Steiner Tree Problem in Graphs and Mixed Integer

Linear Programming-Based Approach in GAMS

MILOˇ

Sˇ

SEDA

Institute of Automation and Computer Science

Brno University of Technology

Technick´a 2, 616 69 Brno

CZECH REPUBLIC

Abstract: - The Steiner tree problem in graphs involves ﬁnding a minimum cost tree which connects a

deﬁned subset of the vertices. This problem generalises the minimum spanning tree problem, in contrast,

it is NP-complete and is usually solved for large instances by deterministic or stochastic heuristic methods

and approximate algorithms. In this paper, however, we focus on a diﬀerent approach, based on the

formulation of a mixed integer programming model and its modiﬁcation for solving in the professional

optimization tool GAMS, which is now capable of solving even large instances of problems of exponential

complexity.

Key-Words: - Steiner tree problem, NP-completeness, heuristic, performance ratio, network ﬂow,

mixed-integer linear programming, GAMS

Received: September 14, 2021. Revised: May 9, 2022. Accepted: June 5, 2022. Published: July 2, 2022.

1 Introduction

In practice, e.g., in telecommunication applica-

tions [7], [9], [2], [18], we frequently solve prob-

lems based on ﬁnding minimal networks. The

criterion of minimality can be represented by the

total costs for the implementation of the network

or by the total length of connections. These prob-

lems include, e.g., the problem of ﬁnding optimal

location of a source with respect to given sinks.

The Network Steiner tree problem (NSTP) (or

Steiner tree problem in graphs) [10], [14], [22] is

concerned with connecting a subset of vertices at

a minimal cost. More precisely, given an undi-

rected connected graph G= (V, E) with vertex

set V, edge set E, nonnegative weights associ-

ated with the edges, and a subset Bof V(called

terminals or customer vertices), the problem is to

ﬁnd a subgraph, T, which connects the vertices in

Bso that the sum of the weights of the edges in

Tis minimised. It is obvious that the solution is

always a tree and it is called a Steiner minimum

tree for Bin G.

If |B|= 2 then the problem reduces to the

shortest path problem and can be solved by Dijk-

stra’s algorithm. In the case of B=Vthe NSTP

reduces to the minimum spanning tree (MST)

problem and can be solved by Jarn´ık’s (Prim’s),

Bor˚uvka’s or Kruskal’s algorithm. All these al-

gorithms are polynomial. However, in the gen-

eral case the NSTP is NP-complete [15], [20] and

it cannot therefore be solved exactly for larger

instances, i.e. heuristic or approximate meth-

ods must be used. A Steiner minimum tree is

not a minimum spanning tree only, it can also

span some nonterminals, called Steiner vertices,

as shown in Fig. 1.

Figure 1: An example of the Streiner tree problem

in graphs, = terminals, •= Steiner vertices)

2 Approximation algorithms

As was mentioned above, the network Steiner tree

problem is NP-complete. It means that solving

the problem requires a computational time that

grows (in the worst case) exponentially with the

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problem size. Therefore we must use approxi-

mation or heuristic approaches for large scaled

instances. Many approaches are based on a re-

formulation of the problem [5], [8] [12], branch

and bound method, dynamic programming, etc.

[11], [19].

Aproximation algorithms are algorithms that

guarantee that the ”distance” of their solutions

from the optimum is in the worst case restricted

by a mathematically proved upper bound. The

quality of an approximation algorithm is mostly

measured by its performance ratio, which is given

by the ratio of then achieved solution and the op-

timum. If a Steiner tree for a set of terminals B

is computed by an approximation algorithm and

has a total length ASMT (B), Steiner minimum

tree for Bis SMT (B), then the Steiner tree prob-

lem performance ratio ϱgraph(B) is given by

ϱgraph(B) = w(ASMT (B))

w(SMT (B))

The simplest approximation algorithms for the

NSTP are distance network approximation [17],

minimum path approximation [24], and contrac-

tion approximation [21]. We will brieﬂy sum-

marise only the ﬁrst one.

A distance network approximation [3] can be

described as follows:

[To determine a Steiner mimimum tree in a

connected weighted graph G= (V, E)with a set

of terminals B]

1. Construct the complete graph KB= (B, F )

in which the weight of {i, j} ∈ Fis the short-

est distance between iand jin G.

2. Obtain a minimum spanning tree MST (KB)

of the graph KB.

3. Replace each edge {i, j}of M ST (KB) by a

shortest path between iand jin G. (The

resulting graph, G′, is a Steiner subgraph of

G since it is connected and contains B.)

4. Obtain a minimum spanning tree MST (G′)

in G′. (MST (G′) is a Steiner tree.)

5. If vis a Steiner vertex of degree 1 in

MST (G′), delete vfrom the tree MST (G′)

with its incident edge. Continue this process

by deleting one Steiner vertex at a time.

Theorem 2.1 The distance network approxima-

tion algorithm runs in O(|B||V|2)time.

Proof. Running times of the steps are:

1. If the distance graph is not known, Step 1 re-

quires time O(|B||V|2) to compute the short-

est paths from each of the |B|vertices.

2. Kruskal’s minimum spanning tree algorithm

requires time O(|B|log |B|).

3. Each of the |B| − 1 edges of MST (KB)

may correspond to a path in Gof up to

|V| − 1 edges. Hence, Step 3 requires time

O(|B||V|).

4. O(|B||V|log(|B||V|)) time using Kruskal’s

algorithm again.

5. The ﬁnal step is done in time O(|V|).

Step 1 is the most expensive and gives the dis-

tance network approximation algorithm a time

complexity of O(|B||V|2). •

Unfortunately, the performance ratio of the

above algorithm is equal to 2, and thus in the

worst case the length of the obtained Steiner tree

is 2 times longer than the length of the minimum

Steiner tree.

The approximation ratios of other determin-

istic heuristic algorithms for solving the Steiner

problem in graphs are only slightly better, so we

will not discuss them further and move on to the

formulation of the mathematical model and its

solution in the optimization tool GAMS.

3 Mathematical formulation

Let V={1,2, . . . , n}and Sbe a set of Steiner

vertices. For every edge (i, j), cij ,cij ≥0 is a

weight of the edge. The aim is to ﬁnd a connected

graph G′= (B∪S, E′) (Steiner tree), E′⊂E, so

that the sum of weights to be minimal.

In other words, the Steiner minimum tree

problem can be described as a problem of ﬁnding

a set of edges that connects terminals. There-

fore we can deﬁne a bivalent variable xij for each

edge (i, j)∈Eindicating whether the edge (i, j)

is included into the Steiner tree (xij = 1) or not

(xij = 0) and similarly a bivalent variable fiindi-

cating whether vertex iis included in the Steiner

tree (fi= 1) or not (fi= 0). For terminals,

i∈B, it is satisﬁed fi= 1, and fi∈ {0,1}for

the other vertices, i∈(V−B).

Each vertex of B∪Smust be reachable via

edges of E′from any other vertex of B∪S. There-

fore, it must be an endpoint of at least one edge

in E′. For the sake of mathematical formulation,

we select a vertex r, called root, with the orienta-

tion of edges to r, and by this way we get a rooted

tree. Due to the selected orientation the root has

no output edges, and thus xrj = 0 for j∈V−{r}.

So as the ﬁnal graph G′would be a tree, i.e. it

does not contain a cycle, just one path must lead

to the root from the vertices included into the

Steiner tree, and thus exactly one edge must leave

these vertices. If a vertex is not included into the

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Steiner tree, then no edge leaves it. The number

of edges leaving vertex i∈(V− {r}) is deﬁned

by Pn

j=1 xij , i.e. it corresponds to fi.

Finally, we must provide the connectedness of

G′, that means the edges of E′in the selected ori-

entation must enable a connection of any vertex

with the root. For this reason we deﬁne a net-

work ﬂow with respect to E′so that all vertices

of (B∪S)− {r}are sources of 1 unit of ﬂow [1],

and the root is a sink, supplied by ﬂows of all

vertices from the set (B∪S)− {r}of G′.

Let yij be a ﬂow through the edge (i, j)∈E′.

If iis a source of 1 unit of ﬂow, then the dif-

ference between the total ﬂow leaving iand the

total ﬂow entering imust be equal to 1. As

vertices unselected into the Steiner tree have

no incident edges, their ﬂows are equal to 0,

and for all vertices i∈(V− {r}) the equation

j=1 yij −Pn

j=1,j=ryji =fiis satisﬁed.

For each edge (i, j)∈ E′, i.e. edge unselected

into the Steiner tree (xij = 0), the ﬂow yij = 0.

On the other hand, if the edge is selected (xij =

1), then since this edge is in the rooted tree is

the only output edge of vertex i, the vertex iis

the source of 1 unit of ﬂow and the total inﬂow

to iis at most equal to the ﬂow from n−2 other

vertices (|V| − |{i, r}|), it follows that the ﬂow yij

through edge (i, j) is at most equal to (n−2) + 1,

i.e., (n−1). Both cases can be expressed by a

single relation as follows: 0 ≤yij ≤(n−1)xij for

i, j ∈V.

Summarizing all the considerations so far, we

can formulate the Steiner problem in graphs as a

mixed integer linear programming problem of the

following form:

Minimise

i=1

j=1

cij xij

subject to

∀j∈(V− {r}) : xrj = 0

∀i∈(V− {r}) :

j=1

xij =fi

∀i∈(V− {r}) :

j=1

yij −

j=1,j=r

yji =fi

∀i, j ∈V: 0 ≤yij ≤(n−1)xij

∀i, j ∈V:xij ∈ {0,1}

∀i∈B:fi= 1

∀i∈(V−B) : fi∈ {0,1}

Another formulation can be found in [16]. A

survey of Steiner tree problem formulations is

presented in [13].

4 Modiﬁcation of the model

and computing in GAMS

The mathematical model derived in the previous

section can be used for computing Steiner trees by

means of the optimisation package GAMS (Gen-

eral Algebraic Modelling System) [6] developed by

Alexander Meeraus and Anthony Brooke at the

World Bank in the 80s especially for the tasks of

linear, nonlinear and mixed integer programming.

The question is how to choose the root from

the given terminals. Since it can be any terminal,

we can choose the ﬁrst terminal in the list of ter-

minals, or the smallest terminal number, as the

root without loss of generality.

From the input data we need to create a ma-

trix with edge lengths C=(cij ), that is, to some-

how deﬁne the weights of the nonexisting edges.

The usual approach, if it is a minimization prob-

lem, is to set these weights equal to ∞. How-

ever, this value is not deﬁned in the computer,

and it is not advisable to choose the largest dis-

playable number in the computer either, as this

may be dependent on the software used and, in

addition, the sum in the objective function could

cause overﬂow of the range. For these reasons, we

assign zero weights to non-existing edges. With-

out further modiﬁcation, however, this setting

would lead to erroneous results, because the min-

imum Steiner tree would be constructed from

zero-ranked edges, i.e., edges that in the speciﬁed

graph do not exist at all. However, this short-

coming is easily remedied by adding constraints

to ensure that for those pairs of vertices that are

not connected by an edge, the values of xij and

yij are zero.

In order to speed up the calculation, it will

also be useful to limit the range of values of the

yij variables. The model is based on the fact that

all vertices except the root are sources of ﬂow of

size 1, and hence those vertices of the root graph

where no edges enter are also sources of ﬂow of

size 1. It follows that all ﬂows must be equal to

integer values.

If we denote the root by r, then the model can

be written as follows:

Minimise

i=1

j=1

cij xij

subject to

r:= min{k|k∈B}

∀j∈(V− {r}) : xrj = 0

∀i∈(V− {r}) :

j=1

xij =fi

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∀i∈(V− {r}) :

j=1

yij −

j=1,j=r

yji =fi

∀i, j ∈V:yij ≤(n−1)xij

∀i, j ∈V:yij ∈Z+

∀i, j ∈V, cij =0: xij := 0

∀i, j ∈V:xij ∈ {0,1}

∀i∈B:fi= 1

∀i∈(V−B) : fi∈ {0,1}

where Z+denotes the set of nonnegative integers.

GAMS is a high level language for formulat-

ing models with concise algebraic statements that

are easily read by modellers and computers alike,

easily modiﬁed, and easily moved from one com-

puter environment to another - it is independent

on the solution algorithms of speciﬁc solvers. The

basic components of a GAMS model are:

•SETS statement declares sets of indices of ar-

rays and assigns values to them.

•In input data section, parameters are de-

clared and values are assigned to them

(SCALAR statement is deﬁned for constants,

PARAMETERS statement for arrays, TA-

BLE statement simpliﬁes matrix declara-

tions). PARAMETERS statement can also

be used for a deﬁnition of a data object by

a formula which expresses how to calculate

object values from the values of the other al-

ready deﬁned objects.

•VARIABLES statement declares decision

variables and objective function and enables

to specify the type of variables (BINARY,

INTEGER, POSITIVE, NEGATIVE, FREE

(default)).

•EQUATIONS section describes constraints of

the optimisation problem. MODEL state-

ment builds a model of the task and assigns

a name to the list of constraints.

•SOLVE statement calls the solver and speci-

ﬁes the way of optimisation (MINIMIZING

or MAXIMIZING), variable optimised and

runs calculations.

•DISPLAY statement is determined to output

of results.

The source code in GAMS for data from Fig.

1 can be expressed by the following way:

$TITLE Network Steiner tree problem

$OFFSYMXREF

$OFFUELLIST

$OFFUELXREF

OPTION ITERLIM=200000

* ITERLIM maximal number of iterations

* Section defining index sets

SETS

K indices of terminals /1*4/

I start vertices of edges /1*13/;

ALIAS(I,J);

* J - end vertices of edges

*****************************************

* Input data section

PARAMETERS

B(K) indices and numbers of terminals

/1 2, 2 4, 3 7, 4 12/;

PARAMETERS

N,R;

N=CARD(I);

R=B("1");

TABLE C(I,J) edge weights

* (0 = corresponding edge does not exist)

1 2 3 4 5 6 7 8 9 10 11 12 13

109000014000000

29080060000000

30807300000000

400704000001000

50034020004000

60600207280000

7140000701300000

800000213000003

90000080001031

100000400010500

1100010000005080

120000000030801

130000000310010;

*****************************************

* Variable section (decision variables

* and objective function)

VARIABLES

X(I,J) equals to 1 if the edge ij is

included into Steiner tree, 0 otherwise

Y(I,J) edge flow

F(I) equals to 1 if the vertex i is

included into Steiner tree, 0 otherwise

St total sum of edge weights in

Steiner tree;

BINARY VARIABLE X;

INTEGER VARIABLE Y;

BINARY VARIABLE F;

* Equation section

EQUATIONS

EQ1(I,J)

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EQ2(I)

EQ3(I)

EQ4(I,J)

EQ5(I,J)

EQ6(I,K)

EQ7(I,J)

DELKA;

EQ1(I,J)$((ORD(I) EQ R)

AND (ORD(J) NE R)) .. X(I,J) =E= 0;

EQ2(I)$(ORD(I) NE R)

.. SUM(J,X(I,J)) =E= F(I);

EQ3(I)$(ORD(I) NE R)

.. SUM(J,Y(I,J))

-SUM(J$(ORD(J) NE R),Y(J,I))

=E= F(I);

EQ4(I,J) .. Y(I,J) =G= 0;

EQ5(I,J) .. Y(I,J) =L= (N-1)*X(I,J);

EQ6(I,K)$(ORD(I) EQ B(K)) .. F(I) =E= 1;

EQ7(I,J) .. X(I,J)$(C(I,J) EQ 0) =E= 0;

DELKA .. St =E= SUM((I,J),C(I,J)*X(I,J));

* Model description, solver execution

* and display of results

MODEL STEINER /ALL/;

SOLVE STEINER USING MIP MINIMIZING St;

DISPLAY X.L, F.L, Y.L, St.L;

Since INTEGER in GAMS is restricted

only for nonnegative integers the declaration

of INTEGER VARIABLE Y; is suﬃcient to specify

constraint assigning yij to Z+domain.

The relational operator =G= has the meaning

of ≥and =E= corresponds to =.

To avoid large changes of GAMS program ac-

cording input data structures, it is more eﬃcient

to prepare the data structures in a separate text

ﬁle and include it in GAMS program by $INCLUDE

directive, e.g. as follows:

$INCLUDE "B1.TXT"

Since more complex instances have (cij ) tables

with hundreds of columns and the line width is

limited, the tables must be expressed in a contin-

ued form. Similarly long sequences of terminals

must be written into several lines. Both these

modiﬁcations were provided by a special conver-

sion program.

5 Statistics and computational

results

Traditional approaches to calculations by heuris-

tic methods use statistical tests, which examine

at a certain signiﬁcance level (e.g. α= 0.05),

to what extent the mean value of the results ob-

tained by diﬀerent methods and diﬀerent settings

of their parameters at a larger number of runs is

the same or diﬀerent (and therefore one of the

methods gives better results). For the t-test, we

assume that the sets of values have a normal dis-

tribution. However, this assumption may not be

met and then one of the non-parametric tests,

such as the Wilcoxon test, must be used.

However, given the validity of the No Free

Lunch Theorem [25], [26], one should not expect

a general conclusion that any of the heuristics for

each problem instance gives better results than

other heuristics.

Here, we do not explore heuristics in this pa-

per and instead use a mixed integer programming

model using the CPLEX software tool built as a

solver in the GAMS environment to ﬁnd an exact

solution by deterministic computation. Statis-

tical evaluations are therefore meaningless here.

What can be said, however, is that the power

of this software today is considerably greater

than it was 20 years ago, when in our expe-

rience for a problem with complexity O(20!)

the system ended up with a runtime error and

the message ”insufficient space to update

U-factor ...”; today it computes an exact so-

lution on a laptop with a 2.4 GHz processor in

about 4.5 minutes.

Table 1 summarizes the results of computing

Steiner minimum trees for test problems with ver-

tex counts of 50 and 100, deﬁned sets of terminals

and computation times, with all tree lengths cor-

responding to the optimal solutions. It can be

seen that even for such large instances GAMS is

able to obtain an accurate solution in a time of a

few seconds at most.

test |V| |B|time [sec] length

B1 50 9 0.11 82

B2 50 13 0.56 84

B3 50 25 0.39 142

B7 75 13 1.14 116

B8 75 19 1.64 105

B9 75 38 0.51 225

B13 100 17 2.00 173

B14 100 25 1.88 244

B15 100 50 1.53 335

Table 1: Computational results

6 Conclusions

The paper proposed a general mathematical

model of the network Steiner tree and its modiﬁ-

cation to use it in GAMS. According to our expe-

rience from another application area, presented in

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[27], this tool is already able to successfully solve

exponentially complex problems with data struc-

tures with thousands of rows and columns. Here,

we veriﬁed that it is able to ﬁnd exact solutions

for graphs with up to 100 vertices.

However, more tests will need to be performed

in the future to compare the results with bench-

mark data from combinatorial optimization li-

braries as [4], [23].

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