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.

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 )

34. Potential move conditions
𝑆𝐴

35. Potential move conditions
𝐵𝐴
𝐼𝑓:
𝑝 𝑆𝑖 𝜏 ∪ 𝐵𝑖 + 𝑝𝑗 ≤ 1 + 𝑅
𝑣𝑎𝑙𝑢𝑒 ∶ (2, 𝜎−1 𝑖 )

36. Potential move conditions
𝐵𝐴

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:

38. Potential move conditions
𝐵𝐴

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:

40. Potential move conditions
𝐵𝐵

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

46. Example
𝐵𝐿
𝑀1 𝑀2 𝑀3

47. Example
𝐵𝐿
𝑀1 𝑀2 𝑀3

48. Example
𝐵𝐿
𝑀1 𝑀2 𝑀3

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!