A New Mathematical Model For the Minimum Linear Arrangement

Problem

Mahdi Moeini

1,2

, Serigne Gueye

and Sophie Michel Loyal

Laboratoire Informatique d’Avignon (LIA), Universit

e d’Avignon et des Pays de Vaucluse,

LIA-CERI 339, chemin des Meinajaries, Agroparc BP 91228, 84911, Avignon, France

Centre de Recherche en Informatique de Lens (CRIL-CNRS UMR 8188), Universit

e d’Artois,

Rue Jean Souvraz,SP 18, F 62307 Lens Cedex, France

Laboratoire de Math

ematique Appliqu

ee du Havre (LMAH), Universit

e du Havre,

25 rue Philippe Lebon, 76058, Le Havre, France

Keywords:

Integer Programming, Minimum Linear Arrangement Problem, Valid Inequality.

Abstract:

This paper addresses a classical combinatorial optimization problem called the Minimum Linear Arrange-

ment (MinLA) Problem. The MinLA problem has numerous applications in different domains of science

and engineering. It is known to be NP-hard for general graphs. The objective of this paper is to introduce

a new mathematical model and associated theoretical results, including novel rank inequalities. Preliminary

computational experiments are reported on some benchmark instances.

1 INTRODUCTION

The Minimum Linear Arrangement (MinLA) is a

challenging problem in combinatorial optimization.

For a given graph, the MinLA consists in arranging

the nodes of the graph on a line in such a way to min-

imize the sum of the distance between the adjacent

nodes. In the literature, the MinLA is known under

different names such as the optimal linear ordering,

the edge sum problem, the minimum-1-sum (see e.g.,

(Amaral et al., 2008), (Horton, 1997), (Caprara and

Gonzalez, 2005), (Caprara et al., 2010), (Petit, 1999),

(Hungerlaender and Rendl, 2012), (Schwarz, 2010)).

In addition to theoretical interests, the MinLA has

many practical applications. A non-exhaustive list in-

cludes the design of VLSI layouts, the graph drawing,

the single machine job scheduling, etc. ((Petit, 1999),

(Hungerlaender and Rendl, 2012), (Schwarz, 2010)).

The MinLA can be solved efﬁciently for some par-

ticular kinds of graphs. For example, there are poly-

nomial time algorithms for the MinLA on trees, out-

erplanar graphs, and certain Halin graphs (Caprara

et al., 2010). But in general, the MinLA is NP-hard

(Garey et al., 1976). Because of the hardness of ﬁnd-

ing optimal values, lower bounding and heuristic al-

gorithms are usually applied to get good approxima-

tions ((Caprara et al., 2010), (Petit, 1999), (Schwarz,

2010)).

In this paper, we present a new 0-1 linear pro-

gramming problem and investigate the associated new

polyhedra for which valid inequalities are provided

using lifting techniques or introducing some rank in-

equalities. The lower bound corresponding to the op-

timal value of the relaxed problem is tested on stan-

dard benchmarks. The results are modest and show

that many facets of the corresponding new polyhedra

has to be discovered.

The structure of the paper is as follows. In Sec-

tion 2, the Minimum Linear Arrangement (MinLA)

problem is reviewed and the new formulation of the

MinLA is given. Section 3 is devoted to theoretical

results on new rank inequalities. The computational

experiments are reported in section 4 and the last sec-

tion includes some conclusions.

2 THE MINIMUM LINEAR

ARRANGEMENT (MinLA)

PROBLEM

The present section includes the basic deﬁnitions and

models of the MinLA problem.

2.1 Deﬁnitions and Preliminaries

Let G = (V, E) be an undirected (connected) graph,

where V (with |V | = n) is the set of nodes and E de-

Moeini M., Gueye S. and Michel Loyal S..

A New Mathematical Model For the Minimum Linear Arrangement Problem.

DOI: 10.5220/0004827800570062

In Proceedings of the 3rd International Conference on Operations Research and Enterprise Systems (ICORES-2014), pages 57-62

ISBN: 978-989-758-017-8

 2014 SCITEPRESS (Science and Technology Publications, Lda.)

notes the set of edges. A layout is deﬁned as an one-

to-one function ϕ :V −→ [1..n]. The Minimum Linear

Arrangement problem (MinLA) is the combinatorial

optimization problem consisting in ﬁnding a layout ϕ

minimizing the following sum

∑

uv∈E

|ϕ(u) − ϕ(v)|.

(see e.g., (Amaral et al., 2008), (Horton, 1997),

(Caprara and Gonzalez, 2005), (Caprara et al., 2010),

(Petit, 1999), (Hungerlaender and Rendl, 2012),

(Schwarz, 2010)).

There are various mathematical formulations for

the MinLA. An overview of the different models can

be found in (Petit, 1999) and (Schwarz, 2010). A ba-

sic formulation of MinLA is obtained by considering

the following variables. Let us deﬁne the variables x

as:



1 if the label k is assigned to the node i,

0 otherwise.

(1)

With these variables, the MinLA can be formulated as

follows

MinLA-Quadratic:

min

∑

(i, j)∈E

∑

|k − l|x

(2)

s.t:

∑

i=1

= 1 : k ∈ {1, · · · , n}, (3)

∑

k=1

= 1 : i ∈ {1, ··· , n}, (4)

∈ {0, 1} : i, k ∈ {1, · · · , n}, (5)

This is a 0-1 quadratic programming problem for

which different reformulation has been proposed (see

(Schwarz, 2010) for an overview of different formu-

lations of MinLA).

We propose a non-standard linearization of

MinLA working as follows:

Let us deﬁne

∑

(i, j)∈E

∈ {0, 1}. (6)

If we apply this deﬁnition on (2)-(5), i.e., MinLA-

Quadratic, the formulation of the MinLA becomes:

A New Linearization for MinLA:

min

∑

k,l

|k − l| f

(7)

s.t:

∑

i=1

= 1 : k ∈ {1, · · · , n}, (8)

∑

k=1

= 1 : i ∈ {1, ··· , n}, (9)

≥ (x

+ x

− 1) : ∀(i, j) ∈ E, ∀k, l ∈ {1, · · · , n},

(10)

∈ {0, 1} : i, k ∈ {1, · · · , n}, (11)

∈ {0, 1} : ∀k, l ∈ {1, · · · , n}. (12)

Unfortunately, the formulation (7)-(12) has two

drawbacks. The ﬁrst one is due to the large number of

constraints (10), that is O(n

), and the second one to

the fact that the resulting bound is very poor. Indeed,

one may observe that the solution

= 0 (for k, l = 1, · · · , n),

(for i, k = 1, · · · , n),

is feasible, with 0 as the objective value. Hence, it is

necessary to strengthen the relaxation by introducing

valid inequalities (see (Amaral et al., 2008), (Amaral,

2009), (Amaral and Letchford, 2011), (Horton, 1997),

(Caprara et al., 2010), (Schwarz, 2010)). We show in

Section 3, how to deal with these two aspects.

Notice that f

can be seen as a binary ﬂow be-

tween the locations k and l. Indeed, f

equal 1 if the

two entities located in k and l are linked by an edge

in G, and 0 otherwise. As a consequence, one may

see that the graph whose the adjacency matrix is rep-

resented by f = { f

} is isomorphic to G. Hence we

transform the initial problem in a new one consisting

in ﬁnding an optimal isomorphic graph for G.

3 VALID INEQUALITIES

In order to reduce the number of constraints (10), we

show in the following theorem how we can produce

O(n

) equivalent inequalities using lifting techniques

(Nemhauser and Wolsey, 1998).

Theorem 1: If we deﬁne A as the adjacency

matrix of the (connected) graph G = (V, E), the

following inequalities are valid for (7)-(12):

(i) For k, l, i

∈ {1, 2, ··· , n}:

≥

∑

j=1

+ α(1 − x

), (13)

where

α := min(A

− A

such that 1 ≤ i

6= i

≤ n and 1 ≤ j

≤ n.

ICORES2014-InternationalConferenceonOperationsResearchandEnterpriseSystems

(ii) For k, l, i

∈ {1, 2, ··· , n}:

≥

∑

j=1

∑

=1,k

6=k

, (14)

where

:= min(A

− A

such that i

6= i

, j and j 6= i

and k

6= k.

Proof: (i) Remember the deﬁnition of f

∑

(i, j)∈E

∈ {0, 1}

For constant values of k and l, and setting i ←− i

, we

can do a lifting by the assumption of x

:= 1. Under

these assumptions, we have f

≥

∑

j=1

. Our

objective consists in ﬁnding an α ∈ R, such that the

above inequality remains true even for x

6= 1. Thus

we have to solve the following problem

α := min

(1 − x

)

( f

−

∑

j=1

such that x

6= 1 and k, l, i

∈ {1, 2, ··· , n}. The cor-

responding optimal value is given by:

α := min(A

− A

where, 1 ≤ i

6= i

≤ n and 1 ≤ j

≤ n.

(ii) This part is similar to the part (i) with the dif-

ference of the assumption on x

= 0 for all k

6= k.

We know that

≥

∑

i=1

∑

j=1

i j

where k < l. We are looking for the real values α

such that

≥

∑

j=1

, j

∑

6=k,k

particularly, we are looking for α

that minimizes

−

∑

j=1

, j

under the constraint x

= 1.

There are two cases:

Case (1) k

= l and k

6= k: in this case α

min

j6=i

, j

Case (2) k

6= k, l: in this case α

min

j6=i

6=i

6= j

, j

− A

, j

By taking the minimum of these cases, we obtain the

result of the theorem. 

Now, we focus on the second drawback of the for-

mulation (7)-(12). In order to ﬁnd better lower bounds

for the continuous relaxation of (7)-(12), we introduce

some sets of valid inequalities.

The ﬁrst set of valid inequalities is related to the

connectivity of G. We note that

∑

is equal to the

degree of any node having the label k, hence the fol-

lowing inequalities are valid for any connected graph

∑

l=1

≥ (minimum vertex degree of the graph G),

(15)

∑

l=1

≤ (maximum vertex degree of the graph G),

(16)

∑

k,l

= 2|E|, (17)

where, |E| is the total number of the edges.

A stronger version of these inequalities is given

through the following theorem:

Theorem 2: For any connected graph G the following

equalities are valid for (7)-(12):

∑

j=1

(degree of node j) x

: l ∈ {1, · · · , n}

(18)

Proof: for any value of l, we have

∑

(i, j)∈E

∑

(i, j)∈E

∑

Since

∑

= 1, hence

∑

(i, j)∈E

. If we deﬁne

A as the adjacency matrix of the graph G, then

∑

i=1

∑

j=1

i j

∑

j=1

∑

i=1

i j

∑

j=1

∑

i=1

i j

Since the degree of the node ( j) is equal to

∑

i=1

i j

consequently,

∑

j=1

(degree of node j) x

where l belongs to {1, ··· , n}. 

A well known set of valid inequalities for different

formulations of the MinLA consists in using triangu-

lar inequalities. The following theorem is based on

this idea.

Theorem 3: The following inequalities are valid for

(7)-(12):

• If k, l, m ∈ {1, ··· , n}, then

|k − l| f

≤ |k − m| + |m − l| : ∀k, l, m, (19)

|k − l| f

≤ |k − m|+|m − l| f

: ∀k < l < m, (20)

ANewMathematicalModelFortheMinimumLinearArrangementProblem

• For all k, l ∈ {1, · · · , n} and for all (i, j) ∈ E, if de-

gree of the node i is equal to 1, then

− x

+ f

≤ 1. (21)

Proof: All of these inequalities can be easily ob-

tained by using the fact that |k − l| ≤ |k −m| + |m − l|

(for all triple k, l, m) and x

, f

∈ {0, 1} (for all

i, k, l ∈ {1, · · · , n}). 

Deﬁnition: For a given graph G, the chromatic

number of G is denoted by χ(G) and is equal to the

minimum number of colors that one needs to color

the nodes of G in such a way that all adjacent nodes

of G get different colors.

The following theorem introduces new valid inequal-

ities that involve the chromatic number of a given

graph G.

Theorem 4 (Chromatic Inequalities): For any

graph G having the chromatic number χ(G) and for

all triple k, l, s, the following inequalities are valid for

(7)-(12):

k,l

+ f

l,s

+ f

k,s

≤ χ(G). (22)

Proof: First of all, we note that the left hand side

of (22) is at most equal to 3. If χ(G) ≥ 3, we are

done; otherwise, if χ(G) = 2, then there is (at least)

one node among the nodes having the labels k, l, s that

is not connected to one of the others. Without loss of

generality, let us suppose that the node having the la-

bel k is not connected to the node with the label s; this

means that there will be no ﬂow between the nodes k

and s. Consequently, the nodes k and s can have the

same colors and hence f

k,s

= 0, which completes the

proof. 

The results of the theorem 4 can be useful when

χ(G) = 2. This corresponds to some special graphs

such as trees and the graphs without any cycles of odd

lengths. This is rather restricting in using the inequal-

ities (22). Notice that ﬁnding the chromatic number

of a given graph is an NP-hard problem. Neverthe-

less, tight upper bound of χ(G) may be found by us-

ing heuristic algorithms.

Deﬁnition: For a given graph G, matching is a

subset of edges in which, no two edges are adjacent

to a same node. A maximum matching of G is de-

ﬁned as a matching with the maximum cardinality.

The matching number of G, noted by ν(G), is the size

of a maximum matching.

As in the case of chromatic number, some valid

inequalities depending on the matching number may

be introduced.

Theorem 5 (Matching Inequalities): For any

graph G having the matching number ν(G) and for

all k and “0 ≤ i and k + 2i + 1 ≤ n”; the following

inequality is valid for (7)-(12):

i:k+2i+1≤n

∑

i=1

k+2i,k+2i+1

≤ ν(G). (23)

Proof: For an arbitrary (connected) path in G, the

left hand side of (23) is (at most) the number of the

edges in the path such that any couple of edges is

separated by at least one edge. By noting this fact,

one can conclude that the number of these edges

cannot exceed the matching number of G. 

4 COMPUTATIONAL

EXPERIMENTS AND

NUMERICAL RESULTS

In this section, we present the preliminary results that

we have obtained by applying the valid inequalities of

the previous section.

The model has been coded in C++ and has been

solved with IBM CPLEX 12.2 in an Intel Core 2 Duo

of 3 GHz and 3.25 GB of RAM. The experiments

have been carried out on some benchmark instances

already used in (Caprara et al., 2010), (Caprara et al.,

2011), and (Schwarz, 2010). Table 1 reports some

characteristics of the instances. In this table, for each

graph, the number of nodes (n), of edges (m), and of

triangles (t) are reported. The absence of triangles can

be useful for chromatic inequalities.

The results are reported in Tables 2 and 3. In Ta-

ble 2, we denote by “Optimal” the known optimal

values in the literature (see (Caprara et al., 2010),

(Caprara et al., 2011), and (Schwarz, 2010)). The

column “Optimal” corresponds to optimal values ob-

tained through exact algorithms, such as Branch-and-

Cut procedures. Concerning our experiments, “LP(2-

3)” is used to denote the optimal value of the relaxed

linear program under the valid inequalities of the the-

orems 2 and 3. Table 3 contains more results on

the smaller sized instances. More precisely, Table 3

presents the optimal value of the relaxed linear pro-

gram under the valid inequalities of the theorems 1-3

(denoted by LP(1-3)) that are compared to the results

of (LP(2-3)). The CPU time (in seconds) of each case

is shown in a side column (i.e., cpu). There is a time

limit of 1200 seconds on CPLEX.

The number of the constraints (10) is huge.

Hence, in our experiments, the constraints (10) have

not been used. We just considered the remaining con-

straints of the model (7)-(12) as well as some of the

rank inequalities. This concerns the valid inequali-

ties that have been introduced in the theorems 1, 2,

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Table 1: The characteristics of the benchmark instances.

Name n m t

bcspwr01 39 46 2

bcspwr02 49 59 3

bcspwr03 118 179 23

bcspwr04 274 669 582

can 24 24 68 60

can 61 61 248 396

can 62 62 78 2

can 73 73 152 32

can 96 96 336 320

can 144 144 576 912

can 161 161 608 592

can 187 187 652 620

can 229 229 774 690

curtis54 54 124 78

dwt 59 59 104 30

dwt 66 66 127 62

dwt 72 72 75 0

dwt 87 87 227 147

dwt 162 162 510 464

dwt 209 209 767 707

dwt 221 221 704 608

dwt 245 245 608 374

lshp − 265 265 744 480

ibm32 32 90 28

will57 57 127 94

and 3. The valid inequalities of the theorems 4 and 5

have been also tested but we have observed that, un-

fortunately, they do not signiﬁcantly help to improve

the bound. That is why, for the sake of conciseness,

we report only the results of the other theorems with

more impact.

According to the numerical results that are re-

ported in Tables 2 and 3, we get the following ob-

servations:

• the relaxed LP model under the rank inequalities

of the theorems 2 and 3 (i.e., LP(2-3)) gives (in

most of the cases) integer values for the variables

• the rank inequalities of the theorem 2 are particu-

larly efﬁcient in approximating the integral enve-

lope of the model (7)-(12).

• inclusion of the lifting inequalities (i.e., theorem

1) has a huge inﬂuence on the size (consequently,

on the computational time) of the problem but

contributes slightly in improvement of the lower

bounds. Due to this fact, in our experiments, we

included only 1500 lifting inequalities in the re-

laxed LP model.

Table 2: The results of the experiments under the valid in-

equalities of the theorems 2 and 3 (LP(2-3)).

Name Optimal cpu LP(2-3) cpu

bcspwr01 106 0.7 54 0.031

bcspwr02 161 1.8 70 0.047

bcspwr03 662 189.5 241 0.359

bcspwr04 3696 limit 1189 3.295

can 24 210 2.8 138 0.015

can 61 1137 538 650 0.094

can 62 210 4.2 95 0.093

can 73 1083 limit 241 0.109

can 96 2105 27786 780 0.188

can 144 2873 1710.6 1460 0.484

can 161 5657 limit 1478 0.577

can 187 3827 limit 1496 0.781

can 229 7461 limit 1732 1.515

curtis54 454 54.5 214 0.062

dwt 59 289 27.4 150 0.078

dwt 66 192 1.7 189 0.093

dwt 72 167 21.2 79 0.124

dwt 87 932 1901.4 424 0.172

dwt 162 2281 limit 1078 0.578

dwt 209 5905 limit 1824 1.421

dwt 221 3603 limit 1500 1.702

dwt 245 3422 limit 1093 2.233

lshp − 265 5497 limit 1441 1.827

ibm32 485 250.5 178 0.031

will57 335 30.5 214 0.062

5 CONCLUSIONS

In this paper, we presented a new mathematical pro-

gram for solving the Minimum Linear Arrangement

(MinLA) problem. This formulation has been fol-

lowed by introduction of some new valid inequali-

ties. We presented some preliminary numerical re-

sults showing that, except for dwt 66, more inves-

tigations are necessary in order to have a better de-

scription of the associated polyhedra. In any case,

due to the fact that the formulation imply only O(n

)

additional variables, the computational time is very

small. As a perspective, additional polyhedral analy-

sis will be done to improve the bound and to develop

a branch-and-cut algorithm. The works in these direc-

tions are currently in progress.

ACKNOWLEDGEMENTS

This work was ﬁnancially supported by the region of

Haute-Normandie (France) and the European Union.

ANewMathematicalModelFortheMinimumLinearArrangementProblem

Table 3: The results of the experiments by taking into account the lifting inequalities.

Name Optimal cpu LP(2-3) cpu LP(1-3) cpu

can 24 210 2.8 138 0.015 150 434.156

curtis54 454 54.5 214 0.062 253 1200.203

dwt 59 289 27.4 150 0.078 154 10.218

dwt 66 192 1.7 189 0.093 191 1.922

will57 335 30.5 214 0.062 263 1200.250

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