Structure and Phase Transition Phenomena in the VTC Problem

1. 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

2. 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

3. Structure vs. Complexity New results

4. 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

5. Complexity Graph Phase transition from almost all solvable to almost all unsolvable Almost all solvable area Almost all unsolvable area Phase Transition Computational Cost Fraction of unsolvable cases Fraction of preassignment

6. Quasigroup Patterns and Problems Hardness Rectangular Pattern Aligned Pattern Balanced Pattern Hardness is also controlled by structure of constraints, not just percentage of holes Tractable Very hard

7. 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

8. Random vs Balanced Balanced Random

9. After Permuting Random bandwidth = 2 Balanced bandwidth = 4

10. Structure vs. Computational Cost Balanced QCP Computational cost QCP Aligned/ Rectangular QCP % of holes Balancing makes the instances very hard - it increases bandwith!

11. 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.

12. Virtual Transportation Company

13. The Allocation Problem Problem: How to allocate the jobs to the companies?

14. Definition of Fairness I j1 j2 j3 j4 j5 c1 100 50 50 50 100 c2 90 30 25 30 95 c3 95 90 25 30 25 • Min-max fairness: min maxi TotalCosti

15. Definition of Fairness II j1 j2 j3 j4 j5 Ordered Cost Vectors: r(S’)=<100,90,80> r(S’’)=<100,95,80> r(S’)<r(S’’) c1 100 50 50 50 100 c2 90 30 25 30 95 c3 95 90 25 30 25 Lex min-max fairness: Very powerful notion - analogous to fairness notion used in load balancing for network design (Kleinberg et al 2000)

16. 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

17. 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) J1 J1 J2 J2 J3 J3 C3 C3 C2 C2 C1 C1

18. Average-Case Complexity: Instance Distributions Generating an instance: Two ways of selecting the companies for each job: Fixed connectivity: For each job select exactlyc 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 

19. Fixed Connectivity Model Complexity and Phase Transition with c=3 Phase Transition with different c

20. Constant-probability Model Complexity and Phase Transition with p=0.18 Phase Transition with different p

21. Comparison of the complexity between the two models Fixed connectivity model is harder insights into the design of bidding models

22. 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 --->

23. 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