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

6. Quasigroup Patterns and Problems Hardness Rectangular PatternAligned PatternBalanced Pattern TractableVery hard Hardness is also controlled by structure of constraints, not just percentage of holes

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 Balanced bandwidth = 4 Random bandwidth = 2

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

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? j1j2j3j4j5 c1 100 50 c2 9590302530 c3 9590253025

14. Definition of Fairness I Min-max fairness: min max i TotalCost i 9095 c3 30253090 c2 50 253025 95 100 c1 j5j4j3j2j1

15. Definition of Fairness II Lex min-max fairness: 9095 c3 30253090 c2 50 253025 95 100 c1 j5j4j3j2j1 Ordered Cost Vectors: r(S’)= r(S’’)= r(S’)<r(S’’) Very powerful notion - analogous to fairness notion used in load balancing for network design (Kleinberg et al 2000)

16. 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

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) C3 C2 C1 J2J1J3 J2J1J3 C3 C2 C1

18. A Decision Algorithm for Min-max Fair Allocation Decision Problem:  allocation:  i, K i < 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 K i

19. 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 

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

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

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

23. 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 --->

24. 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