Batch%20Scheduling%20of%20Conflicting%20Jobs

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Batch Scheduling of Conflicting Jobs
Hadas Shachnai The Technion
Based on joint papers with L. Epstein, M. M.
Halldórsson and A. Levin.
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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 schedule - A multicoloring of G.
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Batch Scheduling of Conflicting Jobs

• Given is an undirected graph G(V,E)
• Each 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

G

• Minimize
• sum of job completion times (SJC)
• sum of batch completion times (SBC)
• Makespan (Max coloring)

SJC(I)221253725
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Batch Scheduling of Conflicting Jobs

• Given is an undirected graph G(V,E)
• Each 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

G

• Minimize
• sum of job completion times (SJC)
• sum of batch completion times (SBC)
• Makespan (Max coloring)

SBC(I)3235728
Max-col (I)7
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Known Results

• For general graphs BSC and Max-coloring are hard
  to approximate within factor n1-e unless NPZPP
  (Bar-Noy 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 (Contd)

• Sum of batch completion times

• A 4?-approximation for SBC for graph classes on
  which Maximum Independent Set can be approximated
  within factor ?, for some ? 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

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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.

- Last guess ? e-1 times optimal guess -
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
• Each group solved separately.
• Subsolutions are pasted together in order to
  produce final solution

Solve efficiently in terms of OPT - Length
based immediate - LP based bound clique
number of the induced subgraph
Each block must be solved with a small makespan
A(V1)
A(V3)
A(V5)..
A(V2)
A(V4)
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Bounds for Perfect and Line Graphs

• Preprocessing the input I
• Pick a random number ? U0,1).
• Partition the jobs into classes by their
  processing times J0 j pj e? and Ji j
  ei-1 ? lt pj ei ?.
• Let k be the largest index of any non-empty
  class.
• For all i0,1, , k, round up the processing time
  of each job j ? Ji to pjei ?. 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 EOPT? eOPT, 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 (Contd)
(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
  non-preemptive 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. Apply the Preprocessing step for partitioning J
   to job classes by rounded processing times.
2. 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'.
3. Solve LP for the input jobs with rounded
   processing times and conflict graph G'.
4. Partition the jobs in the input into blocks
   V0,V1, , VL, by their LP completion times.
5. 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 (Contd)

• 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 Vizings 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