Batch Scheduling of Conflicting
Jobs
Hadas Shachnai
The Technion
Based on joint papers with L. Epstein, M. M. Halldórsson
and A. Levin.
Batch Scheduling Problems
• A batch is a set of jobs that can be processed jointly
• The completion time of a batch is the latest completion
time of a job in the batch.
• In the p-batch model, the length of a batch is the
maximum processing time of any job in the batch.
• The jobs are processed on a batching machine which
can process up to b jobs simultaneously.
• Objective functions
– Sum of completion times of jobs
– Sum of batch completion times
– Makspan
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Batch Scheduling of Conflicting Jobs
• But, what if some jobs cannot be scheduled
simultaneously?
• Real-life examples: Conflicting resource requirements,
compatibility/cooperation among jobs etc.
• Such conflicts are often modeled by an undirected
graph.
A
Lengths =
color
requirements
B
B,C need some
shared resource
E
C
D
A schedule - A multicoloring of G.
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Batch Scheduling of Conflicting Jobs
 Given is an undirected graph G=(V,E)
 Each
G vertex v V has a positive length.
 Find a proper batch coloring of the graph: each batch is
assigned a distinct contiguous set of colors of size equal to
the maximum length of any vertex in the batch.
 Each batch is an indepndent set in G
SJC(I)=2*2+1+2*5+3+7=25
-sum of job completion
times (SJC)
- sum of batch completion
times (SBC)
-Makespan (Max coloring)
Machines
 Minimize
1 2 3 4 5 6 7 time
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Batch Scheduling of Conflicting Jobs
 Given is an undirected graph G=(V,E)
 Each
G vertex v V has a positive length.
 Find a proper batch coloring of the graph: each batch is
assigned a distinct contiguous set of colors of size equal to
the maximum length of any vertex in the batch.
 Each batch is an indepndent set in G
-sum of job completion
times (SJC)
- sum of batch completion
times (SBC)
-Makespan (Max coloring)
SBC(I)=3*2+3*5+7=28
Max-col (I)=7
Machines
 Minimize
1 2 3 4 5 6 7 time
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Known Results
• For general graphs BSC and Max-coloring are hard
to approximate within factor n1-ε unless NP=ZPP (BarNoy et al, 1998; Feige and Kilian, 1998)
• Sum of job completion times
Constant factor approximations for certain
subclasses of conflict graphs (e.g., perfect, interval,
line and bipartite graphs (Epstein, Halldórsson, Levin, S,

2006).
 EPTASs for planar graphs and graphs with bounded
treewidth (Halldórsson and S., 2008)
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Known Results (Cont’d)
• Sum of batch completion times
A 4ρ-approximation for SBC for graph classes on
which Maximum Independent Set can be approximated
within factor ρ, for some ρ ≥1 (Epstein, Halldórsson, Levin,

S, 2006).
• Max coloring
 Constant factor approximation algorithms for bipartite,
planar, interval and perfect graphs (Epstein and Levin,2007;
Escoffier at al., 2006; Pemmaraju et al., 2004; Pemmaraju and Raman,
2005)
 PTASs for graphs with bounded treewidth (Escoffier at al.,
2006; Pemmaraju and Raman, 2005)
 Solvable in ploynomial time on paths (Halldórsson and S., 2008)
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Batch Coloring Problems with Minsum
Objective - a General Technique
• Minimize sum of job completion times
• Unbounded model (b ≥ n)
• Obtain approximation algorithms for SJC on several
classes of conflicts graphs
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A simple guessing game
• Player A decides on a number x.
• Player B tries a sequence x1, x2, ..., of
guesses until it finds xi that Player A says
satisfies xi ≥ x.
• The value of the game is the performance
ratio
x
i

i
x
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A simple deterministic strategy
• Guess 1, 2, 4, 8, 16, ...
• Performance ratio of 4:
– The last number is at most 2x
– The previous numbers are a geometric series, at
most x.
• This is also best possible...deterministic.
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A randomized strategy
• Defeat the worst-case instance by
– changing the base of the geometric series
– randomizing the initial guess  [0,1)
• For this game, set base to be e
– Define guess xi = ei+, i ≥0.
0

+1
+2
+3
- Last guess  e-1 times optimal guess
+4
log length
- Achieves performance ratio of e.
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Geometric Grouping in Coloring
• Each vertex has a real value attached
• Divide the real line into geom. increasing segments
V1
V2
V3
V4
V5
length
Solve
efficiently
terms separately.
of OPT:
• Each
group insolved
- Length based: immediate
Each block must be
are pasted
together in solved
orderwith
to a small
-• LPSubsolutions
based: bound clique
number of
final solution
the produce
induced subgraph
makespan
A(V1)
A(V2)
A(V3)
A(V4)
A(V5)..
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Bounds for Perfect and Line Graphs
Preprocessing the input I:
• Pick a random number  ~ U[0,1).
• Partition the jobs into classes by their processing
times: J0= { j: pj ≤ e} and Ji={ j: ei-1+  < pj ≤ ei+ }.
• Let k be the largest index of any non-empty class.
• For all i=0,1, …, k, round up the processing time of
each job j  Ji to p’j=ei+ . The resulting input is I’.
Lemma (preprocessing): Let OPT, OPT be the sum of
completion times of an optimal solution for I and I’, such
that in I’ the jobs are scheduled in batches, and all jobs in a
batch have a common class. Then E[OPT] ≤ e·OPT, where
the expectation is over the random choices of  .
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Using Non-preemptive Scheduling Scheme
Problem: Given an instance J= {1, …, n} of dependent
jobs, with the conflict graph G=({V= 1, … , n}, E),
schedule the jobs non-preemptively on a set of
(unbounded size of) machines so as to minimize the
sum of completion times of all jobs.
Linear programming formulation:
• For any edge (u,v)E there is a variable uv {0,1};
uv =1 if u precedes v in the schedule, and 0
otherwise.
• Denote by Nv the set of neighbors of v in G.
• Denote by C1, …, CNv the set of maximal cliques in Nv.
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LP formulation (Cont’d)
(LP)
minimize
 fv
vV
subject to:
fv ≥ pv +  puuv,
for all vV, 1 r Nv
uv + vu ≥ 1
for all (u,v) E
uCr
Let fv* denote the completion time of job Jv in the
optimal solution for LP.
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Non-preemptive Scheduling Scheme
• Partition the jobs to blocks of geometrically increasing
sizes by the fv* values.
• Apply to each block Vk an algorithm A for nonpreemptive multicoloring, so as to minimize the total
number of colors used.
• Concatenate the schedules obtained for the blocks:
first the schedule for V0, then the schedule for V1 and
so on…
• Let OPT*= v fv*, w(Vk) is the maximum size of a clique
in Vk, and suppose that A(Vk)  ß w(Vk).
Theorem (Non-pre-scheduling): The LP scheme gives
a non-preemptive schedule in which the sum of start
times of the jobs is at most 3.591 ß OPT* - p(V)/2,
where p(V)= v pv .
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Approximation Algorithm JB for SJC
1.
2.
3.
4.
5.
Apply the Preprocessing step for partitioning J to
job classes by rounded processing times.
For any pair of jobs Ji, Jj that belong to different
classes, add an edge (i,j) in the conflict graph G.
Denote the resulting graph G'.
Solve LP for the input jobs with rounded processing
times and conflict graph G'.
Partition the jobs in the input into blocks V0,V1, …,
VL, by their LP completion times.
Schedule the blocks in sequence using for each
block a coloring algorithm for unit length jobs.
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Analysis of the Algorithm
Theorem 2: JB approximates SJC within a factor of
9.76 ß + (1 – (e-1)/2)  9.76 ß + 0.14
• In general, LP may not be solvable in polynomial time on
G’; can be solved when the maximum weight clique
problem is solvable on G.
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Analysis of the Algorithm (Cont’d)
• In particular, maximum weighted clique and coloring are
polynomially solvable on perfect graphs.
Corollary 1: JB is a 9.9-approximation algorithm for SJC
on perfect graphs.
• In a line graph there are at most n maximal cliques; also,
using Vizing’s theorem, ß= 1+ o(1).
Corollary 2: JB is a 9.9+ o(1)-approximation algorithm for
SJC on line graphs.
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Summary and Open problems
• Interesting features of the current results:
Randomized strategy is combined in many ways
– K-colorable subgraphs (interval,compar.)
– LP values + lengths (line, perfect)
• NP-hardness for partial k-trees, trees, paths?
• Any non-trivial graph classes that are polynomially
solvable (beyond stars)?
• Better ratios…
Thank you
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