RNC Approximation Algorithms for the
Steiner Problem
Hans Jurgen Pr omel
Institut fur Informatik Humb oldt Universit at zu Berlin
Berlin Germany proemel informatik hu berlin de
Angelika Steger
Institut fur Informatik TU Munc hen
Munc hen Germany steger informatik tu muenchen de
Abstract In this pap er we present an RN C algorithm for nding a min
imum spanning tree in a weighted uniform hyp ergraph assuming the edge
weights are given in unary and a fully p olynomial time randomized approxima
tion scheme if the edge weights are given in binary From this result we then
derive RN C approximation algorithms for the Steiner problem in networks with
approximation ratio for all
Intro duction
In recent years the Steiner tree problem in graphs attracted considerable attention as
well from the theoretical p oint of view as from its applicabili ty e g in VLSI layout It
is rather easy to see and has b een known for a long time that a minimum Steiner tree
spanning a given set of terminals in a graph or network can b e approximated in p olyno
mial time up to a factor of cf e g Choukhmane or Kou Markowsky Berman
After a long p erio d without any progress Zelikovsky Berman and Ramaiyer
Zelikovsky and Karpinski and Zelikovsky improved the approximation factor
step by step from to
In this pap er we present RN C approximation algorithm for the Steiner problem
with approximation ratio for all The running time of these algo
rithms is p olynomial in and n Our algorithms also give rise to conceptually much
easier and faster though randomized sequential approximation algorithms than the
Karpinski Zelikovsky approach which almost match their approximation factor
The core of our algorithm is a new RN C algorithm for nding a minimum spanning
tree in uniform hyp ergraphs The problem of nding a minimum spanning tree in
a given graph is well studied and known to b e sequentially solvable almost in linear
time Moreover this problem can also b e solved e ciently in parallel On the
other hand the problem of nding a minimum spanning tree in a k uniform hyp ergraph
is known to b e NP hard whenever k The status of the case k however is 1
not yet completely decided For the unweighted case Lov asz provided a very com
plicated O n algorithm which was later improved to a O n algorithm by Gab ow
and Stallmann Here n denotes the numb er of vertices of the hyp ergraph Also
Lov asz presented a conceptionally simple randomized algorithm for the unweighted
case by reducing it to a sequence of computations of determinants of appropriate matri
ces For the weighted case Camerini Galbiati and Ma oli generalized the approach
of Lov asz to obtain a randomized pseudo p olynomial time algorithm
In this pap er we present an RN C algorithm for nding a minimum spanning tree in
a weighted uniform hyp ergraph with edge weights in unary This algorithm simpli es
to an RN C algorithm for deciding the existence of a spanning tree in an unweighted
uniform hyp ergraph and implies a fully p olynomial approximation scheme if the edge
weights are given in binary
To achieve this result we combine ideas from Mulmuley Vazirani Vazirani
which they used to obtain an RN C algorithm for nding a p erfect matching in a given
graph with ideas from Lov asz resp Camerini Galbiati and Ma oli where
they presented a random pseudo p olynomial time algorithm for the general problem of
nding a base of sp eci ed value in a weighted represented matroid sub ject to parity
conditions
Minimum Spanning Trees in Uniform Hyp ergraphs
A hypergraph H V F is a generalization of graphs where F is an arbitrary family
of subsets of V and not just a family of element subsets An r uniform hyp ergraph
is a hyp ergraph all of whose edges have cardinality exactly r
Many notions and results of graph theory generalize to hyp ergraphs Here we just
need cycles and trees A cycle of length l in H is a sequence x e x e
l l
x e of vertices and edges such that the x are distinct vertices the e are distinct
l l i i
edges x e e and x e e for all i l A hyp ergraph H is a tree
l i i i
i H is connected and contains no cycles A spanning tree of a hyp ergraph H is a
subhyp ergraph T of H that is a tree and satis es V T V H In contrary to
graphs not every connected hyp ergraph contains a spanning tree
Minimum Spanning Tree Problem MST
Input A weighted hyp ergraph H V F w where w F N
Output Find a spanning tree of H of minimum weight
Restricted to uniform hyp ergraphs that is to graphs the minimum spanning tree
problem is easily solved e ciently sequentially as well as in parallel On the other hand
a trivial reduction from Exact Cover by Sets shows that for unweighted uniform
hyp ergraphs even deciding whether there exists a spanning tree is NP complete The
status of the minimum spanning tree problem for uniform hyp ergraphs is not yet
completely resolved Lov asz showed that the problem is in P for unweighted
uniform hyp ergraphs His algorithm however is quite complicated and not very
e cient Later Gab ow and Stallmann reduced the complexity from O n to
O n In Lov asz presented a conceptionally simple randomized algorithm for
the unweighted case by reducing it to a sequence of computations of determinants of
appropriate matrices Camerini Galbiati and Ma oli generalized this approach 2
of Lov asz to obtain a randomized pseudo p olynomial time algorithm for the weighted
case
In this section we develop a randomized parallel algorithm for the minimum span
ning tree problem in uniform hyp ergraphs It is also based on the algebraic approach
of Lov asz The parallelization is based on ideas from Mulmuley Vazirani Vazi
rani We start with some de nitions
T
Let A a b e a skew symmetric matrix i e A A of size n n and
ij
let P b e the set of all partitions of f ng into two element sets For an element
p ffi i g fi i gg of P we denote by p the sign of the p ermutation
n n
n n
i i i i
n n
and by p the pro duct
n
Y
p a
i i
2j 1 2j
j
One easily veri es that p p is indep endent of the order of the classes and the
order within the classes of p Therefore
X
pf A p p
p P
is well de ned It is called the pfa an of A A well known result from linear algebra
cf e g says
Lemma If A is a skew symmetric matrix A of size n n and B an arbitrary
n n matrix then
T
det A pf A and pf B AB det B pf A
2
n
Lemma Let m n and a b a b be vectors in R and let x x
m m m
be m indeterminants Then the n n matrix
m
X
T T
A x a b b a
i i i
i i
i
is skew symmetric and satis es
X
pf A x x det a jb ja jb
i i i i i i
1 n 1 1 n n
i i m
1 n
With the notation j we just mean the concatenation of the column vectors a and b
i i
j j
Let H V F b e a uniform hyp ergraph on n vertices For every edge
f fi j k g in F we pick one vertex arbitrarily say i and let e fi j g ande fi k g
f f
Let G V E b e a multi graph on the same vertex set as H and with edge set
E fe e j f F g
f f
In fact neither Lov asz nor Camerini Galbiati and Ma oli study the minimum
spanning tree problem directly They b oth consider the matroid parity problem which contains
the minimum spanning tree problem in uniform hyp ergraphs as a sp ecial case 3
Fact A set ff f g F forms a spanning tree in H if and only if fe e
n f f
1 1
e e g forms a spanning tree in G 2
f f
n n
That is the problem of nding a minimum spanning tree in a uniform hyp ergraph
is equivalent to the problem of nding a minimum spanning tree in a multi graph
where the edges are paired i e either b oth are in the tree or none
For every pair of edges e e we de ne two n dimensional vectors a and b as
f f f f
follows
if i e if i e
f f
a b
f i f i
otherwise otherwise
Note that the a s and b s are essentially the incidence vectors of e and e except
f f f f
that the n st comp onent has b een cut o The following fact resembles a well
known prop erty of the incidence matrix of a graph
Fact Let f f be n edges in F Then
n
if f f form a spanning tree in H
n
j det a jb j ja jb j
f f f f
1 1 n n
2
otherwise
Finally let A denote the n n matrix
X
w f T T
a b b a A
f f
f f
f F
With the ab ove facts and notation at hand it is now relatively straightforward to design
an algorithm for constructing a minimum spanning tree whenever this tree is unique
Lemma Assume H is a uniform hypergraph on n vertices and w F N is
spanning tree T of minimum weight say a weight function such that H has a unique
w
0
is the highest power of that divides det A Moreover w Then det A and
w f T T
if we let A A a b b a for al l f F then
f f f
f f
det A
f
is even f T if and only if
w
0
Proof Combining Lemma and Fact we deduce that
X
w T
det A pf A
T
T
where the sum is over all spanning trees T of H and f g for all such trees
T
Hence
X
w i
0
det A c with c and appropriate c c Z
i
i
The rst part of the theorem follows For the second part just observe that A is the
f
matrix corresp onding to the hyp ergraph H f The reasoning ab ove therefore implies
that
w w
0 0
c if f T
f
det A
f
w
0
c if f T
f
for appropriate constants c Z 2
f 4
To achieve the uniqueness of a minimum spanning tree in general hyp ergraphs we
use randomization
Lemma Let H V F be a hypergraph on n vertices For every vertex v V
choose uniformly and independently at random an integer r v from n and de ne
P
the weight w f of an edge f F as w f r v Then
v f
Prob There exists a unique edge of minimum weight
2
Corollary Let H V F w be a weighted uniform hypergraph on n vertices
containing at least one spanning tree For every edge f F choose uniformly and
n
independently at random an integer r f from and de ne a weight w E
N as fol lows
w f n w f r f
Then
Prob There exists a unique minimum spanning tree with respect to w
Proof Construct a hyp ergraph H as follows The vertex set of H consists of all edges
of H and the edges of H corresp ond to all spanning trees of H Apply Lemma
with resp ect to the hyp ergraph H Then the weight of an edge in H is at most
n
n n and with probability at least there exists a minimum weight edge
On the other hand after scaling the weights w f of the edges in H by n the
weight of a spanning tree in H is a multiple of n That is by adding the values r f
we maintain the order relation on the spanning trees of H of di erent weight We just
disturb the order of the spanning trees in H which have the same weight a little
just enough to reach uniqueness with resp ect to w 2
Algorithm Minimum Spanning Trees in Uniform Hypergraphs
Input A weighted uniform hyp ergraph H V F w on n vertices
Output A spanning tree T of H or Failure
Compute the weight function w as de ned in Corollary
Compute the matrix A as de ned in and let w b e the largest integer such
w
0
that divides det A
Let T
For every edge f F do in parallel
det A
f
Compute
2w
0
det A
f
is even then T T ff g if
2w
0
if T is a spanning tree of H then return T
else return Failure
Observe that step involves most of the computational e ort Here one can use
e g Pan s randomized matrix inversion algorithm which requires O log n time
and O n l pro cessors for computing the determinant of an n n matrix whose
entries are l bit integers As the entries of the matrix A are of size exp onential
in O n max w f step needs thus in total O log n time and O m n
f F
max w f pro cessors Combining this observation with Lemma and Corollary
f F
we obtain the following result 5
Theorem For al l uniform hypergraphs which contain at least one spanning tree
The run Algorithm returns a minimum spanning tree with probability at least
ning time of the algorithm is O log n and it uses at most O m n max w f
f F
processors In particular if the weight function w is polynomial ly bounded in n or if
the weights are given in unary this yields an RN C algorithm for nding a minimum
spanning tree in a uniform hypergraph 2
Corollary For every there exists a randomized paral lel algorithm with run
ning time O log n and O m n processors that returns for al l weighted
uniform hypergraphs H with at least one spanning tree a spanning tree T such that
w T mst H with probability at least Here mst H denotes the length of
a minimum spanning tree in the hypergraph H
Proof We apply the usual scaling technique Let H V F w b e a weighted
uniform hyp ergraph on n vertices and let w max w f For a given we
max f F
set t w n and de ne a new hyp ergraph H V F w with the same vertex
max
and edge set and weight function w given by
w f
w f for all f F
t
Observe that by construction
n
mst H mst H
t
n
and that w max w f O Hence by Theorem Algorithm returns
f F
max
in O log n time using a minimum spanning tree T in H with probability at least
m n pro cessors If T is indeed a minimum spanning tree in H we can b ound O
its weight in H as follows
w T t w T t mst H mst H tn mst H w
max
That is if we could guarantee that mst H w we would b e home Unfortunately
max
this is not true in general But another trick helps here
Let w w b e the o ccurring weights in H sorted in increasing order For
s
every i s we de ne a hyp ergraph H V F w by deleting all edges from
i i
H which have weight larger than w That is we let F ff F j w f w g
i i i
Furthermore let T b e an arbitrary minimum spanning tree in H and let i b e
opt
de ned such that w is the weight of an edge of maximum weight in T Then
i
0
opt
clearly w mst H mst H That is if use the scaling technique outlined ab ove
i i
0 0
to compute in parallel a minimum spanning tree in all hyp ergraphs H and return
i
of the at most s spanning trees those with minimum weight this tree T will b e with
a spanning tree in H such that w T mst H The probability at least
running time of this mo di ed algorithm is of course still O log n while the numb er
of pro cessors increases by a factor of at most m to O m n 2
Corollary There exists a ful ly polynomial randomized paral lel and sequential ap
proximation scheme for nding a minimum spanning tree in uniform hypergraphs
2 6
Remark It is not true that if Algorithm outputs a spanning tree that this tree
is then necessarily a minimum spanning tree The following hyp ergraph H V F
provides an example 1 2
6 7 3
5 4
Here the solid edges f g f g and f g have weight while the dashed
edges f g f g and f g have weight H has exactly minimum spanning
trees of weight Hence the algorithm correctly determines w Now consider H
minus an edge f If f is an edge of weight then H f contains minimum spanning
w
0
is even for all solid edges On the other hand if f is an edge of trees So det A
f
w
0
weight then H f contains only one minimum spanning tree So det A is o dd
f
for all dashed edges Hence with resp ect to the weight function w w the algorithm
returns the three solid edges which do form a spanning tree but not a minimum one
Approximation Algorithms for the Steiner Problem
Let G V E b e a graph and K V b e a subset of the vertex set A subgraph T of
G is a Steiner tree for K if T is a tree containing all vertices of K i e K V T
such that all leaves of T are elements of K A Steiner minimum tree for K in G is a
Steiner tree T such that jE T j in unweighted graphs resp w T in weighted graphs
is minimum
Steiner Problem in Networks SPN
Input A network a weighted graph N V E w and a set K V
Output A Steiner minimum tree for K in N that is a Steiner tree T such that
w T min fw T j T a Steiner tree for K in N g
If the input is restricted to unweighted graphs G V E and sets K V we sp eak
of the Steiner Problem in Graphs SPG Given a graph G V E or network
N V E w and a terminal set K we denote by smt G K resp smt N K the
length of a Steiner minimum tree for K in G resp N Note that for K V the Steiner
problem is exactly the minimum spanning tree problem In the following we denote the
length of a minimum spanning tree in a hyp ergraph H by mst H
The computational complexity of the Steiner tree problem in graphs and networks
varies considerably with the structure of the underlying graph and the cardinality of
the terminal set K While it can b e easily solved whenever jK j or K V and for
certain classes of graphs it is rather di cult in general
The Steiner tree problem in networks was among the rst problems shown to b e
NP hard in the seminal pap er of Karp Bern and Plassmann then proved
that even the sp ecial problem with edge weights restricted to the values and is 7
MAX S N P hard A consequence of the new characterization of the class NP by
Arora Lund Motwani Sudan and Szegedy is thus that there exists no p olynomial
time approximation scheme for the Steiner tree problem unless P NP Hence unless
P NP the b est p erformance ratio attainable by a p olynomial time algorithm is a
constant larger than It remains a challenging question how close to this p erformance
ratio can b e
Let N V E w b e a network and K V a terminal set With resp ect to this
Steiner problem we de ne for all r a weighted hyp ergraph H N K K F w
r r r
on the vertex set K as follows The edge set F consists of all subsets of K of cardinality
r
at most r and the weight w f of an edge f F is the length of a Steiner minimum
r r
tree for f in the network N
Fact Let N V E w be a network with terminal set K and let H N K be as
r
de ned above Then
mst H N K smt N K
r
Proof Let T b e a minimum spanning tree in H N K For every edge f in T cho ose
r
a Steiner minimum tree T for f in N Then the fact that T is a spanning tree in
f
S
T is a connected subgraph of V E which contains H N K implies that S
f r
f T
all vertices in K 2
Reconsidering the pro of of Fact we observe that given a spanning tree T in
H N K one can easily determine a Steiner tree for K in the original network N of at
r
most the same length We just take the union of Steiner minimum trees for all f T
and delete edges until the obtained graph is a tree in which all leaves are terminals
Let denote the least upp er b ound of the ratio mst H N K smt N K for
r r
all networks N V E w and terminal sets K V One easily sees that
cf e g or Obtaining is considerably more di cult Zelikovsky shows
that and considering appropriate binary trees one deduces easily that in fact
r
For general r N Du Zhang and Feng proved that implying
r
in particular that for r and nally Borchers and Du proved that
r
t t t t
t l t l for r l and l
r
Observe that for all constants r N the hyp ergraph H N K can b e constructed
r
r
in p olynomial time There are less than k subsets of K of cardinality at most r
where k jK j For each of these subsets a Steiner minimum tree can b e found in
O n log n nm with the algorithm of Dreyfus and Wagner A plausible approach
for designing an approximation algorithm is therefore to solve the minimum spanning
tree problem in H N K and deduce from that a Steiner tree for K in N as outlined
r
ab ove This would give an approximation algorithm with ratio As however nding
r
minimum spanning trees in r uniform hyp ergraphs is NP complete for all r this
reduction to the spanning tree problem is not really helpful except for the case k
Zelikovsky showed that in the sp ecial case of H N K one can use a greedy
approach to nd a spanning tree T in H N K of length at most mst H N K
In fact this is not quite obvious at this p oint as the reduction from the Steiner tree problem
generates only sp ecial spanning tree problems It is however easy to see that also these sp ecial
spanning tree problems are NP hard to solve cf e g 8
times the mst H N K and thus a Steiner tree of size at most
length of a Steiner minimum tree
Berman and Ramaiyer found a di erent pro cedure which makes also use of the
hyp ergraphs H N K for r They obtained an algorithm with p erformance ratio
r
h
X
i i
i
i
for all h Zelikovsky invented a so called relative greedy heuristic for ap
proximating mst H N K that yields an approximation algorithm for the length of
r
a Steiner minimum tree with p erformance ratio ln A slight further im
provement then led to a ratio of for any p ositive see Karpinski and
Zelikovsky
In order to use the algorithm of the previous section for solving the spanning tree
problem in H N K we have to reduce the spanning tree problem in hyp ergraphs
with edges containing at most three vertices to a corresp onding problem in a uniform
hyp ergraph i e a hyp ergraph where all edges consist of exactly three vertices This
however can easily b e achieved
Reducing the spanning tree problem to uniform hypergraphs Let H V F w b e a
weighted hyp ergraph on n vertices such that every edge contains at most three vertices
Construct a weighted uniform hyp ergraph H V F w as follows The vertex set
V consists of all vertices of V plus n new vertices z z and
n
F F fe fz g j e F jej i n g
i
fv fz z g j v V i j n g
i j
New triples containing exactly one z vertex will b e called typ e I triples while those
containing two z vertices are of typ e I I The weight function w is de ned as follows
w f if f F
w f
w e M if f is a typ e I triple with e F and e f
M if f is a typ e I I triple
Fact The above reduction has the fol lowing properties
i Every spanning tree T in H gives rise to a spanning tree T in H of weight w T
w T n M by replacing every edge of cardinality in T by a type I triple by
adding di erent z vertices and adding type II triples until every z vertex is contained
in exactly one triple
ii If T is a spanning tree in H of weight w T nM then every z vertex is covered
by exactly one triple of T Thus T corresponds to a spanning tree T of H of weight
w T w T n M
iii Let M n max w f Then a minimum spanning tree in H of length less than
f F
nM corresponds to a minimum spanning tree of H whereas the fact that the length of
a minimum spanning tree in H is at least nM indicates that H contains no spanning
tree Furthermore if H is the complete hypergraph e g the hypergraph H N K in
the reduction from the Steiner tree problem then choosing M max w f su ces
f F
to achieve these properties 2 9
Algorithm Steiner Tree Problem in Graphs
Input A connected network N V E w and a terminal set K V
Output A Steiner tree for K or Failure
Compute the hyp ergraph H N K
Transform the hyp ergraph H N K to the corresp onding uniform hyp ergraph
H
Use the algorithm of the previous section to nd a spanning tree T in H
If this algorithm returns Failure then stop
Transform T to a spanning tree T in H N K according to Fact
Transform T into a Steiner tree S for K in N return S
For an implementation of step observe rst that a Steiner minimum tree for a two
element set fx y g is just a shortest x y path Furthermore a Steiner minimum tree for
a three element set fx y z g is the union of three shortest paths Namely a shortest
x w path plus a shortest y w path plus a shortest z w path where w is an appropriate
vertex in V As the all pairs shortest path problem can b e solved in O log n time
on O n pro cessors we conclude that step can certainly b e achieved in O log n
time and O n pro cessors Step is easily implemented within the same time b ound
Note that the weight of an edge in the hyp ergraph H N K and hence also in the
hyp ergraph H is less than n max w e Theorem thus implies that O log n
e E
time and O n max w f pro cessors su ce for step As steps and are again
f F
easily implemented within this time b ound we obtain the following theorem
Theorem Algorithm is a randomized paral lel algorithm which returns with prob
ability at least a Steiner tree S for K such that w S smt N K The algorithms
runs in O log n time using O n max w f processors In particular if the
f F
weight function w is polynomial ly bounded in n or if the weights are given in unary
this yields an RN C approximation algorithm for the Steiner Problem in Networks
with performance ratio 2
Corollary There exists an RN C approximation algorithm with performance ratio
for the Steiner Problem in Graphs 2
For networks we can also use the algorithm of Corollary instead of those of
Theorem to obtain an RN C approximation scheme for p erformance ratio
Corollary For every there exists a randomized paral lel algorithm that given
a network N V E on n vertices and a terminal set K returns in O log n steps
processors with probability at least a Steiner tree T such that using pol y n
smt N K T
2
Of course the algorithm can also b e implemented as a sequential algorithm Here
one can also use the fact that the determinant of an n n matrix containing l bit
integers can b e computed on a random access machine in O n l log l bit op erations
where O n is the numb er of arithmetic op erations required to multiply two n n
matrices Currently the b est know value is 10
Corollary There exists a randomized sequential O n log n approximation al
gorithm with performance ratio for the Steiner Problem in Graphs 2
1
log
n log n Corollary For every there exists a randomized sequential O
approximation algorithm with performance ratio for the Steiner Problem in
Networks 2
In comparison the b est deterministic sequential approximation algorithm for the
Steiner Problem in Networks of Karpinski and Zelikovsky has a p erformance
ratio of which is slightly b etter than However this p erfor
mance guarantee is only achieved in the limit More precisely building on the work
of Zelikovsky Karpinski and Zelikovsky de ne a class A of approximation
k
k
algorithms such that the running time of A is b ounded by O n and the p erformance
k
ratio of A tends to for k tending to in nity For reasonable small k say k up
k
to the p erformance ratio is however still larger than
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