/2-c)-approximation algorithm for some fixed c>0, improving upon the long-standing bound of 3/2. 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. Second, we give a new general bipartite-rounding procedure that produces an assignment with certain strong negative correlation properties.

[1]  Gerhard J. Woeginger,et al.  A PTAS for minimizing the weighted sum of job completion times on parallel machines , 1999, STOC '99.

[2]  Maxim Sviridenko,et al.  Approximating the Configuration-LP for Minimizing Weighted Sum of Completion Times on Unrelated Machines , 2013, IPCO.

[3]  David B. Shmoys,et al.  Scheduling to Minimize Average Completion Time: Off-Line and On-Line Approximation Algorithms , 1997, Math. Oper. Res..

[4]  Nikhil Bansal,et al.  The Santa Claus problem , 2006, STOC '06.

[5]  Amin Saberi,et al.  An approximation algorithm for max-min fair allocation of indivisible goods , 2007, STOC '07.

[6]  Wayne E. Smith.  Various optimizers for single‐stage production , 1956 .

[7]  Gerhard J. Woeginger,et al.  Polynomial time approximation algorithms for machine scheduling: ten open problems , 1999 .

[8]  Maxim Sviridenko,et al.  Approximation Algorithms for Maximum Coverage and Max Cut with Given Sizes of Parts , 1999, IPCO.

[9]  Sanjeev Khanna,et al.  Approximation Algorithms for Minimizing AverageWeighted Completion Time , 2004, Handbook of Scheduling.

[10]  Jan Karel Lenstra,et al.  Approximation algorithms for scheduling unrelated parallel machines , 1987, 28th Annual Symposium on Foundations of Computer Science (sfcs 1987).

[11]  Ola Svensson,et al.  Santa Claus schedules jobs on unrelated machines , 2010, STOC '11.

[12]  David B. Shmoys,et al.  Scheduling to minimize average completion time: off-line and on-line algorithms , 1996, SODA '96.

[13]  Rajiv Gandhi,et al.  Dependent rounding and its applications to approximation algorithms , 2006, JACM.

[14]  Aravind Srinivasan,et al.  A unified approach to scheduling on unrelated parallel machines , 2009, JACM.

[15]  Jan Vondrák,et al.  Multi-budgeted matchings and matroid intersection via dependent rounding , 2011, SODA '11.

[16]  Martin Skutella,et al.  Scheduling Unrelated Machines by Randomized Rounding , 2002, SIAM J. Discret. Math..

[17]  Alan M. Frieze,et al.  A new rounding procedure for the assignment problem with applications to dense graph arrangement problems , 2002, Math. Program..

[18]  Uriel Feige,et al.  Santa claus meets hypergraph matchings , 2008, TALG.

[19]  Sanjeev Khanna,et al.  A PTAS for Minimizing Weighted Completion Time on Uniformly Related Machines , 2001, ICALP.

[20]  Cynthia A. Phillips,et al.  Task Scheduling in Networks , 1997, SIAM J. Discret. Math..

[21]  Evripidis Bampis,et al.  Approximation schemes for minimizing average weighted completion time with release dates , 1999, 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039).

[22]  Rajeev Motwani,et al.  Approximation techniques for average completion time scheduling , 1997, SODA '97.

[23]  Martin Skutella,et al.  Convex quadratic and semidefinite programming relaxations in scheduling , 2001, JACM.

[24]  Sanjeev Khanna,et al.  On Allocating Goods to Maximize Fairness , 2009, 2009 50th Annual IEEE Symposium on Foundations of Computer Science.

[25]  Fabián A. Chudak.  A min‐sum 3/2‐approximation algorithm for scheduling unrelated parallel machines , 1999 .

[26]  Yossi Azar,et al.  Convex programming for scheduling unrelated parallel machines , 2005, STOC '05.

[27]  Aravind Srinivasan,et al.  Minimum Weighted Completion Time , 2008, Encyclopedia of Algorithms.

[28]  Amin Saberi,et al.  An Approximation Algorithm for Max-Min Fair Allocation of Indivisible Goods , 2010, SIAM J. Comput..

[29]  Maxim Sviridenko,et al.  Solving Optimization Problems with Diseconomies of Scale via Decoupling , 2014, 2014 IEEE 55th Annual Symposium on Foundations of Computer Science.

[30]  Jeff Kahn,et al.  On the stochastic independence properties of hard-core distributions , 1997, Comb..

[31]  Jan Karel Lenstra,et al.  Approximation algorithms for scheduling unrelated parallel machines , 1987, 28th Annual Symposium on Foundations of Computer Science (sfcs 1987).

[32]  Uriel Feige,et al.  On allocations that maximize fairness , 2008, SODA '08.

[33]  Han Hoogeveen,et al.  Non-Approximability Results for Scheduling Problems with Minsum Criteria , 1998, INFORMS J. Comput..

[34]  Mark S. Squillante,et al.  Optimal scheduling of multiclass parallel machines , 1999, SODA '99.

[35]  Martin Skutella,et al.  Single Machine Scheduling with Release Dates , 2002, SIAM J. Discret. Math..

[36]  Cynthia A. Phillips,et al.  Minimizing average completion time in the presence of release dates , 1998, Math. Program..