1. A Unified Approach to Approximating Resource Allocation and Scheduling
Amotz Bar-Noy.……...AT&T and Tel Aviv University
Reuven Bar-Yehuda….Technion IIT
Ari Freund……………Technion IIT
Seffi Naor…………….Bell Labs and Technion IIT
Baruch Schieber…...…IBM T.J. Watson
Slides and paper at:

2. Summery of Results: Discrete
Single Machine Scheduling
Bar-Noy, Guha, Naor and Schieber STOC 99: 1/2 Non Combinatorial*
  Berman, DasGupta, STOC 00: /2
  This Talk, STOC 00(Independent)      1/2
Bandwidth Allocation Albers, Arora, Khanna SODA 99: O(1) for |Activityi|=1* Uma, Phillips, Wein SODA 00:        1/4 Non combinatorial. This Talk STOC 00 (Independent)       1/3 for w  1/ This Talk STOC 00 (Independent)       1/5 for w  1   
Parallel Unrelated Machines:
Bar-Noy, Guha, Naor and Schieber STOC 99: 1/3
  Berman, DasGupta STOC 00: /2
  This Talk, STOC 00(Independent)      1/2

3. Summery of Results: Continuous
Single Machine Scheduling
Bar-Noy, Guha, Naor and Schieber STOC 99: 1/3 Non Combinatorial
  Berman, DasGupta STOC 00: /2·(1-)
  This Talk, STOC 00: (Independent) /2·(1-)
Bandwidth Allocation Uma, Phillips, Wein SODA 00:       1/6 Non combinatorial
This Talk, STOC 00 (Independent)       1/3 ·(1-) for w  1/2        /5 ·(1-) for w  1
Parallel unrelated machines:
Bar-Noy, Guha, Naor and Schieber STOC 99: 1/4

4. Summery of Results: and more…
General Off-line Caching Problem
 Albers, Arora, Khanna SODA 99: O(1) Cache_Size += O(Largest_Page)    O(log(Cache_Size+Max_Page_Penalty))
This Talk, STOC 00:
Ring topology:
Transformation of approx ratio from line to ring topology 1/  1/(+1+)
Dynamic storage allocation (contiguous allocation):
Previous results: none for throughput maximization
Previous results Kierstead 91 for resource minimization: 6
This paper: 1/35 for throughput max using the result for resource min.

5. The Local-Ratio Technique: Basic definitions
Given a profit [penalty] vector p.
Maximize[Minimize] p·x
Subject to: feasibility constraints F(x)
x is r-approximation if F(x) and p·x [] r · p·x*
An algorithm is r-approximation if for any p, F
it returns an r-approximation

6. The Local-Ratio Theorem:
x is an r-approximation with respect to p1
x is an r-approximation with respect to p- p1 
x is an r-approximation with respect to p
Proof: (For maximization)
p1 · x  r × p1*
p2 · x  r × p2*
p · x  r × ( p1*+ p2*)
 r × ( p1 + p2 )*

7. Special case: Optimization is 1-approximation
x is an optimum with respect to p1
x is an optimum with respect to p- p1
x is an optimum with respect to p

8. A Local-Ratio Schema for Maximization[Minimization] problems:
Algorithm r-ApproxMax[Min]( Set, p )
If Set = Φ then return Φ ;
If  I  Set p(I)  0 then return r-ApproxMax( Set-{I}, p ) ;
[If I  Set p(I)=0 then return {I}  r-ApproxMin( Set-{I}, p ) ;]
Define “good” p1 ;
REC = r-ApproxMax[Min]( S, p- p1 ) ;
If REC is not an r-approximation w.r.t. p1 then “fix it”;
return REC;

9. The Local-Ratio Theorem: Applications
Applications to some optimization algorithms (r = 1):
( MST) Minimum Spanning Tree (Kruskal)
( SHORTEST-PATH) s-t Shortest Path (Dijkstra)
(LONGEST-PATH) s-t DAG Longest Path (Can be done with dynamic programming)
(INTERVAL-IS) Independents-Set in Interval Graphs Usually done with dynamic programming)
(LONG-SEQ) Longest (weighted) monotone subsequence (Can be done with dynamic programming)
( MIN_CUT) Minimum Capacity s,t Cut (e.g. Ford, Dinitz)
Applications to some 2-Approximation algorithms: (r = 2)
( VC) Minimum Vertex Cover (Bar-Yehuda and Even)
( FVS) Vertex Feedback Set (Becker and Geiger)
( GSF) Generalized Steiner Forest (Williamson, Goemans, Mihail, and Vazirani)
( Min 2SAT) Minimum Two-Satisfibility (Gusfield and Pitt)
( 2VIP) Two Variable Integer Programming (Bar-Yehuda and Rawitz)
( PVC) Partial Vertex Cover (Bar-Yehuda)
( GVC) Generalized Vertex Cover (Bar-Yehuda and Rawitz)
Applications to some other Approximations:
( SC) Minimum Set Cover (Bar-Yehuda and Even)
( PSC) Partial Set Cover (Bar-Yehuda)
( MSP) Maximum Set Packing (Arkin and Hasin)
Applications Resource Allocation and Scheduling : ….

10. Maximum Independent Set in Interval Graphs
Activity9
Activity8
Activity7
Activity6
Activity5
Activity4
Activity3
Activity2
Activity1
time
Maximize s.t For each instance I:
For each time t:

11. Maximum Independent Set in Interval Graphs: How to select P1 to get optimization?
Activity9
Activity8
Activity7
Activity6
Activity5
Activity4
Activity3
Activity2
Activity Î
time
Let Î be an interval that ends first;
1 if I in conflict with Î
For all intervals I define: p1 (I) =
0 else
For every feasible x: p1 ·x  1
Every Î-maximal is optimal.
For every Î-maximal x: p1 ·x  1
P1=1
P1=0
P1=0
P1=0
P1=1
P1=0
P1=1
P1=1

12. Maximum Independent Set in Interval Graphs: An Optimization Algorithm
P1=P(Î )
Activity9
Activity8
Activity7
Activity6
Activity5
Activity4
Activity3
Activity2
Activity Î
time
Algorithm MaxIS( S, p )
If S = Φ then return Φ ;
If I  S p(I) 0 then return MaxIS( S - {I}, p);
Let Î  S that ends first;
I  S define: p1 (I) = p(Î)  (I in conflict with Î) ;
IS = MaxIS( S, p- p1 ) ;
If IS is Î-maximal then return IS else return IS  {Î};
P1=0
P1=0
P1=0
P1=0
P1=P(Î )
P1=0
P1=P(Î )
P1=P(Î )

13. Maximum Independent Set in Interval Graphs: Running Example
P(I5) =
P(I6) =
P(I3) =
P(I2) =
P(I1) =
P(I4) = -4 -5 -2

14. Single Machine Scheduling :
Bar-Noy, Guha, Naor and Schieber STOC 99: 1/2 LP
Berman, DasGupta, STOC 00: /2
This Talk, STOC 00(Independent)      1/2
Activity9
Activity8
Activity7
Activity6
Activity5
Activity4
Activity3
Activity2
Activity ?????????????
time
Maximize s.t For each instance I:
For each time t:
For each activity A:

15. Single Machine Scheduling: How to select P1 to get ½-approximation ?
Activity9
Activity8
Activity7
Activity6
Activity5
Activity4
Activity3
Activity2
Activity Î
time
Let Î be an interval that ends first;
1 if I in conflict with Î
For all intervals I define: p1 (I) =
0 else
For every feasible x: p1 ·x  2
Every Î-maximal is 1/2-approximation
For every Î-maximal x: p1 ·x  1
P1=1
P1=0
P1=0
P1=0
P1=0
P1=0
P1=0
P1=0
P1=1
P1=0
P1=1
P1=0
P1=1
P1=0
P1=1
P1=1
P1=1
P1=1

16. Single Machine Scheduling: The ½-approximation Algorithm
Activity9
Activity8
Activity7
Activity6
Activity5
Activity4
Activity3
Activity2
Activity Î
time
Algorithm MaxIS( S, p )
If S = Φ then return Φ ;
If I  S p(I)  0 then return MaxIS( S - {I}, p);
Let Î  S that ends first;
I  S define: p1 (I) = p(Î)  (I in conflict with Î) ;
IS = MaxIS( S, p- p1 ) ;
If IS is Î-maximal then return IS else return IS  {Î};

17. Bandwidth Allocation Albers, Arora, Khanna SODA 99: O(1) |Ai|=1*
Uma, Phillips, Wein SODA 00: /4 LP.
This Talk 1/3 for w  1/ and 1/5 for w  1  
Activity9
Activity8
Activity7
Activity6
Activity5
Activity4
Activity3
Activity2
Activity I w(I)
s(I) e(I) time
Maximize s.t For each instance I:
For each time t:
For each activity A:

18. Bandwidth Allocation time Bandwidth time Activity 9 Activity 8

19. Bandwidth Allocation for w  1/2 How to select P1 to get 1/3-approximation?
Activity9
Activity Î
Activity7
Activity6
Activity5
Activity4
Activity3
Activity2
Activity I w(I)
s(I) e(I) time
if I in the same activity of Î
For all intervals I define: p1 (I) = *w(I) if I in time conflict with Î
else
For every feasible x: p1 ·x  3
Every Î-maximal is 1/3-approximation
For every Î-maximal x: p1 ·x  1

20. Bandwidth Allocation The 1/5-approximation for any w  1
Activity9
Activity w > ½
Activity w > ½ w > ½
Activity6
Activity w > ½
Activity4
Activity w > ½ w > ½
Activity2
Activity w > ½ w > ½ w > ½
Algorithm:
GRAY = Find 1/2-approximation for gray (w>1/2) intervals;
COLORED = Find 1/3-approximation for colored intervals
Return the one with the larger profit
Analysis:
If GRAY*  40%OPT then GRAY  1/2(40%OPT)=20%OPT else
COLORED*  60%OPT thus COLORED  1/3(60%OPT)=20%OPT

21. Single Machine Scheduling with Release and Deadlines
Activity 9
Activity 8
Activity 7
Activity 6
Activity 5
Activity 4
Activity 3
Activity 2
Activity 1
time
Each job has a time window within which it can be processed.

22. Single Machine Scheduling with Release and Deadlines
Activity 9
Activity 8
Activity 7
Activity 6
Activity 5
Activity 4
Activity 3
Activity 2
Activity 1

23. Continuous Scheduling{
w(I)
 d(I) 
s(I) e(I)
Single Machine Scheduling (w=1)
Bar-Noy, Guha, Naor and Schieber STOC 99: 1/3 Non Combinatorial
  Berman, DasGupta STOC 00: /2·(1-)
  This Talk, STOC 00: (Independent) /2·(1-)
Bandwidth Allocation Uma, Phillips, Wein SODA 00:       1/6 Non combinatorial
This Talk, STOC 00 (Independent)       1/3 ·(1-) for w  1/2        /5 ·(1-) for w  1

24. Continuous Scheduling: Split and Round Profit (Loose additional (1-) factor)
If currant p(I1)   original p(I1) then delete I1
else Split I2=(s2,e2] to I21=(s2, s1+d1] and I22=(s1+d1,e2]
 d(I1) 
 d(I2) 
I11
I12
 d(I1) 
I21
I22
 d(I2) 

25. Minimization problem: General Off-line Caching Problem

26. The Demand Scheduling Problem
Resource
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
w(I)
0.0
Minimize s.t. For each instance I:
For each time t: t
Demand(t)

27. Special Case: “Min Knapsack”
Demand = 1
For all intervals I define: p1 (I) = Min {w(I), 1}
For every feasible x: p1 ·x  1
minimal is 2-approximation
For every minimal x: p1 ·x  2

28. From Knapsack to Demand Scheduling
max demand=1
at time t’
For all intervals I intersecting time t’ define: p1 (I) = Min {w(I), 1}
p1 (others) = 0
p1 (all “right-minimal”) is at most 2
p1 (all “left-minimal”) is at most 2
For every minimal x: p1 ·x 
For every feasible x: p1 ·x  1
Every minimal is 4-approximation

29. General Off-line Caching Problem
Albers, Arora, Khanna SODA 99:
O(1) Cache_Size += O(Largest_Page) O(log(Cache_Size+Max_Page_Penalty))
This Talk 4
General Off-line Caching Problem
0.9
0.8
w(page2)=0.7
0.6
w(page1)=0.5
0.4
w(page3)=0.3
0.2
0.1
0.0
page page page page page page2 page3
w(pagei) = size of pagei
p(pagei) = the reload cost of the page
Cache size

30. 4-Approximation for Demand Scheduling
Algorithm MinDemandCover( S, p )
If S = Φ then return Φ ;
If there exists an interval I  S s.t. p(I) = 0 ;
then return {I}+MinDemandCover( S - {I}, p) ;
Let t’ be the time with maximum demand k;
Let S’ be the set of instances intersects time t ;
Let δ = MIN {p(I)/w(I) : I S’} ;
MIN {w(I) ,k} if I  S’
For all intervals I  S define: p1 (I) = δ ×
0 else
C = MinDemandCover( S, p- p1 ) ;
Remove elements form C until it is minimal and return C ;

31. Application: 4-Approximation for the Loss Minimization Problem
Resource
The cost of a schedule is the sum of profits of instances not in the schedule.
For the special case where Ai is a singleton {Ii} the problem is equivalent
to the Min Demand Scheduling where:

32. END?

33. jobs time d d d d d d d d d d Parallel Unrelated Machines: Continous
Bar-Noy, Guha, Naor and Schieber STOC 99: 1/ /4
  Berman, DasGupta STOC 00: / /2·(1-)
  This Talk, STOC 00(Independent)      1/ /2·(1-) d d d d
jobs
time d d d d d d

34. Parallel unrelated machines:k A i c h c c k d
time d h i A
machine c h d d

35. i k A i k i A time machine c c c c
Parallel unrelated machines: 1/5-approximation (not in the paper) Each machine resource 1 p1(Red ) = p1(orange d ) = 1; p1 (Yellow d ) = 2width; p1 (All others) = 0; i k A i c d h c c k d
time d h i d A
machine c h d d d

36. END!

37. Preliminaries Activity9 s(I) e(I) time Activity8 Activity7 Activity6
Activity I w(I)
s(I) e(I) time