Using Personal Condor toSolve Quadratic AssignmentProblems

Jeff LinderothAxioma, Inc.

jlinderoth@axiomainc.com

Partners in Crime

Kurt Anstreicher

Nate Brixius

University of Iowa

Jean-Pierre Goux

MCS Division, ANL

LOTS of people in this room!

University of Wisconsin

Our Mission

1. Find the best possible solution to large quadratic assignment problem (QAP) instances

2. Prove that the solution is indeed optimal

3. Show how to exploit the Computational Grid offered by Personal Condor to make it happen

What’s a QAP?

Can be thought of as a facility location problem

The QAP is NP-REALLY-Hard TSP: Solve

n=13509 QAP: Solve n=25

Q: Why Is This Important? Answer #1: Practical applications

Facility Location Hospital Design Flight Instrument Layout

Answer #2: Similarity Comparable to other practically

important combinatorial optimization problems

TSP, MIP

The REAL Answer – It’s NOT!

“The Journey Is The Reward”

What can we learn about solving complexnumerical problems on Computational Grids?

The Perfect Marriage

While my wife likes this slide, really it’s the QAP and Condor that make the perfect marriage!

+

Making the Perfect Marriage Something Old Something New Something Borrowed Something Blue

Something Old: Branch-and-Bound1. Bound

Solve “auxiliary” problem that gives a lower bound on the optimal solution to the problem

Any assignment of facilities to locations gives an upper bound on the optimal solution

What if lower bound < upper bound?

Branch Divide-and-

Conquer! Recursively make

problem smaller by assigning each facility to a fixed location

Without the bounding, this is complete enumeration. (n!)

This is not “pleasantly parallel” computing!

* Something VERY old

Something New: A convex quadratic programming

relaxation

Solved with the Frank-Wolfe Algorithm*. Each iteration is one linear

assignment problem

Something Borrowed: With Condor it is easy to “borrow”

CPU cycles1. Call your friends and colleagues and

flock with their Condor pools2. Write an NPACI proposal and Glide-

In to supercomputer resources3. If all else fails (Condor/Globus not

installed), hobble in!

My Personal GridNumber Type Location Method

414 Intel/Linux Argonne Hobble-In96 SGI/Irix Argonne Glide-In

1024 SGI/Irix NCSA Glide-In

16 Intel/Linux NCSA Flocked

45 SGI/Irix NCSA Flocked

246 Intel/Linux Wisconsin Flocked

146 Intel/Solaris Wisconsin Flocked

133 Sun/Solaris Wisconsin Flocked

190 Intel/Linux Georgia Tech Flocked

94 Intel/Solaris Georgia Tech Flocked

54 Intel/Linux Italy (INFN) Flocked

25 Intel/LinuxNew Mexico (AHPCC)

Flocked

5 Intel/Linux Columbia U. Flocked

10 Sun/Solaris Columbia U. Flocked

12 Sun/Solaris Northwestern Flocked

* My sincerest apologies for the terrible pun

Something Blue? You could work until you’re blue in the

face and not solve QAP instances*

Instance Arch. Wall Time

Person Date

Nug22 Ultra 360MHz 56 Hours Hahn 1999

Nug24 Ultra 360MHz 9 days Hahn 1999

Nug25 Ultra 360MHz 66 days Hahn 1999

Nug22 48-96 Cenju-3 9 days Marzetta 1998

Nug25 64-128 Paragon

30 days Marzetta 1998

The Holy Grail We want to solve nug30! Extrapolating results and using an idea

of Knuth*, we conjecture that we will need roughly 10-15 years of CPU time

How can we be sure to use 10-15 years of CPU time somewhat efficiently?

We have the additional burden of working in Condor’s extremely dynamic environment!

* Something Old

Making the Marriage Work The MW runtime support library helps us

cope with the dynamic nature of our platform MW – Master Worker paradigm

Must deal with contention at the master Search/ordering strategies at both master and

worker are important! Parallel Efficiency improves from 50% to 90% Lots more details!

Paper available at www.optimization-online.org

Mission Accomplished!

Wall Clock Time 6:22:04:31

Avg. # Machines 653

Max. # Machines 1007

CPU Time Approx. 11 years

Nodes 11,892,208,412

LAPs 574,254,156,532

Parallel Efficiency 92%

Solution Characteristics

Number of Workers

The Ups & Downs1. Human (read Jeff) error

• Master compiled for <= 1000 workers

2. Condor schedd bug (Gasp!!!!)3. Master shut down to fix NFS problems4. Condor schedd bug5. Human (read Jeff) error

• Incorrect editing of configuration files resulting in many incorrect submissions

Number of Workers on June 12

Number of Workers at Three Biggest Contributors

Number of Workers at Three Next Largest Contributors

KLAPS

The Moral of the Story A good wedding/marriage requires four

key ingredients There were also four key ingredients to

solving nug301. Powerful mathematics for producing a lower

bound2. Innovative branching techniques3. An EXTREMELY powerful computing platform4. “Marrying” the algorithm to the platform in

an appropriate manner

The TRUE Moral It is possible to do complex

numerical calculations on the Computational Grid using Condor!

It opens the doors to attacking heretofore unsolved problems!

http://www.mcs.anl.gov/metaneos