Vickrey Prices and Shortest Paths What is an Edge Worth?

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Vickrey Prices and Shortest PathsWhat is an Edge
Worth?

• By John Hershberger and Subhash Suri

Carlos Esparza and Devin Low 3/5/02
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Motivation

• TCP is self-regulating, vulnerable to selfish
  protocol-breakers
• Can we define a strategyproof packet routing
  protocol with efficient allocations?
• Can Vickrey auctions be applied usefully to this
  domain?
• Can Nisans shortest path method be improved?
• Are marginal contributions of packet routers
  efficiently computable?

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Outline

• Domain and Problem Statement
• Main Algorithm Simplest Case
• Computational Improvements
• Extension to Complex Cases
• Critiques
• Discussion Questions

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Domain Routing Graph

• Apply Vickrey Pricing scheme
• Strategyproof Rational link owners must
  truthfully express link cost

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Domain Strategyproof Analysis

• An agent can deviate by over-reporting or
  under-reporting its edge cost C
• If the agent over-reports, then either there is
  no effect, or the agent is not included in the
  shortest path as a result of the over-report
• If the agent is not included in the shortest
  path, the agent loses the opportunity to get a
  transfer payment, so its utility decreases
• If the agent under-reports with a false cost F lt
  C, then either there is no effect or the agent is
  included in the shortest path as a result of the
  under-report (receiving transfer payment based on
  F)
• F lt C, so if the agent is included in the
  shortest path, it is paid less than its edge
  cost, so its utility is negative
• So an agent can never profit from untruthful cost
  reporting

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Problem Statement Shortest Path with Edge
Deletion
Shortest path
What is new shortest path?

• Truthful valuation equals marginal contribution
• Computing marginal contribution is
  computationally prohibitive
• Need efficient method for computing shortest path
  with deleted links

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Simplest Case All Vertices on Path
Figure 1

• Vx are all left of cut, Vy are all right of cut
• Ei denotes set of edges crossing cut
• d(x,y G\ei) min d(x,u) c(u,v) d(v,y)

(u,v) ? Ei (u,v) ! ei
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Simplest Case All Vertices on Path

• One can iterate across ei with great
  computational efficiency
• The difference between Ei and Ei1 is simply
• Add to Ei all edges whose left endpoint is vi1
• Remove from Ei all edges whose right endpoint is
  vi1
• Total time complexity is thus O(m log m) for
  now!

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Simplest Case All Vertices on Path
Path Algorithm

• Part 1 Setup
• L R are k-element arrays whose elements are
  edges
• Q a priority queue of (weight, edge) pairs
  indexed by weight
• Part 2 Initialization
• For each e ? E \ path(x,y)
• If left(e) lt right(e), put e into Lleft(e)
  and Rright(e)

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Simplest Case All Vertices on Path
Path Algorithm

• Part 3 Minimum weight
• For i 1 to k 1
• (a) For each e (u,v) ? Li Insert (w,e)
  into Q, with weight w d(x,u G \ ei) c(u,v)
  d(v,y G \ ei)
• (b) Remove from Q all (w,e) pairs with e ? Ri
• (c) Report the minimum weight in Q as the d(x,y
  G \ ei )

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Computational Improvements
Structural Inefficiency Priority Queue

• Priority Queue only operations that require
  non-constant time
• Naïve priority queue implementation leads to a
  running time of O(m log m)
• Can this be improved? YES!

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Computational Improvements
Structural Optimization Fibonacci Heap

• Use Fibonacci Heap instead of Priority Queue
• The heap only has a subset of the edges that
  would be in a naïve construction of Q
• But it is guaranteed to have the minimum-weight
  element of E
• Time complexity improved to O(n log n m)!
• What was Nisans 99 result?
• O(nm log n)

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Extension to Complex Cases
Undirected Networks

• The shortest path tree X with source x is the
  union of all the shortest paths from x to other
  vertices in V
• The shortest path tree Y with sink y is the
  union of all the shortest paths from vertices in
  V to y
• Before X Y path(x,y) gt not generally the
  case

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Extension to Complex Cases
Undirected Networks

• The vertices connected to vi in the remaining
  forest form block Bi when all edges in
  path(x,y) are removed
• Vx ?(j1 to i) Bj
• Vy ?(ji 1 to k) Bj

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Extension to Complex Cases
Reducing Undirected Graph to Simplest Case

• We know that all u ? Vx , path(x,u) is in Vx
• Thus, d(x,u) d(x,u G \ e.I)
• Not so obvious that path (v,y) is in Vy for all v
  ? Vy
• Lemma 1 d(v,y) d(v,y G \ e.I)
• Algorithm can now be applied after making
  adjustments to right and left indices!

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Extension to Complex Cases
Undirected Networks

• Theorem 3
• Given a directed network G with m edges and a
  pair of vertices (x,y), we can compute d(x,y G \
  e) for each edge e ? path(x,y) in total time O(n
  log n m) plus the time to compute a shortest
  path tree in G

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Critiques

• Assumptions do not reflect real world networks
  Model Inaccuracy
• Multiple Sources Sinks Mirror Sites
• Sending Several Users information over one link
  Multicasting
• Users sending multiple packets Capacity Modeling
• Intentionally exceeding stable bandwidth Alinear
  Cost Functions
• Multiple Edge ownership Combinatorial Auctions
• No empirical results Graph Simulation, Auctions,
  Networks
• Improved strictly worst-case analysis
  Average-Case Analysis?
• VCG are expensive what is the mechanism cost?
  VCG not BB
• Complexity claims are asymptotic What is
  real-world n?

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Additional Discussion Questions

• Can this algorithm be applied effectively to
  other domains, such as re-routing around crashing
  links? Conversely, can the literature on
  crashing links be imported effectively to this
  domain?
• Can this algorithm be adapted to distributed
  computing, giving up a central mechanism
  organizer?
• How can traditional distributed computing survive
  selfish manipulations? TCP is decidedly not
  strategyproof!
• For what cost functions is exploitable TCP
  optimal?