On the Configuration-LP of the Restricted Assignment Problem
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This document summarizes a research paper on the configuration linear programming (LP) approach for approximating the restricted assignment problem. The paper presents the configuration LP formulation, which has an exponential number of variables. It proves that the configuration LP has an integrality gap of at most 2 - 1/6, allowing one to estimate the restricted assignment problem within a factor of less than 2 in polynomial time. The paper also describes an iterative algorithm that maintains feasibility while inserting jobs, and analyzes its running time and ability to always find a valid move. Future work directions are identified, such as improving the approximation bound.
On the Configuration-LP of the Restricted Assignment Problem
- 1. On the Configuration-LP of the Restricted Assignment Problem Ali Asghar Gorzin Aras Pourdamghani Klaus Jansen, Lars Rohwedder (SODA 2017) University of Kiel, Germany
- 2. Outline • Definition • Related works • Configuration LP • Algorithm • Analysis • Conclusion & Future works
- 3. Definition
- 4. Assignment Problem How to distribute jobs among a set of machines to minimize the makespan The highest load among all machines 𝑚𝑎𝑥𝑖∈𝑀 𝑗∈𝜎−1(𝑖) 𝑝𝑖𝑗
- 5. Assignment Problem 3 4 1 2 3 21 {5,8,5} {3,2,4} {8,4,8} {3,6,1} M J
- 6. Assignment Problem 3 4 1 2 3 21 {5,8,5} {3,2,4} {8,4,8} {3,6,1} M J
- 7. Assignment Problem 1 3 2 4makespan 3 4 1 2 3 21 {5,8,5} {3,2,4} {8,4,8} {3,6,1} M J
- 8. Restricted Assignment Problem Each job is allowed only on a subset of machines, but the processing time on any of them is identical. 𝑝𝑖𝑗 ∈ {𝑝𝑖, ∞}
- 9. Restricted Assignment Problem 3 4 1 2 3 21 {∞,8,∞} {∞,4,4} {4,4,∞} {3,∞,3} M J
- 10. Restricted Assignment Problem 3 4 1 2 3 21 {∞,4,4} {4,4,∞} {3,∞,3} M J {∞,8,∞}
- 11. Restricted Assignment Problem 3 1 3 makespan 3 4 1 2 3 21 {∞,4,4}{4,4,∞}{3,∞,3} M J {∞,8,∞} 2
- 12. Approximation vs. Estimation 1. A c-estimation algorithm computes a value 𝑬 with 𝑂𝑃𝑇 < 𝐸 < 𝑐. 𝑂𝑃𝑇 2. A c-approximation algorithm provides a solution too Feige, Jozpeh (2015) 1. There are NP optimization problems for which estimation is easier than approximation • Unless P = NP or TFNP = FP
- 13. Related works
- 14. Identical Machines Graham (1966) Gave Factor 2 approximation Algorithm Graham(1969) Gave Factor 4/3 approximation Algorithm Hochbaum, Shmoys(1986) Gave a PTAS
- 15. Unrelated Machines Lenstra, Shmoys, Tardos (1990) 1. gave an approximation guarantee of 2 2. there is no approximation better than 3/2
- 16. Restricted Assignment Problem Svensson (2012) Gave an estimation guarantee of 2 − 1 7 Jansen, Land, Maack (2016) Gave an instance with integrality gap 3 2 Jansen, Rohwedder (2017) Polynomial & Qusi-Polynomial bound on number of operation
- 17. Applications Svensson (2012) Gave an estimation guarantee of 2 − 1 7 Jansen, Land, Maack (2016) Gave an instance with integrality gap 3 2 Jansen, Rohwedder (2017) Polynomial & Qusi-Polynomial bound on number of operation
- 18. Configuration LP
- 19. Configuration LP A configuration for a machine is a set of jobs that would not exceed the target makespan T.
- 20. Configuration LP Primal of the Configuration-LP The linear programming system is as follows:
- 21. Configuration LP Primal of the Configuration-LP A mixture of configurations is assigned to a machine, where 𝑥𝑖𝑐shows which fraction of the configuration is assigned.
- 22. Configuration LP Primal of the Configuration-LP But the Configuration-LP has an exponential number of variables!
- 23. Configuration LP Dual of the Configuration-LP But the dual of Configuration-LP has an exponential number of constraints!
- 24. Configuration LP Dual of the Configuration-LP Could be solved using ellipsoid method to any desired accuracy.
- 25. Configuration LP Theorem 1.1. The configuration-LP for the Restricted Assignment problem has an integrality gap of at most 2 - 1/6. ✤ One can estimate Restricted Assignment problem by a factor less than 2 in polynomial time.
- 26. Configuration LP Theorem 1.1. The configuration-LP for the Restricted Assignment problem has an integrality gap of at most 2 - 1/6.
- 27. Algorithm
- 28. Iterative approach Inserting 𝑗 𝑛𝑒𝑤 at each step, maintaining the makespan condition meanwhile.
- 29. Algorithm • While 𝑗 𝑛𝑒𝑤 has not been assigned • If possible • performs a valid move in the blocker tree • else • adds a new blocker
- 30. Potential move conditions • (𝑗, 𝑖) is not already in 𝛕 • The size of 𝑗 corresponds to the type • 𝑗 is not undesirable on 𝑖 • The load on 𝑖 meets certain conditions depending on the type
- 31. Partioning Small and Big jobs: Jobs are divided into two parts. A job is small if 𝑝𝑖 ≤ 1 2 and big otherwise.
- 32. Blocker A blocker is a move shown by (𝑖, 𝑗, 𝜃) which shows moving job 𝑗 to machine 𝑗 with type 𝜃. 𝜃 ∈ {𝑆𝐴, 𝐵𝐴, 𝐵𝐵, 𝐵𝐿}
- 33. Potential move conditions 𝑆𝐴 𝑣𝑎𝑙𝑢𝑒 ∶ (1, 𝜎−1 𝑖 )
- 35. Potential move conditions 𝐵𝐴 𝐼𝑓: 𝑝 𝑆𝑖 𝜏 ∪ 𝐵𝑖 + 𝑝𝑗 ≤ 1 + 𝑅 𝑣𝑎𝑙𝑢𝑒 ∶ (2, 𝜎−1 𝑖 )
- 37. Potential move conditions 𝐵𝐿 𝑣𝑎𝑙𝑢𝑒 ∶ (3, − min 𝐵𝑖) 1. 𝑝 𝑆𝑖 𝜏 ∪ 𝐵𝑖 𝑚𝑖𝑛 + 𝑝𝑗 > 1 + 𝑅 2. 𝑝 𝑆𝑖 𝜏 + 𝑝𝑗 ≤ 1 + 𝑅 𝑖𝑓:
- 39. Potential move conditions 𝐵𝐵 1. 𝑝 𝑆𝑖 𝜏 ∪ 𝐵𝑖 + 𝑝𝑗 > 1 + 𝑅 2. 𝑝 𝑆𝑖 𝜏 ∪ 𝐵𝑖 𝑚𝑖𝑛 + 𝑝𝑗 ≤ 1 + 𝑅 𝑣𝑎𝑙𝑢𝑒 ∶ (4, 𝐵𝑖) 𝑖𝑓:
- 41. Valid move 𝑗, 𝑖 Be a valid move 𝐵 𝑘 Be the blocker that activate 𝑗 Algorithm assigns 𝑗 to 𝑖 and discards 𝐵 𝑘, 𝐵 𝑘+1, …
- 42. Valid move 𝑗, 𝑖 Be a valid move 𝐵 𝑘 Be the blocker that activate 𝑗 Algorithm assigns 𝑗 to 𝑖 and discards 𝐵 𝑘, 𝐵 𝑘+1, …
- 43. Valid move 𝑗, 𝑖 Be a valid move 𝐵 𝑘 Be the blocker that activate 𝑗 Algorithm assigns 𝑗 to 𝑖 and discards 𝐵 𝑘, 𝐵 𝑘+1, …
- 44. Example 𝑆𝐴 𝑆𝐴 𝐵𝐴 𝐵𝐿 𝑀1 𝑀2 𝑀3 𝑀1
- 45. Example 𝐵𝐴 𝐵𝐿 𝑀1 𝑀2 𝑀3 𝑀1
- 49. Analysis
- 50. Two Main Conditions 1. At each iteration the blocker tree either contains a valid move or there is a potential move that can be added to the blocker tree 2. The algorithm terminates after a bounded number of steps
- 51. Two Main Conditions 1. Algorithm does not get stock. 2. The algorithm terminates.
- 52. Theorem 1 • Def: At the start of each iteration of the loop, there is a valid move in the blocker tree or a potential move of a job in 𝐴 𝜏 . • Prove by contradiction: Showing dual LP is unbounded. Therefore, the primal LP is infeasible.
- 53. Proof (Thm 1) For all jobs 𝑗 and machines 𝑖 let: It is well defined because 𝑀 𝜏 𝑆𝐴 ∩ 𝑀 𝜏 𝐵𝐴 = 𝜙
- 54. Claim 1 𝑗∈𝐽 𝑧𝑗 ∗ > 𝑧𝑗 𝑛𝑒𝑤 + 𝑖∈𝑀 𝑧∗ 𝜎−1 𝑖 > 0 + 𝑖∈𝑀 𝑦∗ + 1 6 𝑀 𝜏 𝑆𝐴 − 1 6 𝑀 𝜏 𝐵𝐴 ≥ 𝑖∈𝑀 𝑦∗ 𝑗∈𝐽 𝑧𝑗 ∗ > 𝑖∈𝑀 𝑦∗ The objective value of dual is negative:
- 55. Theorem 2 • Def: After finite steps Algorithm will terminate. • #𝑆𝑡𝑒𝑝𝑠 𝑏𝑜𝑢𝑛𝑑𝑒𝑑 𝑏𝑦 𝐽 , 𝑀 𝑎𝑛𝑑 𝑑𝑢𝑟𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝐽𝑜𝑏𝑠 • Prove by Invariance: (𝑣𝑎𝑙(𝐵!), 𝑣𝑎𝑙(𝐵2), 𝑣𝑎𝑙(𝐵3), … . 𝑣𝑎𝑙(𝐵𝑙),∞) will decreasing lexicographically
- 56. Future works
- 57. Open Directions 1. Improving the approximation bound in either Restricted or General version of the Assignment Problem. Since efficient variants of respective local search algorithm were discovered for Restricted Max-Min Fair Allocation.
- 58. Open Directions 2. The estimation bound found in this paper is not tight. Proposed approach: A restricted situation where for every job either or
- 59. Thank you!
Editor's Notes
- #5: Each Job to exactly one