A Truthful 2-approximation
Mechanism for the Steiner
Tree Problem
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)
An example
a
T
c
2
4
3
1
1
1
5
2
2
1
b
N={a,b,c}
5
3
2
3
2
T: optimal
Steiner tree
About Steiner Tree problem


It is NP-hard
It is approximable within 1.55

Robins, Zelikovsky (2000)
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(٠)
Hands payments pe to each agent Ae using some
payment function p
More formally

Feasible solutions:


Type of agent Ae:



e: weight of edge e
Intuition: e is the cost Ae incurs whenever she
uses e
Ae’s valuation of TF:


F: set of all trees in G spanning N
ve(e,T)= e if eE(T), 0 otherwise
SCF: minimum Steiner tree of G=(V,E,)
with terminals N
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
VCG mechanism

M= <g(r), p(x)>:
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)
g(r) should compute an
optimal solution!!!
What to do?
…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, COCOON’05
Our goal: to design a mechanism
satisfying:

1.
2.
3.
g(٠) is monotone
Solution returned by g(٠) is a “good”
solution, i.e. an approximated solution
g(٠) and p(٠) computable in polynomial
time
A 2-approximation algorithm
1.
Build the weighted complete graph D with
node set N
1.
2.
3.
4.
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
N={a,b,c}
An example
5
a
2
3
a
1
10
15
b
3
1
12
2
4
D
8
2
13
2
c
7
7
b
8
c
N={a,b,c}
An example
5
a
2
3
a
1
10
15
b
3
1
12
2
4
D
8
2
13
2
c
7
7
b
8
M
c
N={a,b,c}
An example
5
a
2
H
3
a
1
10
15
b
3
1
12
2
4
D
8
2
13
2
c
7
7
b
8
M
c
N={a,b,c}
An example
5
a
2
T
3
a
1
10
15
b
3
1
12
2
4
D
8
2
13
2
c
7
7
b
8
M
c
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
Definition of g(٠)
We modify the 2-apx algorithm in
order to garantee

1.
2.
Monotonicity
Efficiency w.r.t. the computation to the
threshold values
Idea:
we can guarantee an acyclic
expansion of M
High level description of g(٠)
1. Compute D; M=MST(D)
2. Let r be any terminal node. T=({r},). N(T)={r}
3. At each step g(٠) reaches a new terminal node b (until
N(T)=N)
1. Choose an edge (a,b) of M s.t. a  N(T) and b N(T)
2. Try to expand (a,b) whithout forming cycles
3. If this is not possible, look for an edge (a’,b) of D s.t.
1. (a’,b) admits an acyclic expansion
2. a’  N(T)
3. w(a’,b)=w(a,b)
4. M=M\{(a,b)}  {(a’,b)}
5. Expand (a’,b); N(T) = N(T)  {b}
4. Return T and M
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’))
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.:
1.
2.
3.



4.
5.
a’,b’  N(T)
’=PT(a’,b’) passes through x
a  N(a’)
N(T)=N(T)  {b}; T=T [x,b];
if (aa’) then M=M\{(a,b)}  {(a’,b)}
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:


a=a’
a  a’
Case 1 (a’=a)
M
T
c b’
b’ c
a a’

x
d
b
a
a’
d
=PG(a,b)
’=PT(a,x)  [x,b] is an alternative
shortest path from a to b
we expands (a,b) with ’ (acyclic expansion)
b
Case 2 (a’  a)
T
x
b’
d
a’ c
c
a’
M
a

b
it must be
dG(a,x)=dG(a’,x)
a
b’
d
 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)
b
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
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?
Example 1
M
a’
a’
a
a’’
w(a’,b’) w(a,b) +be
b’
b
a’’
T
a
PG(a,b)
w(a’’,b’’) +be
b’’
e
PG(a’,b’)
b
b’ b’’
Ae raises her bid of be
(a’,b’) becomes lighter than (a,b)
PG(a’’,b’’)
Example 1
M
a
a’
a’’
a’
w(a,b)+be
w(a’,b’)
e
w(a’’,b’’)+be
b’
a’’
a
b
b’’
g(٠) selects (a’,b’)
and e exits form the solution
b’
b
b’’
T
Example 2
M
a’
a’
a
a’’
b’
b
T
a
PG(a,b)
w(a’’,b’’) +be
w(a,b)
(a,b)
w(a’,b’) dG-e
a’’
b’’
e
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
PG(a’’,b’’)
Example 2
M
a’
a’
a
a’’
w(a’,b’) dG-e(a,b)
b’
b
a’’
T
a
w(a’’,b’’) +be
b’’
PG-e(a,b)
e
PG(a’,b’)
b
b’ b’’
g(٠) still selects (a,b) but
e exits from the solution
PG(a’’,b’’)
…more formally…
The image of e on M
Im(e)={(a,b)  E(M) | e  PT(a,b)}
(a,b)
The threshold e
is defined as
for e w.r.t. the edge (a,b)  Im(e)
(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)
Threshold of e
If Im(e)={(a1,b1), … , (ah,bh)}
(ai,bi)
e = max { e
i=1,..,h
}
Easy to see:
e can be computed in polynomial time
Theorem
The running time of the mechanism is
O((n+k2)m log (m,n). The space used is O(n2).