Structure and Phase Transition Phenomena

in the VTC Problem

C. P. Gomes, H. Kautz, B. Selman

R. Bejar, and I. Vetsikas

IISIIISI

Cornell UniversityCornell University

University of WashingtonUniversity of Washington

Outline

I - Structure vs. complexity - new results II - VTC Domain - The allocation problem

Definitions of fairness Boundary Cases Results on average case complexity

fixed probability model constant connectivity model

III - Conclusions and Future Work

Structure vs. Complexity New results

Quasigroup Completion Problem (QCP)

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

Phase Transition

Almost all unsolvable area

Fraction of preassignmentFra

ctio

n o

f u

nso

lvab

le c

ases

Almost all solvable area

Complexity Graph

Phase transition from almost all solvableto almost all unsolvable

Co

mp

uta

tio

nal

Co

st

Quasigroup Patterns and Problems Hardness

Rectangular Pattern Aligned Pattern Balanced Pattern

Tractable Very hard

Hardness is also controlled by structure of constraints, not just percentage of holes

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

BalancedRandom

After Permuting

Balanced bandwidth = 4

Random bandwidth = 2

Structure vs. Computational Cost

Balanced QCP

QCP

% of holes

Com

pu

tati

on

al

cost

Balancing makes the instances very hard - it increases bandwith!

Aligned/ Rectangular QCP

Structural FeaturesStructural 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.

Virtual Transportation Company

The Allocation Problem

Problem: How to allocate the jobs to the companies?

j1 j2 j3 j4 j5

c1 100

100

50 50 50

c2 95 90 30 25 30

c3 95 90 25 30 25

Definition of Fairness I

Min-max fairness:min maxi TotalCosti

9095c3

30253090c2

505050

253025

95

100100c1

j5j4j3j2j1

Definition of Fairness II

Lex min-max fairness:

9095c3

30253090c2

505050

253025

95

100100c1

j5j4j3j2j1 Ordered Cost Vectors:

r(S’)=<100,90,80>

r(S’’)=<100,95,80>

r(S’)<r(S’’)Very powerful notion - analogous to fairness notion used in load balancing for network design (Kleinberg et al 2000)

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

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 alljobs (identical jobs) Polynomial worst case complexity: O(NxM)

C3C2C1

J2J1 J3

J2J1 J3C3C2C1

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

companies/jobs

Branches% of Solutions

100%

50%

0%

75%

25%

1200

600

0

companies/jobs

c=2c=3c=4

75%100%

100%

50%

25%

0%

Complexity and Phase Transition with c=3

Phase Transition with different c

Constant-probability Model

-50

50

150

250

350

450

550

650

750

companies/jobs

Branches

% of Solutions

100%

75%

50%

25%

0%

companies/jobs

p=0.16

p=0.17

p=0.18

100%

75%

50%

25%

0%

Complexity and Phase Transition with p=0.18

Phase Transition with different p

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