Edge Deletion and VCG Payments in Graphs

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Edge Deletion and VCG Payments in Graphs

• (True Costs of Cheap Labor Are Hard to Measure)
• Edith Elkind
• Presented by Yoram Bachrach

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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

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The Big Picture
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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 buyers expected payments in
  shortest path auctions

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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

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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?

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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

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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

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VCG for Purchasing Paths and Edge Deletion
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Purchasing Paths

• Consider a graph GltV,Egt 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.

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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

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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.
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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/51). 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.
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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

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How rewarding can it be?

• If e0 isnt 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

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MIN-VCG-PAYMENT

• We are given a network ltGltV,Egt,s,tgt, 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
  Cis are 1
• We prove it by a reduction from LONGEST-PATH

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LONGEST-PATH

• Gets an unweighted graph GltV,Egt 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

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MIN-VCG is NP-complete

• Given an instance ltGltV,Egt,s,t,kgt of
  EXACT-LONGEST-PATH we construct input for MIN-VCG
  as in the following example, with target value
  Tnk
• 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 nk, so the VCG payments are
  nk

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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 kltk the shortest s-t path (P1) has length of
  kn, and all of its edges win, and is paid
  1(kn)-(kn)gt2, so we pay at least 2ngtnk.
• If kgtk, the shortest s-t path is the lower P2.
  Each edge is paid 1(kn)-(nk), so the total
  payment is at least 2(kn)gtkn

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Approximating MIN-VCG

• Unless PNP, 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

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MIN-VCG and Longest-Path

• We use the same construction as before, and try
  every combination of s,t in V, and any value of k
  1,,n as inputs to the MIN-VCG approximation
  oracle (total of calls)
• Each call returns a sub graph with an s-t path
  (or the VCG payments would be infinite)
• We pick the shortest s-t path from the sub graph
  returned on each call, and add it to the list
• We return the longest path in the list
• We now show that this is an -approximation for
  LONGEST-PATH

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LONGEST-PATH Approximation

• Suppose Gs 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 Ln by deleting
  all edges not on this path (Ln 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 Ln
• If it is exactly Ln, were done (found a path
  of length L in G)

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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

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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
• Weve 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?

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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

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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 wis 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.

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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

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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 AgtM the lower edge wins, and is paid A
• If AltM a top path is chosen, and the two last
  edges have a threshold of M-A, so the payment is
  at least 2M-AgtM
• Either case, the total payment exceeds M

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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.

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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

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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

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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