Edge Deletion and VCG Payments in Graphs

1. Edge Deletion and VCG Payments in Graphs (True Costs of Cheap Labor Are Hard to Measure) Edith Elkind Presented by Yoram Bachrach

2. Agenda • VCG payments for purchasing routes • Effects of edge deletion on VCG payments • Upper and lower bounds • How hard is it to figure what to delete? • General graphs • Series-Parallel graphs • Series-Parallel graphs with fixed edge costs • General distributions of edge costs

3. The Big Picture

4. Shortest Path Auctions • A buyer wants to purchase a path from s to t in a graph • Selfish agents own the edges • Edge costs are private • Only the selfish agents know the true costs • Eliciting this information is not trivial • We want to lower the buyer’s expected payments in shortest path auctions

5. VCG for Shortest Path Auctions • All edges submit their bids • The buyer chooses the cheapest path • The buyer pays each winning agent its threshold bid • The highest amount that agent can bid and remain on the chosen path • Truthful and individually-rational • Detail free • Does not make any assumptions on the underlying distributions of costs • May have extremely high costs • Bad behavior when we have long vertex-disjoint paths

6. Reducing VCG Payments • How do we modify the graph? • Adding new edges is not possible • These are resources we do not have • Removing edges can be performed by prohibiting some agents from participating • Counter-intuitive: we are reducing the competition. Why would this reduce the payments?

7. The Problem Setting • The mechanism designer knows the underlying graph and edge cost distributions • But does not know the exact costs • The designer must choose which edges to delete, before running VCG on the remaining graph • The edge deletion is an offline process • We do not use the distributions on every run

8. The Main Results • Deleting edges can reduce the expected payments by a factor of • Finding which edges to delete is hard • The problem is NP-hard • Even if all edge costs are constants (degenerate distributions) • The problem is hard to approximate • The problem is tractable for a specific subclass of graphs – series-parallel graphs with constant edge costs • Even for series-parallel graphs, arbitrary edge cost distributions has a bad performance ratio

9. VCG for Purchasing Paths and Edge Deletion

10. Purchasing Paths • Consider a graph G=<V,E> with a source vertex s and target vertex t • Each edge ei has a cost ci, drawn from a distribution Fi • The cost of a path P, |P|, is the sum of the costs of the edges on the path • Edges announce their costs (bids), and the mechanism chooses the winning paths • VCG mechanism select the path with the lowest cost, and pay each winning edge its threshold bid. Losing edges get nothing.

11. VCG Payments in Graphs • The threshold bid of a winning edge is the sum of the actual bid and a bonus (the maximal extra cost the edge can have until the chosen path would not include it) • The bonus is the difference between the cost of the cheapest path that does not include that edge, and the cost of the winning path

12. Deleting can be rewarding… The m edges path wins. The threshold bid of each edge is 1. We pay a total of m-2. If we remove any edge on the m edges path, one of the lower edges wins. Its threshold bid is 1, so it is payed 1. Edge deletion can give a performance ratio of m.

13. Even with constant costs… The m/5 path wins. Each edge is paid its cost plus a bonus of m/5. The total payment is m/5 * (m/5+1). Deleting any edge on that path causes one of the longer paths to win. None of the edges gets any bonus, so the total payment is 2m/5. Again, we get a ratio with a magnitude of m.

14. How rewarding can it be? • On graphs with constant edge costs, and L edges on the shortest s-t path, the ratio of payments before and after edge deletion is less than L. • Let be the cost of the cheapest path in • Let e0 be the edge that maximizes • We have • Consider a subgraph G’, with shortest path P’

15. How rewarding can it be? • If e0 isn’t in P’, we have • Otherwise, e0 wins, and the length of the shortest path in G’ without it is at least , so it gets a bonus of , so the total payment is at least • Either way, we pay at least • So, deleting no edges is an L-approximation for choosing which edges to delete…

16. MIN-VCG-PAYMENT • We are given a network <G=<V,E>,s,t>, and ci, the costs of the edges, and a target value t • We are asked whether there is a subset of the edges to delete, so the VCG payments would be less than t • Boolean version of the optimization problem • MIN-VCG-PAYMENT is NP-complete, even if all the Ci’s are 1 • We prove it by a reduction from LONGEST-PATH

17. LONGEST-PATH • Gets an unweighted graph G=<V,E> and a target L, and deceives if G contains a simple path with length of at least L • The problem is NP-hard • A modified version, EXACT-LONGEST-PATH also gets a source vertex s and a target vertex t, and checks if there is a path with length of exactly L • If this problem can be solved in polynomial time, so can LONGEST-PATH: simply try all possible s,t vertices, and any value between the original input L and |V|

18. MIN-VCG is NP-complete • Given an instance <G=<V,E>,s,t,k> of EXACT-LONGEST-PATH we construct input for MIN-VCG as in the following example, with target value T=n+k • If G has a path of length k, we can keep just this path and remove the rest of G. This leaves 2 s-t’ paths of length n+k, so the VCG payments are n+k

19. MIN-VCG is NP-complete • If G has no such path, consider any subset of edges to keep. • The shortest s-t path in G now has length of k’. • If k’<k the shortest s-t’ path (P1) has length of k’+n, and all of its edges win, and is paid 1+(k+n)-(k’+n)>2, so we pay at least 2n>n+k. • If k’>k, the shortest s-t’ path is the lower P2. Each edge is paid 1+(k’+n)-(n+k), so the total payment is at least 2(k+n)>k+n

20. Approximating MIN-VCG • Unless P=NP, LONGEST-PATH cannot be approximated within • Consider looking for a subset of edges that minimizes VCG payments. • We show that if we had an appoximation algorithm with an approximation ratio of we could construct an approximation algorithm of LONGEST-PATH with an approximation ratio of at most

21. LONGEST-PATH Approximation • Suppose G’s longest path has length of L’. Let s’,t’ be the first and last vertices on it • Consider the call to MIN-VCG with the inputs of s’,t’,L’ • We can achieve VCG payments of L’+n by deleting all edges not on this path (L’+n edges with bid of 1, and bonus of 0)0 • The MIN-VCG approximation returns a sub graph with payments smaller than • The shortest s-t’ path in the upper part is at most L’+n • If it is exactly L’+n, we’re done (found a path of length L’ in G)

22. LONGEST-PATH Approximation • Otherwise all edges in P1 are on the winning path. Each has the same threshold bid, so each is paid at most or the total payment would be at least • But the threshold bid of any edge on P1 is • So , or • If we get that , so P is a 2-approximation

23. Restricting inputs to MIN-VCG • MIN-VCG is NP-hard, and unlikely to have an approximation algorithm • We may still be able to deal with restricted types of inputs • We’ve shown it is hard to approximate using a reduction for LONGEST-PATH • Series-Parallel graphs have trivial algorithms for finding the LONGEST-PATH • Is MIN-VCG tractable for Series-Parallel graphs?

24. Series-Parallel Graphs • A single edge (s-t) is a SPN • A serial connection of two SPNs is an SPN • Serial connection unifies the last vertex of the first SPN with the first vertex of the second SPN • A parallel connection of two SPN is an SPN • Parallel connection unifies the first vertex of the first SPN with the first vertex of the second SPN, and the last vertex of the first SPN with the last vertex of the second SPN

25. MIN-VCG is NP-complete for SPNs • By a reduction from SUBSET-SUM • SUBSET-SUM • Gets a list of positive integers w1,…,wk, and a target integer M • Decides if there exists a subset of the wi’s with a sum of exactly M • The reduction from SUBSET-SUM builds a simple SPN, with 2 edges per each wi, two extra edges, and one edge connecting s-t. • The target payment T is set to M.

26. MIN-VCG and SUBSET-SUM • If no edges are deleted, VCG chooses the 0 path. The two last edges have a threshold bid of M, so the payment is at least 2M • To reduce the payment, we need to lower the threshold bids of these edges – by closing the gap between the top and bottom paths • If we have a ‘yes’ instance of subset sum, we can delete the 0 edges of the appropriate indices, and have 2 paths of cost M. The bonus would be 0, so the total cost would be M

27. MIN-VCG and SUBSET-SUM • If we have a ‘no’ instance of subset sum, no matter which edges we delete, we have one path that is longer than the other • Let the cost of the cheapest s-t path in the top part be A • If A>M the lower edge wins, and is paid A • If A<M a top path is chosen, and the two last edges have a threshold of M-A, so the payment is at least 2M-A>M • Either case, the total payment exceeds M

28. MIN-VCG for SPNs with fixed edge costs is tractable • The paper suggests a dynamic programming algorithm. • A subroutine takes an SPN composed of two sub-SPNs and computes a family of candidate solutions • Solutions built from the sub-SPNs’ candidate solutions. • The family is guaranteed to contain the correct solution. • The algorithm is based on testing , the sub-graph that minimizes the VCG payments, assuming the shortest path must have a length of i, and the bonus to each edge is at most k-i.

29. MIN-VCG for fixed cost SPNs • We cap the bonus by adding an s-t edge • For serial connection • For all choices of j, the total bonuses paid do not increase • For one choice of j, where j is the length of the shortest path on the left side, and i-j the length of the shortest path on the right side, the bonuses are at least as much as they were

30. General Distributions of Edge Costs • An algorithm that only receives the expectancy of edge cost distributions can always fail miserably compared to one that gets the full information about the distributions • There is always an example with a VCG ratio of • Example given with either • The constant distribution: constant price of ½ • The Bernuli distribution: price of 0 with probability ½, and price of 1 with probability ½.0

31. Conclusions • VCG prices for shortest path auctions • Deleting edges can reduce VCG payments • But it is hard to decide what to delete • Hard even to approximate • Remains hard for some input restrictions • General SPNs • Is tractable for very restricted inputs • SPNs with constant payments • Using just the expected edge costs is not enough • Open questions • Other sub-classes of restricted inputs? • Not knowing the distribution, but just the first k moments