Transcript Variations of the Prize- Collecting Steiner Tree Problem

Variations of the PrizeCollecting Steiner Tree
Problem
Olena Chapovska and Abraham P. Punnen
Networks 2006
Reporter: Cheng-Chung Li
2006/08/28
Outline
Introduction
Polynomial Algorithms
A linear Algorithm for P(4)
NP-Hard Problems
Conclusion
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Introduction to PCSTP
Let G be an undirected graph with node set V(G) and
edge set E(G)
Let F be the family of all trees in G
For each edge eE(G), a nonnegative cost ce is
prescribed; for each node iV(G), a nonnegative
weight wi is also prescribed
Then the prize collecting Steiner tree problem(PCSTP)
is to
– Minimize
c
eE (T )
e

w
iV (T )
i
1
– Subject to TF
2
PCSTP=10
3
4
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About PCSTP
PCSTP has applications in the design of fiber-optic
networks
PCSTP was first considered by Goemans and Williamson
in 1995
If wi=M, a large number for all iV(G), PCSTP reduces to
the minimum spanning tree
However, PCSTP in general is NP-hard, since the Steiner
tree problem is a particular case of it
Some people developed polynomial time approximation
algorithms to solve PCSTP with guaranteed performance
bound
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In This Paper
O(m+nlogn)
f1 (T )  max {ce } 
The paper considers seven
new variations of the
PCSTP. The ith problem
P(i), 1i7, in this class
can be defined as
eE (T )
w
iV (T )
i
f 2 (T )  max {ce }  max {wi }
eE (T )
iV (T )


f 3 (T )  max max {ce },  wi 
iV (T )
 eE (T )



f 4 (T )  max max {ce }, max {wi }
iV (T )
 eE (T )

f 5 (T )   ce  max {wi }
– P(i): Minimize fi(T)
– Subject to TF
eE (T )
The objective function of
problems P(1) to P(7) can
be given as follows:
iV (T )


f 6 (T )  max   ce ,  wi 
eE (T ) iV (T ) 


f 7 (T )  max   ce , max {wi }
eE (T ) iV (T )

NP-hard
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Related to Other Problems
If wi=M, for all iV(G), P(1), P(2), P(3),
and P(4) reduce to the bottleneck spanning
tree problem(BSP)
Let V*V(G) be a set of prescribed nodes
in G. Choose wi=M for all iV* and wi=0
for all iV(G)\V*, P(1), P(2), P(3), and P(4)
reduce to the bottleneck Steiner tree
problem(BSTP)
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Polynomial Algorithms
Let S be a subtree of V(G) and T be a spanning
tree of G. A subtree T’ of T is said to be cover S if
SV(T’). A subtree T’’ of T that covers S is
minimal, if there is no subtree T’’’ of T that covers
S such that V(T’’’)V(T’’)
Let T0 be a tree in G containing at least one edge
and T* be a minimal spanning tree of G. Let T^ be
a minimal subtree of T* that covers V(T0). Note
that V(T0)V(T^) and V(T^)\V(T0) may or may
not be empty
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Polynomial Algorithms
Lemma 1: maxeT0{ce}maxeT^ {ce}
– Choose an edge edT^ such that ced= maxeT^
{ce}, if edT0, the result follows immediately.
– Otherwise, let P[a,b](respectively, [a,b]) be
the unique path of node a, b in T^(respectively ,
T0). There exists two nodes u and v in V(T0)
such that edP[u,v] but ed[u,v], i.e.,
P[u,v][u,v]
P[u,v]
– So
u
v
[u,v]
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Polynomial Algorithms
Theorem 2: Let T* be a minimum spanning tree of G. Then,
there exists an optimal solution to P(i), i=1,2,3,4 which is a
subtree of T*
– We first prove the case for i=1
– Suppose no subtree of T* is an optimal solution to P(1) and let T0
be an optimal solution to P(1). Let T^ be the minimal subtree of T*
covering V(T0). By lemma 1, maxeT0{ce}maxeT^ {ce}
– Because V(T0)V(T^) and wi0, sum_{i\nin V(T^0)}\geq
\sum_{i\nin V(T^0)}
– So, f1(T^)f1(T0) and hence, T^ is also an optimal solution to P(1).
This contradicts the fact that no subtree of T* is an optimal
solution to P(1)
– Also, V(G)/V(T^)  V(G)/V(T0), P(2), P(3), P(4) follows using
similar arguments.
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Greedy Algorithm for P(1)
Let T* be a minimum spanning tree of G and
t1<t2<…<t|E| be an ascending arrangement of all
distinct edge costs of T*
Initially, consider isolated nodes as “current forest”
and designate it F*; Choose a node with largest
weight in T* and designate it as the “best solution
sofar”
Introduce the edges that cost t1<t2<…<t|E to F*,
update F* and the “best solution sofar”(if
necessary)
The process is continued until F* becomes T*
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The Time Complexity of Algorithm
Once T* is given, the greedy algorithm can be
implemented in O(nlogn) using appropriate data structures.
Thus P(1) can be solved in O((m,n)+nlogn), where (m,n)
is the complexity of computing the minimum spanning tree
Note that a minimum spanning tree of G can be identified
in O(m+nlogn) time, and hence, P(1) can be solved in
O(m+nlogn) time.
Of course, it is possible to incorporate the above greedy
algorithm within the minimum spanning tree computations
itself, without explicitly computing a minimum spanning
tree first.
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NP-hard Problems
Theorem P(5) and P(7) are NP-hard
– Let V’V(G). By choosing each nodes in V’
have weight M, and nodes in V(G)/V’ have
weight 0
– So, an optimal solution to P(5) solves the
Steiner tree problem on G and the Steiner tree
problem is NP-hard
– A similar proof can be given for the case of P(7)
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NP-hard Problems
v1
vn
an
a1 v
0
a2
v2
Theorem P(6) is NP-hard
– We consider the PARTITION problem first: Given n
numbers a1,a2,…,an, the PARTITION problem is to find
a partition S1,S2 of {1,2,…n} such that
\sum_{jS1}aj=\sum{jS2}aj or declare that no such
partition exists
– It can be verified that optimal objective funcation value
of P(6) on this instance is ½ \sum_{i=1}^{n}ai
precisely when the required partition exists
– Because PARTITION is NP-hard, P(6) is NP-hard too
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NP-hard Problems
Theorem P(1) and P(3) are NP-hard for
arbitrary node weights
100
1
10
-300
1
100
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NP-hard Problems
W first consider P(1) and reduce the node-weighted Steiner
tree problem (NWSTP) to it. The NWSTP can be defined
as follows
– Let G’ be a graph and for each iG’ a weight xi is prescribed. Let
V’ be a given subset of V(G’). Then the NWSTP is to find a tree T’
in G’ with V’V(T’) such that \sum_{iV(T’)\V’}xi is minimized.
Choose G=G’ and wi=M for all iV’. Set c(e)=1 for all
eE(G) and wi=-xi for each iV(G)-V’. It can be verified
that an optimal solution to this instance of P(1) will solve
NWSTP
Using similar arguments, it can be shown that P(3) is NPhard for arbitrary node weights
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Conclusion
P(1)~P(4), O(m+nlogn)
P(5)~P(7), NP-hard
In problems P(1) to P(7) if trees are replaced by st paths of G, the resulting problems are NP-hard. It
follows from the fact that by choosing wi=M for
all iV(G) where M is a large number, optimal s-t
paths are forced to include all nodes of G and
hence solving the Hamiltonian path problem on G
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