A Truthful 2-approximation Mechanism for the Steiner Tree Problem

1
A Truthful 2-approximation Mechanism for the
Steiner Tree Problem
2
The Steiner Tree problem

• INPUT
• Undirected, weighted graph G(V,E,c)
• N ? V set of terminal nodes
• OUTPUT
• a tree T spanning N of minimum total cost, i.e.,
  which minimizes c(T)?e?Tc(e)

3
An example
Na,b,c
5
T
c
a
2
3
1
4
1
T optimal Steiner tree
1
5
2
2
2
1
3
2
b
3
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About Steiner Tree problem

• It is NP-hard
• It is approximable within 1.55
• Robins, Zelikovsky (2000)

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Steiner Tree game

• Each edge e is controlled by a selfish agent Ae
• Only Ae knows te (weight of the edge e)
• We want to compute a good solution w.r.t. the
  true costs
• We do it by designing a mechanism
• Our mechanism
• Asks each agent to report her cost
• Computes a solution using an output algorithm
  g(?)
• Hands payments pe to each agent Ae using some
  payment function p

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More formally

• Feasible solutions
• F set of all trees in G spanning N
• Type of agent Ae
• ?e weight of edge e
• Intuition ?e is the cost Ae incurs whenever she
  uses e
• Aes valuation of T?F
• ve(?e,T) ?e if e?E(T), 0 otherwise
• SCF minimum Steiner tree of G(V,E,?) with
  terminals N

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How to design a truthful mechanism for the
problem?
Notice that the (true) total weight of a
feasible T is
?e?E ve(?e,T)
the problem is utilitarian!
VCG mechanisms apply
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VCG mechanism

• M ltg(r), p(x)gt
• g(r) arg minx?F ?j vj(rj,x)
• pe(x) for each e?E
• pe ?j?e vj(rj,g(r-e)) -?j?e vj(rj,x)

What to do?
g(r) should compute an optimal solution!!!
we look for (approximated) one-parameter
mechanisms!
L. Gualà, G. Proietti, A Truthful
(2-2/k)-Approximation Mechanism for the Steiner
Tree Problem with k Terminals, COCOON05
9

• Our goal to design a mechanism satisfying
• g(?) is monotone
• Solution returned by g(?) is a good solution,
  i.e. an approximated solution
• g(?) and p(?) computable in polynomial time

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A 2-approximation algorithm

• Build the weighted complete graph D with node set
  N
• For every pair of nodes in N, w(a,b) dG(a,b)
• Compute an MST M of D
• Expand any edge of M with the corresponding
  shortest path. This defines a subgraph H of G
• Return any feasible tree T which is a subgraph of
  H

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An example
Na,b,c
5
D
a
2
3
1
3
1
10
15
2
12
2
4
8
2
b
c
13
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An example
Na,b,c
5
D
a
2
a
3
1
3
M
1
7
10
7
15
2
12
2
4
8
2
8
b
c
b
c
13
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An example
Na,b,c
5
D
a
2
a
3
1
H
3
M
1
7
10
7
15
2
12
2
4
8
2
8
b
c
b
c
13
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An example
Na,b,c
5
D
a
2
a
3
1
T
3
M
1
7
10
7
15
2
12
2
4
8
2
8
b
c
b
c
13
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Theorem Takahashi, Matsuyama,80
The algorithm is a 2-approximation algorithm for
the Steiner tree problem, i.e. it returns a
solution with cost at most twice the cost of the
optimal solution.
Is the algorithm monotone?
..good question!
It depends on Steps 3 and 4
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Definition of g(?)

• We modify the 2-apx algorithm in order to
  garantee
• Monotonicity
• Efficiency w.r.t. the computation to the
  threshold values

Idea we can guarantee an acyclic expansion of M
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High level description of g(?)

1. Compute D MMST(D)
2. Let r be any terminal node. T(r,?). N(T)r
3. At each step g(?) reaches a new terminal node b
   (until N(T)N)
4. Choose an edge (a,b) of M s.t. a ? N(T) and b
   ?N(T)
5. Try to expand (a,b) whithout forming cycles
6. If this is not possible, look for an edge (a,b)
   of D s.t.
7. (a,b) admits an acyclic expansion
8. a ? N(T)
9. w(a,b)w(a,b)
10. MM\(a,b) ? (a,b)
11. Expand (a,b) N(T) N(T) ? b
12. Return T and M

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Expanding (a,b)

• An edge (a,b) admits an acyclic expansion w.r.t.
  a current tree T if there is a shortest path
  ?PG(a,b) s.t.
• ?a,x is already in the current tree
• ?x,b passes through no node of T (except x)
• Notation
• Let e(a,b) ? E(M)
• Removing e splits M into two subtrees
• N(a) the node set of the subtree containig a
• N(b) remaining nodes
• CM(a,b) non-tree edges crossing the cut
  (N(a),N(b))

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Expanding (a,b)

• Let ?PG(a,b)
• Let x be the first node of T encountered along ?
  (traversing ? from b to a)
• Let (a,b) be any edge of the current M s.t.
• a,b ? N(T)
• ?PT(a,b) passes through x
• a ? N(a)
• N(T)N(T) ? b TT? ?x,b
• if (a?a) then MM\(a,b) ? (a,b)

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Correctness (sketch)

• T is acyclic
• At any time M is an MST of D
• we have to show that the expanding step is
  correct
• There are 2 cases
• aa
• a ? a

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Case 1 (aa)
M
T
c
b
a
c
a
b
?
x
a
d
d
a
b
b
?PG(a,b)
?PT(a,x) ? ?x,b is an alternative shortest
path from a to b
we expands (a,b) with ? (acyclic expansion)
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Case 2 (a ? a)
M
a
a
T
c
a
c
x
?
b
d
b
d
a
b
b
it must be dG(a,x)dG(a,x)
? w(a,b)w(a,b)
?PT(a,x) ? ?x,b is a shortest path from a
to b
we swap (a,b) and (a,b) in M
and we expands (a,b) with ? (acyclic
expansion)
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Lemma
The algorithm g( ) is monotone
proof
It suffices to prove that any non-selected edge e
it still non-selected when Ae raises her bid
Notice e does not belong to any shortest path
selected in M
Thus, if Ae raises her bid, the only edges in D
which increase their weight are edges in E(D)\E(M)
M remains an MST of D
the solution computed by g() is the same and e
is not selected
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Computing the payments
we have to pay each selected edge e as its
threshold value

• How much can Ae raise her bid
• before exiting from the computed solution?

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Example 1
T
M
a
a
a
a
a
a
PG(a,b)
PG(a,b)
w(a,b)
?be
w(a,b)
?be
w(a,b)
e
b
b
b
PG(a,b)
b
b
b
Ae raises her bid of ?be
(a,b) becomes lighter than (a,b)
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Example 1
M
a
T
a
a
a
a
a
w(a,b)?be
w(a,b)
e
w(a,b)?be
b
b
b
b
b
b
g(?) selects (a,b) and e exits form the solution
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Example 2
T
M
a
a
a
a
a
a
PG(a,b)
PG(a,b)
w(a,b)
?be
w(a,b)
dG-e(a,b)
w(a,b)
e
b
b
b
PG(a,b)
b
b
b
Ae raises her bid of ?be
PG-e(a,b) becomes shorter than PG(a,b)
(a,b) is still a lightest edge crossing the cut
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Example 2
T
M
a
a
a
a
a
a
PG(a,b)
dG-e(a,b)
PG-e(a,b)
?be
w(a,b)
w(a,b)
e
b
b
b
PG(a,b)
b
b
b
g(?) still selects (a,b) but e exits from the
solution
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more formally
The image of e on M Im(e)(a,b) ? E(M) e ?
PT(a,b)
(a,b)
?e
The threshold for e w.r.t. the edge (a,b)
? Im(e) is defined as
(a,b)
?e
be min (dG-e(a,b) dG(a,b)), (swap(a,b)(e)
dG(a,b))
where
swap(a,b)(e) min dG-e(a,b)
(a,b) ? CM(a,b)
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Threshold of e
If Im(e)(a1,b1), , (ah,bh)
(ai,bi)
?e max
?e
i1,..,h
Easy to see ?e can be computed in polynomial time
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Theorem
The running time of the mechanism is O((nk2)m
log ?(m,n). The space used is O(n2).