The Auction Algorithm

1. The Auction Algorithm Shahar Paz

2. Assignment Problem • Given: • Persons, Objects • – Objects that may be assigned with person • – Benefit of assigning object with person • Target: • Assign each person with unique object • Maximize total benefit

3. Assignment Problem 4 2 1 1 1 -2 3 2

4. Assignment Problem 4 2 Assignment benefit: 2 + 1 – 2 + 3 = 4 1 1 1 -2 3 2

5. Assignment vs. Min-Cost-Flow • Special case of Min-Cost-Flow • Nodes: • Persons have (1 supply unit) • Objects have (1 demand unit) • Arcs: • (1 capacity unit) • (cost is -benefit)

6. Assignment vs. Min-Cost-Flow • Min-Cost-Flow can turn into Assignment • Arcs => new nodes b(i) b(i)-u(i,j) i i c=0 c(i,j) u(i,j) ij b=u(i,j) c=c(i,j) b(j) j j b(j)

7. Assignment vs. Min-Cost-Flow • Min-Cost-Flow can turn into Assignment • Arcs => new nodes • New nodes split by supply (old arc capacity) ij1 b=1 b(i)-u(i,k) b(i)-u(i,k) i i c=0 ij b=u(i,j) b=1 ij2 c=c(i,j) b(j)-u(j,k) b(j)-u(j,k) j j b=1 ij3

8. Assignment vs. Min-Cost-Flow • Min-Cost-Flow can turn into Assignment • Arcs => new nodes • New nodes split by supply (old arc capacity) • Old nodes split by demand b=-1 ij1 ij1 i1 b=1 b=1 b(i)-u(i,k) i b=1 b=1 i2 b=-1 ij2 ij2 b(j)-u(j,k) j b=1 b(j)-u(j,k) b=1 ij3 ij3 j

9. Assignment vs. Min-Cost-Flow • Min-Cost-Flow can turn into Assignment • Arcs => new nodes • New nodes split by supply (old arc capacity) • Old nodes split by demand • #new nodes • #old nodes

10. Auction Algorithm • Init • Empty assignment , zero prices • Iterate until : • Bidding phase • Some unassigned persons bid for most desired objects • Bids must be greater then current objects’ prices • Assignment phase • Prices of objects increased to max bid • Assignment change accordingly

11. Auction Algorithm • Bidding phase: • Select subset of unassigned persons • For each person : • (most desired object) • (object’s benefit) • (second-desired benefit) • Set bid of person for object :

12. Auction Termination • Observations • Objects receiving bids get assigned • Assigned objects remain assigned • Assigned persons might get unassigned • Only on termination no unassigned persons and objects

13. Auction Termination • Observation • Each time object receives a bid, increase by at least . • Conclusion • Object receives bids get price • Proof

14. Auction Termination • Observation • If person bid times,all objects in get price • Proof • As of some iteration: • Person only bid for objects in • All objects in has price • But has higher benefit (const)

15. Auction Termination • Proposition: • Feasible solution Auction algorithm terminates • Proof: • As of some iteration: • All objects in are assigned • Must be assigned with persons from (price ) • Only persons from bid • Not all persons in are assigned • , but

16. Duality & CS • Dual problem: • Minimize • Over all price vectors • Complementary Slackness: • Given assignment and prices

17. Duality & CS • Observation: • Any Primal  Any Dual • Proof:

18. Duality & CS • Observation: • Any Primal  Any Dual • Conclusion: • Primal  Optimal Primal  Optimal Dual  Dual • Primal = Dual  optimal Primal & Dual • Primal = Dual  CS

19. Duality & CS • Observation: • CS  Primal = Dual • Proof:

20. Auction Optimality • -CS: • Observation: • Auction preserves -CS • Proof:

21. Auction Optimality • Proposition: • Auction algorithm is -optimal • Proof:

22. -Scaling • Choosing  • If data is integer, guarantees optimality • Small  causes small changes, bad performance • -Scaling • Run Auction with large and prices • Compute -CS prices • Run Auction with and prices • Compute -CS prices

23. -Scaling • Adjustments for -Scaling: • On each iteration only 1 person bid • First scaling phase starts with initial prices 0 • Other scaling phases start with previous prices • On scaling phase , use as benefits • So is integer multiple of • Final prices of scaling phase k satisfy-CS with , and -CS with .

24. -Scaling • Candidate-list of person : • Holds pairs of potential next-bids • Modified bidding phase: • search for 2 pairs with • If not found: • Recalculate and bid normally • Else: • Remove all encountered pairs except second match • Select first match and set bid of

25. -Scaling Complexity • Candidate-list complexity: • Calculation of takes • But only happen when decrease • Using takes until recalculation • Auction complexity • Each decrease of is of at least

26. -Scaling Complexity • Proposition: • Running Auction with prices satisfying -CSgives • Proof: • – initial prices,– corresponding assignment • – partial assignment & prices,generated by Auction algorithmjust before last bid of some person

27. -Scaling Complexity • Proof cont: • Find path • , • unassigned under • Observe:

28. -Scaling Complexity • Proof cont: • Find path • , • unassigned under • Observe:

29. -Scaling Complexity • Proof cont: • We got: • Summing: • isn’t assigned under • didn’t receive any bid, has initial price

30. -Scaling Complexity • What is r? • -CS:

31. -Scaling Complexity • Conclusion: • Scaling phase Complexity: • Number of phases: • , – first & last values • Total -Scaling Complexity:

32. Infeasibility • Auction never terminates for infeasible input • One cannot tell if the algorithm won’t terminate,or just haven’t yet • Ways to detect infeasibility: • Adding arcs with highly negative benefit • Checking against lower bound • Checking for augmenting path

33. Questions? Thank you