Structure and Phase Transition Phenomena in the VTC Problem

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Structure and Phase Transition Phenomena in the
VTC Problem

• C. P. Gomes, H. Kautz, B. Selman
• R. Bejar, and I. Vetsikas

IISI Cornell University University of Washington
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Outline

• I - VTC Domain - The allocation problem
• Definitions of fairness
• Boundary Cases
• Results on average case complexity
• fixed probability model
• constant connectivity model
• II - Conclusions and Future Work

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Virtual Transportation Company
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The Allocation Problem
Problem How to allocate the jobs to the
companies?
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Definition of Fairness I

• Min-max fairness
• min maxi TotalCosti

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Definition of Fairness II
Lex min-max fairness
Very powerful notion - analogous to fairness
notion used in load balancing for network design
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Allocation ProblemWorst-Case Complexity

• min-max fairness version of problem
• Equivalent to Minimum Multiprocessor Scheduling
• Worst-case complexity NP-Hard
• Lex min-max fairness version
• At least as hard as min-max fairness

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Boundary Cases

• Uniform bidding
• All companies declare the same cost for a given
  job (same values in all cells of a given column)
• NP-hard equivalent to Bin Packing
• Uniform cost
• A company declares the same cost for all
• jobs (identical jobs)
• Polynomial worst case complexity O(NxM)

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A Decision Algorithm for Min-max Fair Allocation

• Decision Problem
• ? allocation ?i, Ki lt Fixed Cost ?
• Backtrack Search algorithm
• Branching Heuristic
• Pick as next job the one which can be done by the
  smallest number of companies
• Value Ordering Heuristic
• Order companies by decreasing Ki

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Average-Case Complexity Instance Distributions

• Generating an instance
• Two ways of selecting the companies for each job
• Fixed connectivity For each job select exactly c
  companies
• Constant-Probability For each job each company
  is selected with probability p
• The costs for the selected companies are chosen
  from a uniform distribution
• The cost for the non-selected companies is ?

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Fixed Connectivity Model
Complexity and Phase Transition with c3
Phase Transition with different c
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Constant-probability Model
Complexity and Phase Transition with p0.18
Phase Transition with different p
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Comparison of the complexity between the two
models
Fixed connectivity model is harder
insights into the design of bidding models
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Conclusions

• Importance of understanding impact of structural
  features on computational cost
• VTC Domain
• Definitions of fairness
• Boundary cases
• Structure of the cost matrix
• Average complexity
• Critical parameter companies/jobs ---gt

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Future work

• I - Further study structural issues (e.g., effect
  of balancing, backbone in the VTC domain)
• II - Further explore Lex Min Max fairness - very
  powerful! Other notions of fairness.
• III - Consider combinatorial bundles instead of
  independent jobs
• IV - Game Theory issues -
• Strategies for the DOD to provide incentives for
  companies to be truthful and to penalize high
  declared costs

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• Structure vs. Complexity
• New results

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Quasigroup Completion Problem (QCP)
Given a matrix with a partial assignment of
colors (32colors in this case), can it be
completed so that each color occurs exactly once
in each row / column (latin square or
quasigroup)? Example
32 preassignment
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Phase Transition
Computational Cost
Fraction of unsolvable cases
Fraction of preassignment
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Quasigroup Patterns and Problems Hardness
Hardness is also controlled by structure of
constraints, not just percentage of holes
Tractable
Very hard
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Bandwidth
Bandwidth permute rows and columns of QCP to
minimize the width of the narrowest diagonal band
that covers all the holes. Fact can solve QCP in
time exponential in bandwidth


swap
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Random vs Balanced


Balanced
Random
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After Permuting
Balanced bandwidth 4
Random bandwidth 2
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Structure vs. Computational Cost
Balanced QCP
Computational cost
QCP
Aligned/ Rectangular QCP
of holes
Balancing makes the instances very hard - it
increases bandwith!
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Structural Features

• The understanding of the structural properties
  that characterize problem instances such as
  phase transitions, backbone, balance, and
  bandwith provides new insights into the
  practical complexity of many computational tasks.