Grid Computing and Parallel BranchandBound Tree Searches

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Grid Computing and Parallel Branch-and-Bound
Tree Searches
C. Roucairol, V-D. Cung (1) and P. Hahn (2)
1 OPALE team, PRiSM Lab.- CNRS University of
Versailles - Saint Quentin en Yvelines
2 University of Pennsylvania in Philadelphia
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Discrete Optimization Challenges on GRIDDéfis
en Optimisation Combinatoire sur Grille de
machines
A three year software project
www.prism.uvsq.fr/DOC-G/
PARADISEO OPAC
BOB/BOB OPALE
Athapascan P3 O2
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Application challenges
Generic applications
Specific applications

• FranceTelecom
• Multiobjective design
• of mobile cellular network
• Cost min. nb. of sites
• Quality of services
• Min. interferences
• Min. the losts of communication
• Subject to
• Geography area covering
• Handover
• 600.109 possibilities
• for each antenna (its location
• and parameters)

Traveling Salesman Size gt d15112
Quadratic Assignment Size gt Nug30
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Solving methods and libraries
Search range
Exact methods (BOB/BOB)
Metaheuristics (PARADISEO)

• Independant from implicit enumeration
• (Branch-and-X, DP, A, DC, )
• Generic components (operators of
• search, generation, branching,
• bounding, )
• Parallelism node evaluation,
• tree traversal, searching range

• Independant from the searching
• methods (SA, TS, GA, )
• Generic components (operators
• of search, selection, replacement,
• termination,)
• Parallelism solution evaluation,
• neighborhood partitionning,
• cooperative searching

Parallelism (on GRID-Athapascan)
faster/bigger/robust/efficient
CO applications fine to middle grain
parallelism, to be mapped on the grid Library for
parallel programming independent from architecture
All these libraries are in C Open Source
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Quadratic Assignment Problem
Mathematical formulation Koopmans Beckmann
1957
Applications VLSI, airplane cockpits, hospital
services, ergonomic keyboards...

• NP-complete, even for an epsilon-approximated
  solution
• Existence of a QAPLIB (up to size 256),
• Drezner (dre) Taillard (tai and tur) still
  provide new data instances.

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Evolution of Branch-and-X methods

• Impressive progresses since the beginning of
  90s.
• Increasing power of the machines (CPU and
  memory).
• Improvements on the lower bounds column
  generation (Price), polyhedron approach (Cut),
  quadratic programming, lift-and-project,
  semidefinite programming (SDP), etc.
• ? Trends more complexe, longer evaluation time,
  but also better candidate for parallelization.
• Symmetry and dominance detections.
• Massively parallel tree searches.

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New LB RLT level-1
Adams Sherali 1986 Resende, Ramakrishnan
Drezner 1995
Setting yijklxijxkl and CijklFikDjl , the
relaxed LP at level-1
linear assignment contraints and
Complementary pairs
n2(n-1)2/2 variables yijkl, n2 variables xij,
2n2(n-1)2n contraints. n30, 379 350 variables
and 52 260 contraints.
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New LB RLT level-2 Ramakrishnan, Resende,
Ramachandran Pekny 1996
Setting zijklpqxijxklxpq and Dijklpq
CijklCklpqCijpq , The relaxed LP level-2 (3
firms and sites at the same time)
Dijklpq
linear assignment contraints, (2.6 - 2.9) and
Compl. level-2
n2(n-1)2(n-2)2/2 variables zijklpq, n2(n-1)2/2
variables yijkl, n2 variables xij,
n2(n-1)2(n-2)2n2(n-1)2n contraints. n30, 99
280 950 variables and 21 245 460 contraints !
Remark value n-1 times the QAP value.
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Lower bound Super leader
Cijkl
cij
Peter Hahns DUAL procedure Based on Hungarian
method O(n3)
Dijklpq
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Resolution with Peter Hahns DUAL procedure
(Iterated Lagrangian decomposition) Slightly
worse solutions,but faster than CPLEX.
Nugent 12 1995, CPLEX 3.0, Silicon Graphics
250Mhz, 1.2GB RAM ADP (conjugate gradient based
interior point method) spent 6504 seconds. The
best CPLEX 3.0 methods have not been able to
solve instances for size ngt8 and ADP is reported
to be at least 200 times faster than CPLEX 3.0
barrier.
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Parallel Tree Searches Cung 1994 Gendron
Crainic 1994

• 1st type data-parallelism, fine grain on the
  evaluation and the generation of the successors
• Potential parallelism important
• But often synchrone and dependent on the
  application
• 2nd type Medium to coarse grain on the tree
  search a node or a sub-tree per processor
• Completely application independent
• Granularity versus consistency to the serial tree
  search
• Bad speedup if the number of nodes is small
• 3rd type Coarse grain - decomposition of the
  search range
• Easy to implement (independent searches) and
  application independent
• Search redondancy (some critical nodes are
  explored by different processors)

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3 types of parallelization
X10
X11
X10
X11
f15
f15
5,11
12,13
15,16
17,20
X10
X10
X10
X10
X11
X11
X11
X11
f12
f11
f17
f15
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Anomalies in Parallel Tree Searches
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Irrégularity ? dynamic scheduling
load balancing
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Granularity versus Consistency to serial tree
search
5
7
9
10
Envelop of the critical tree
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15
18
v(P)20
BSgt20
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BOB/BOB

• BOB created in 1994 to capitalize the know-how
  in parallel BX.
• An unique API for the applications.
• Only one program for serial or parallel
  exceution.
• Object oriented and re-entrance for BOB.
• Comm. through shared mem., PVM/MPI, Charm,
  Athapascan

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Generic data and execution model
Global upper bound (feasible solution)
Global priority queue
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Enqueue (send the nodes or threads back to
the queue)
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Results on small and medium size problemsCung,
Guignard-Spielberg, Hahn, Mautor Roucairol

• BOB Charm/Converse (charm.cs.uiuc.edu)
• Simple branching (no look-ahead) symmetry
  detection
• RLT level-1 lower bounds

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• Results on larger size problems
• 100 PCs 733Mhz 256 Mbytes, Linux
  (icluster.imag.fr)
• Number of nodes
• Wall clock time in seconds

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Branch-and-Bound method (parallel tree search,
master-slaves, fault tolerant), Convex Quadratic
Relaxation. 2510 machines (MPP included) Runtime
support Condor PVM Globus http//www-unix.mc
s.anl.gov/metaneos/ Anstreicher, Brixius,
GouxLinderoth, 2000
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Impact of parallelismtackle the combinatorial
burst !

• Huge critical tree
• Nug24 48,455,496 nodes in 3488,3490
• Nug30 5448, 6124 12.109 nodes for 677 values
• Parallel tree search is more or less mastered
• if good upper bound is known.
• New evaluation techniques are time consuming
• Grid computing issues (dynamic and heterogeneous)
• Metaheuristic Nug30, Scatter SearchTabou,
• Note optimal value is found at each run in
  lt60s.

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On-going works

• Application part
• Hybridations
• Parallel node evaluation (when it is application
  independent) parallel tree search
• Metaheuristics exact methods (having new
  neighborhood e.g. POPMUSIC and MIMAUSA)
• Parallel pricing and cutting
• Grid computing part
• High performance task scheduling and fault
  tolerant mechanism