Edge Deletion and VCG Payments in Graphs

(True Costs of Cheap Labor Are Hard to Measure)

Edith Elkind

Presented by Yoram Bachrach

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

The Big Picture

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

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

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?

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

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

n v

VCG for Purchasing Paths and Edge Deletion

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.

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

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.

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.

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’

et { }G e

et

0( ) | | ( | |)vcg e ee P

T G P t P L t

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…

0| ' | eP t

0et

0 | ' |et P

0| ' | ( | ' |)eP t P

0 ( ) /e VCGt T G L

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

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|

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

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

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

1n

2

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)

( ' ) 2L n n

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

22 n

1 | 2 | (| | | 1|) 1 ( ' ) (| | )P P P L n P n

1 ' | | 2L P | | ' 2P L

' 4L ' 2 '/ 2L L

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?

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

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.

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

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

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.

,i kG

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

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

1/ 4

n

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