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
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
Virtual Transportation Company
The Allocation Problem
j1
j2
j3
j4
j5
c1 100 100
c2 95 90
c3 95 90
50
50
50
30
25
30
25
30
25
Problem: How to allocate the jobs to the companies?
Definition of Fairness I
• Min-max fairness:
min maxi TotalCosti
j1
j2
j3
j4
j5
c1 100 100 50
c2 95 90 30
c3 95 90 25
50
50
25
30
30
25
Definition of Fairness II
Lex min-max fairness:
j3
j4
j5
Ordered Cost Vectors:
c1 100 100 50
c2 95 90 30
c3 95 90 25
50
50
r(S’)=<100,90,80>
25
30
r(S’’)=<100,95,80>
30
25
r(S’)<r(S’’)
j1
j2
Very powerful notion - analogous to fairness notion used
in load balancing for network design
Allocation Problem
Worst-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
Boundary Cases
• Uniform bidding
J1 J2 J3
C3
C2
C1
• Uniform cost
J1 J2 J3
C3
C2
C1
– 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
– A company declares the same cost for all
jobs (identical jobs)
– Polynomial worst case complexity: O(NxM)
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 
Fixed Connectivity Model
Complexity and Phase
Transition with c=3
1200
Phase Transition with
different c
100%
Branches
% of Solutions
75%
c=2
c=3
c=4
100%
75%
100%
50%
600
50%
25%
25%
0%
0%
companies/jobs
0.
26
0.
36
0.
46
0.
56
0.
66
0.
76
0.
86
0.
87
0.
93
0.
95
1.
02
1.
12
1.
22
0 .2
6
0 .3
0 .3
4
0 .3
8
0 .4
2
0 .4
6
0 .5
0 .5
4
0 .5
8
0 .6
2
0 .6
6
0 .7
0 .7
4
0 .7
8
0 .8
2
0
companies/jobs
Constant-probability Model
Complexity and Phase
Transition with p=0.18
750
Phase Transition with
different p
100%
650
Branches
% of Solutions
550
100%
p=0.16
75%
p=0.17
75%
p=0.18
450
350
50%
50%
25%
25%
0%
0%
250
150
50
companies/jobs
0.8
4
0.7
8
0.6
6
2
0.6
0.5
0.5
4
0.4
2
8
0.3
0.3
0.2
6
-50
26 . 32 . 38 . 44
0.
0
0
0
5
6
2
8
4
0. 0. 5 0. 6 0. 6 0. 7
companies/jobs
8
0.
Comparison of the complexity
between the two models
Fixed connectivity
model is harder
insights into the design of bidding models
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 --->
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
BLANK
Structure vs. Complexity
New results
Quasigroup Completion
Problem (QCP)
Given a matrix with a partial assignment of colors
(32%colors 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
Computational Cost
Fraction of unsolvable cases
Complexity
Graph
Phase
Transition
Phase transition
from almost all solvable
to almost all unsolvable
Almost all solvable
area
Almost all unsolvable
area
Fraction of preassignment
Quasigroup Patterns and Problems
Hardness
Hardness is also controlled by structure of
constraints, not just percentage of holes
Rectangular Pattern
Aligned Pattern
Tractable
Balanced Pattern
Very hard
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
Random vs Balanced
Random
Balanced
After Permuting
Random
bandwidth = 2
Balanced
bandwidth = 4
Structure vs. Computational Cost
Computational
cost
Balanced QCP
QCP
Aligned/ Rectangular
QCP
% of holes
Balancing makes the instances very hard - it increases bandwith!
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.