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Discrete Applied Mathematics

journal homepage: www.elsevier.com/locate/dam

Characterizing acyclic graphs by labeling edges✩

Sebastián Urrutia a,∗, Abilio Lucena b,c

a Computer Science Department, Federal University of Minas Gerais, Belo Horizonte, Brazilb Business Administration Department, Federal University of Rio de Janeiro, Rio de Janeiro, Brazilc PESC/COPPE, Federal University of Rio de Janeiro, Rio de Janeiro, Brazil

a r t i c l e i n f o

Article history:Received 18 November 2011Received in revised form 6 June 2013Accepted 25 June 2013Available online xxxx

Keywords:Acyclic graph characterizationsFormulationsHomogeneous probabilistic minimumspanning tree problem

a b s t r a c t

In this paper two new characterizations of acyclic graphs are introduced. Additionally,restricted versions of them are also proposed to address some important special cases.These restricted characterizations, in turn, were used to obtain new integer programmingformulations for some associated relevant NP-hard problems. Resulting formulations arecompact, in the sense that the number of variables and constraints they contain arepolynomially bounded. One of them, in particular, that for the homogeneous version ofthe Probabilistic Minimum Spanning Tree Problem, under a MILP solver, is used here toobtain, for the first time, proven optimal solutions to that problem.

© 2013 Elsevier B.V. All rights reserved.

1. Introduction

Let G = (V , E) be an undirected and not necessarily connected graphwith a set V of n = |V | nodes and a set E ofm = |E|edges. G is called acyclic if it contains no cycles, i.e., if no sequence of nodes v1, v2, . . . , vp exists such that p > 2, v1 = vpand, for every 1 ≤ i < p, vi and vi+1 are the end nodes of an edge of G. Acyclic graphs are fundamental structures in GraphTheory and are at the core of various relevant practical applications.

In this paper two new characterizations of acyclic graphs are introduced. Additionally, to address some important specialcases, restricted versions of them are also described. Finally, as a result of latter characterizations, new polynomial sizeinteger programming formulations are obtained for some NP-hard problems.

Throughout the paper, an edge of E is denoted by e = {i, j}, where end nodes i, j ∈ V are such that i < j applies. Edgesthat are incident to i ∈ V define a set δi while δe

i identifies, for any e ∈ δi, those edges in δi \ {e}. If G is acyclic, removal ofany edge e = {i, j} ∈ E gives rise to two connected components, C e

i and C ej . The former containing i, the latter containing j.

Finally, Nei and Ne

j denote the number of nodes respectively in C ei and C e

j .The paper is organized as follows. In Section 2 we introduce two theorems characterizing acyclic graphs. Still in

that section, corollaries of these theorems are suggested to characterize two important special cases. Namely, diameterconstrained and edge betweenness centrality constrained acyclic graphs. In Section 3 the corollaries are used to obtainnew integer programming formulations for the following problems: the Diameter Constrained Minimum Spanning TreeProblem, the Capacitated Minimum Spanning Tree Problem and the homogeneous version of the Probabilistic MinimumSpanning Tree Problem. Next, in Section 4, computational results are presented for the latter problem. Finally, the paper isclosed in Section 5 with some concluding remarks and suggestions for future work.

✩ This researchwas partially supported by CNPq grants 303442/2010-7 and 310561/2009-4, respectively awarded to Sebastián Urrutia and Abilio Lucena.∗ Corresponding author. Tel.: +55 3134095882.

E-mail address: surrutia@gmail.com (S. Urrutia).

0166-218X/$ – see front matter© 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.dam.2013.06.019

2 S. Urrutia, A. Lucena / Discrete Applied Mathematics ( ) –

2. Edge labeling characterizations for acyclic graphs

Two different characterizations of acyclic graphs are introduced in this section. Restricted versions of them, implyingtailor made characterizations for some special acyclic graphs, then follow.

Theorem 2.1 (First Characterization of Acyclic Graphs). Assume that a graph G′ = (V ′, E ′) is given and that real valued numbers{αe : e ∈ E ′} are to be assigned to its edges. GraphG′ is acyclic if and only if an assignment existswhere, for any edge e = {i, j} ∈ E ′,αe > αa for all a ∈ δe

i or αe > αa for all a ∈ δej .

Proof. → Assume that G′ is acyclic and consider any edge e = {i, j} ∈ E ′. Accordingly, let Pei (resp. Pe

j ) be the longest pathin C e

i (resp. C ej ) having i (resp. j) as an end node. Additionally, denote by ne

i and nej the number of vertices respectively in Pe

iand Pe

j and take αe = min{nei , n

ej }. Without loss of generality, assume that αe = ne

i . Then, αe > αa results for any a ∈ δei . To

show that, take an edge a ∈ δei and apply the assignment procedure to it. Denote by k ∈ V the end node of a other than i and

notice that nai = ne

j + 1 and that nak ≤ ne

i − 1. Then, since nak < ne

i it follows that αa < αe. As such the suggested assignmentsatisfies the requirements imposed by the theorem.←Now assume that G′ contains a cycle and show, in this case, that no assignment satisfying the conditions set out above

is possible. To reach a contradiction, assume that such an assignment exists and consider the cycle implied by an orderedsequence of edges e1, e2, . . . , ec , for c ≥ 3, where e1 and ec are adjacent to each other while the same applies to ei and ei+1,for i = 1, . . . , c− 1. For the desired property to hold, αe1 must be larger than αec or larger than αe2 . Assume, without loss ofgenerality, thatαe1 > αe2 . Sinceαe2 is smaller thanαe1 itmust therefore be larger thanαe3 . Proceeding in thisway onewouldend up having αec > αe1 , a contradiction, since αe1 > αe2 · · · > αec . Therefore, the proposed assignment is not possible. �

The above theorem can be adapted to characterize special cases of acyclic graphs. The two corollaries that followcharacterize diameter constrained acyclic graphs.

Corollary 2.2. Let D be an even number and G′ = (V ′, E ′) be an acyclic graph. G′ has diameter at most D if and only if anassignment of integers {αe ∈ {1, 2, . . . ,D/2} : e ∈ E ′} is possible where, for any edge e = {i, j} ∈ E ′, αe > αa for all a ∈ δe

i orαe > αa for all a ∈ δe

j .

Proof. → Assume that G′ has diameter at most D and assign numbers {αe : e ∈ E ′} to its edges, exactly as previouslysuggested in the proof for Theorem 2.1. Accordingly, for any edge e = {i, j} ∈ E, min{ne

i , nej } ≤ D/2 must necessarily hold.

This condition applies since, otherwise, the simple path Pei ∪{i, j}∪ Pe

j would contain more than D edges, thus contradictingour assumption that G′ has diameter at most D.← Now assume that G′ has diameter larger than D and that an assignment of integers to the edges of G′ exists satisfying

the conditions imposed by the corollary. Accordingly, consider a simple path P implied by an ordered sequence of edgese1, e2, . . . , et−1, et , where t > D. Path P thus has more than twice as many edges as the largest integer value available to beassigned to it. Therefore, a given integer a ∈ {1, . . . ,D/2} must be assigned to at least three different edges of P , i.e., ei, ejand ek, where i < j < k. Given that this assignment should satisfy the conditions imposed by the corollary, amust be greaterthan b, where b is the integer value assigned to an edge of P adjacent to ej. As such, for one of the two sub-paths ei, . . . , ej andej, . . . , ek, integers assigned to its edges must decrease from a to b and then increase from b to a. Therefore, the least valueinteger assigned to an edge in that sub-path would not satisfy the conditions imposed by the corollary. A contradiction isthus reached and therefore if an acyclic graph has diameter larger than D, the suggested assignment of integers to its edgesis not possible. �

Corollary 2.3. Let D be an odd number and G′ = (V ′, E ′) be an acyclic graph. G′ has diameter at most D if and only if anassignment of integers {αe ∈ {1, 2, . . . , ⌈D/2⌉} : e ∈ E ′} exists where at most one edge per connected component is assignednumber ⌈D/2⌉ and, for any edge e = {i, j} ∈ E ′, αe > αa for all a ∈ δe

i or αe > αa for all a ∈ δej .

Proof. → As for the even case, assume that G′ has diameter at most D and assign numbers {αe : e ∈ E ′} to the edges of G′,exactly as previously suggested in the proof for Theorem 2.1. Observe that, in this case, since the diameter of G′ is less thanor equal to D, min(|Pe

i |, |Pej |) must necessarily be smaller than or equal to ⌈D/2⌉ = (D+ 1)/2. This condition applies since,

otherwise, Pei ∪ {i, j} ∪ Pe

j would be a path containing more than D edges, thus contradicting our assumption that G′ hasdiameter at most D. Now assume that at least two distinct edges, e = {i, j} and e′ = {u, v}, belonging to a same connectedcomponent, were given a value of ⌈D/2⌉ in the proposed assignment. Further assume, without loss of generality, that thesetwo edges are connected to each other through a path R = j, p1, p2, . . . , pt , u that does not contain edges e and e′. Observethat R may be an empty path if j = u and notice, irrespectively of this, that path Pe

i ∪ {i, j} ∪ R ∪ {u, v} ∪ Pe′v would have

more than D edges. Therefore, value ⌈D/2⌉must be assigned to at most one edge per connected component.← Now assume that G′ has diameter larger than D and that an assignment satisfying the conditions imposed by the

corollary do exists. Consider a simple path P implied by the ordered sequence of edges e1, e2, . . . , et−1, et , where t > D.Given that only one of these t edges may be assigned number ⌈D/2⌉, the remaining t − 1 edges may only be assignednumbers in the interval [1, ⌊D/2⌋]. Once again, one would have more than twice as many edges as the number of different

S. Urrutia, A. Lucena / Discrete Applied Mathematics ( ) – 3

integers available to be assigned to them. Therefore the same arguments previously used in the proof of Corollary 2.2 wouldapply here. Accordingly, if an acyclic graph has diameter larger than D, the suggested assignment of integers to its edges isnot possible. �

Corollaries 2.2 and 2.3 can be used to formulate problems where feasible solutions imply diameter or hop constrainedacyclic graphs. In particular, in Section 3, we propose a formulation for such a problem, i.e., the DCMSTP.

The theorem that follows characterizes acyclic graphs by labeling edges differently from what was suggested before. Acorollary of it, more geared into characterizing an additional particular type of acyclic graph, then follows.

Theorem 2.4 (Second Characterization of Acyclic Graphs). Assume that a graph G′ = (V ′, E ′) is given and that non-negative realvalued numbers {αe : e ∈ E ′} are to be assigned to its edges. Graph G′ is acyclic if and only if an assignment exists where, for anyedge e = {i, j} ∈ E ′, αe >

a∈δei

αa or αe >

a∈δejαa.

Proof. → Assume that G′ is acyclic. Set αe = min{Nei ,N

ej } for every edge e ∈ E ′. Considering any edge e′ = {i, j} ∈ E ′ and

assuming without loss of generality, that αe′ = Ne′i , then αe′ = 1+

a∈δe

′

iαa holds. To show this, we consider the computa-

tion of αak for each edge ak in δe′i . Each edge ak joins iwith a distinct node vk. For each of them Nak

i ≥ Ne′j + 1 holds because

Caki includes at least all nodes in C e′

j and node i. Also, since Cakvk is a proper subgraph of C e′

i , for each of edge ak, Nakvk < Ne

i holdsand in consequence αak is equal to Nak

vk . Now, observing that subgraphs Cakvk are disjoint and that the union set of their nodes

is equal to the set of nodes of C e′i minus node i, αe′ = 1+

a∈δe

′

iαa follows. As such the suggested assignment satisfies the

requirements imposed by the theorem.← Let us now assume that G′ contains a cycle and show, in this case, that no assignment satisfying the conditions im-

posed by the theorem is possible. Firstly notice that the conditions being imposed here on the value of αe are stronger thantheir counterparts in Theorem 2.1. Indeed, αa ≥ 0 for any a ∈ E ′ and αe >

a∈δei

αa or αe >

a∈δejαa imply that αe > αa

for all a ∈ δei or αe > αa for all a ∈ δe

j . Then, as Theorem 2.1 holds, the only if claim also holds for this theorem. �

The definition and lemma to be presented next lead to a corollary of Theorem 2.4, characterizing yet another special typeof constrained acyclic graphs.

Definition 2.5. Assume that an acyclic graph G′ = (V ′, E ′) is given and consider an edge e = {i, j} ∈ E ′. The betweennesscentrality of that edge is then defined as be = min(Ne

i ,Nej ).

The Edge Betweenness Centrality of an Edge (EBCE), defined for general weighted graphs in [8], is an important qualitymeasure for network design. Assume, as an example, that G′ is a spanning tree and that it is the desired topology for atelecommunications network. In that case, let us single out one (any) spanning tree edge, say edge e. The number of node-to-node paths using e directly depends on its betweenness centrality and equals be · (n− be). That function is increasing inthe interval [1, n/2]. In one extreme case, for a spanning tree leaf and its accompanying edge, (n − 1) node-to-node pathsare associated with that edge. On the other extreme, ⌈n/2⌉ · ⌊n/2⌋ node-to-node paths exist for a spanning tree edge ofmaximum possible betweenness centrality, ⌊n/2⌋. Therefore, while limiting EBCE one implicitly limits potential bottleneckcongestion in the network.

Lemma 2.6. Assume that an acyclic graph G′ = (V ′, E ′) is given and that positive integers {αe : e ∈ E ′} are to be assigned to itsedges. Additionally, for any edge e = {i, j} ∈ E ′, assume that these numbers must be such that αe >

a∈δei

αa or αe >

a∈δejαa.

Then, it is possible to assign to each edge its betweenness centrality and this is the smallest integer that may be used for that role.

Proof. FromTheorem2.4, an assignment satisfying the conditions imposed by the lemmadoes exist. For such an assignment,assume that there exists an edge e = {i, j} ∈ E ′ for which αe < be and i and j are not leaf nodes. Without loss of generality,further assume that be = Ne

i . We will then show that if αe >

a∈δeiαa (resp. αe >

a∈δej

αa) then at least one edge in δei

(resp. δej ) is assigned an integer smaller than its betweenness centrality.

Firstly assume that αe >

ak∈δeiαak and recall that each edge ak joins node i with a distinct node vk. Observe that

ak∈δeiαak < αe < Ne

i and, as a consequence,

ak∈δeiαak < Ne

i − 1 applies. On the other hand, since each Cakvk is a distinct

proper subgraph of C ei , bak = Nak

vk holds for every edge ak ∈ δei . It then follows that

ak∈δei

bak = Nei − 1 >

ak∈δei

αak and,therefore, at least one edge in δe

i must have been assigned an integer smaller than its betweenness centrality.Now assume thatαe >

ak∈δej

αak and notice, in this case, thatNej ≥ Ne

i > αe >

ak∈δejαak holds. Recall aswell that each

edge ak ∈ δej joins node j with a distinct node vk and has betweenness centrality bak = min(Nak

vk ,Nakj ). If bak = Nak

j applies,at least half of the nodes in the connected component containing e = {i, j}must be in Cak

vk and therefore this condition mayonly apply to at most a single edge of δe

j . Conversely, if bak = Nakvk holds for every ak ∈ δe

j , one would have

ak∈δejαak <

Nej − 1 =

ak∈δej

Nakvk =

ak∈δej

bak . Consequently, at least one edge in δej must have been assigned an integer smaller than

4 S. Urrutia, A. Lucena / Discrete Applied Mathematics ( ) –

its betweenness centrality. If there exists an edge as ∈ δej such that bas = Nas

j applies, then Nasj would have to be at least as

large asNei +1. Accordingly, ifαas is set to a number not smaller than bas , then it would be impossible to have

a∈δej

αa < Nei .

Therefore, in both cases, the integer assigned to an edge in δej must be smaller than its betweenness centrality.

From the results above, assuming that there exists an edge e = {i, j} ∈ E ′ such that αe < be holds and i and j are not leafnodes, we have thus shown, after exploring all possible cases, that there exists an edge adjacent to e which is assigned aninteger smaller than its betweenness centrality and also smaller thanαe. Proceeding in thisway, and taking into account thatthe graph is acyclic, one ends up concluding that an edge incident to a leaf is assigned an integer smaller than its between-ness centrality. Since the betweenness centrality for that edge must be 1, such an integer could not possibly be positive anda contradiction is thus reached.

Conversely, the first part of the proof for Theorem 2.4 shows that there always exist an assignment where every edge isassigned its corresponding betweenness centrality and the proof is thus concluded. �

Corollary 2.7. Assume that an acyclic graph G′ = (V ′, E ′) is given, together with a positive integer K . Graph G′ has no edge withbetweenness centrality larger than K if and only if an assignment of integers {αe ∈ {1, 2, . . . , K} : e ∈ E ′} exists where, for anyedge e = {i, j} ∈ E ′, αe >

a∈δei

αa or αe >

a∈δejαa.

Proof. → Assume that every edge of G′ has betweenness centrality at most K and assign integer valued numbers {αe : e ∈E ′} to its edges, exactly as previously suggested in the proof for Theorem2.4. Observe that, in this case, for any edge e = {i, j},min(Ni,Nj) must necessarily be smaller or equal to K . This condition applies since, otherwise, the betweenness centrality ofedge {i, j}would have to be larger than K .←Now assume that at least one edge of G′ exists with betweenness centrality larger than K . To reach a contradiction, as-

sume that an assignment of integers, as proposed above, exists in this case. However, Lemma 2.6 shows that the least integerthat could be assigned to an edge equals its betweenness centrality, a value that lies outside the range of integer values hereallowed for the assignment; a contradiction. Therefore, in this case, the suggested assignment of integers is not possible. �

Corollary 2.7 may be used to obtain formulations for ECBE constrained problems defined over acyclic graph. An exampleis the Capacitated Minimum Spanning Tree Problem [6]. The corollary may also be used to formulate problems such as thehomogeneous version of the Probabilistic Minimum Spanning Tree Problem [4]. In that case, the contribution of an edge tothe objective function is defined by its betweenness centrality.

3. Formulations for integer programming problems

Assume that a connected undirected graph G = (V , E) is given and that one wishes to formulate the problem ofgenerating a spanning tree for it, i.e., the problem of identifying a connected acyclic subgraph T = (V , ET ) of G. Provided acycle free guarantee is enforced, the connectivity and node spanning requirements for T are met by simply picking n − 1edges of G [5]. As such, T could be seen as an acyclic graph containing exactly n− 1 edges.

In order to introduce spanning tree formulations based on the acyclic graph characterizations proposed here, let usfirst define variables {xe ∈ {0, 1} : e ∈ E}, {ye ∈ R+ : e ∈ E} and

ske ∈ {0, 1} : e ∈ E, k ∈ {1, 2, 3}

. Variables x are used to

identify spanning tree edges while the same indirectly applies to auxiliary variables s1. Variables y, on the other hand, areused to assign appropriate valuesα to the edges of T , thus ensuring that the solution obtained is cycle free. Finally, additionalauxiliary variables s2 and s3 are used to impose the number values demanded by either Theorem 2.1 or Theorem 2.4.

As mentioned before, exactly (n − 1) x variables are to assume a value of 1. Accordingly, whenever xe = 1, edgee = {i, j} ∈ E will be part of T and, in that case, s1e = 0 will be enforced. Furthermore, through a constraint s1e + s2e + s3e = 1,either s2e = 1 or else s3e = 1 must result. If s2e = 1, the theorem conditions are to be enforced via the end node i of e.Conversely, if s3e = 1, those will be enforced via end node j. Based on Theorem 2.1, a formulation for T is given by

e∈E

xe = n− 1 (1)

xe ≤ 1− s1e , ∀e ∈ E (2)

ye +M(1− s2e ) ≥ ya + 1, ∀e = {i, j} ∈ E, a ∈ δei (3)

ye +M(1− s3e ) ≥ ya + 1, ∀e = {i, j} ∈ E, a ∈ δej (4)

s1e + s2e + s3e = 1, ∀e ∈ E (5)

xe ∈ {0, 1}, ∀e ∈ E (6)ye ≥ 0, ∀e ∈ E (7)

ske ∈ {0, 1}, ∀e ∈ E, k ∈ {1, 2, 3}. (8)

Constant M must be fixed to a value at least as large as the maximum value a variable y may take, plus 1. A tight upperbound for, say variable ye, could be established after investigating spanning trees of G that contain edge e. For any such tree,

S. Urrutia, A. Lucena / Discrete Applied Mathematics ( ) – 5

let us count the number of nodes in a path connecting one of the end nodes of e to a leaf. It is not difficult to establish thattherewill always exist one such path containing nomore than n/2 nodes. Indeed, at worst, onewill be facedwith a spanningtree that corresponds to a Hamiltonian path of G, with e placed right at the middle of it. It thus follows that ye ≤ n/2 is validand that M could be set to n/2+ 1. One should also notice that a variable ye may be set to 0 whenever edge e is not part ofthe spanning tree. This fact prevents undesirable conflicts to occur in relation with inequalities (3) and (4). Finally, one mayuse equalities (5) to get rid of variables s1 and constraints (2) and (5) that would then be dropped in favor of

s2e + s3e ≥ xe, ∀e ∈ E.

Let us now turn to Theorem 2.4. Accordingly, an alternative formulation for T is obtained after replacing (3) and (4) informulation (1)–(8) with inequalities

ye +M(1− s2e ) ≥ 1+a∈δei

ya, ∀e = {i, j} ∈ E (9)

and

ye +M(1− s3e ) ≥ 1+a∈δej

ya, ∀e = {i, j} ∈ E. (10)

In this case, constantM must be fixed to a value at least as large as themaximum value the right hand side of inequalities(9) and (10)may take. For a given node v,

a∈δv ba must be smaller than or equal to the sumof the number of nodes available

in the disjoint connected components obtained after v is removed from G. Since these components contain all nodes of Gwith the exception of v,

a∈δv ba is upwards limited by n−1. Therefore, since ya may take value ba for any feasible solution

(as indicated by Lemma 2.6),M could be set to a value of n. One should also notice, as was the case before, that a variable yemay be set to 0 whenever edge e is not part of the spanning tree, thus preventing undesirable conflicts to occur in relationwith inequalities (9) and (10).

3.1. The diameter constrained minimum spanning tree problem

The diameter of a spanning tree T of G equals the number of edges in the longest path of T . Accordingly, for a given graphGwith edge weights {we ≥ 0 : e ∈ E}, DCMSTP [9] is to find a minimumweight spanning tree with diameter no larger thana given integer D. DCMSTP is NP-Hard when D ≥ 4, as shown in [7]. In this subsection, formulation (1)–(8) is adapted toDCMSTP. This is carried out by enforcing the conditions imposed by Corollary 2.2, whenever D is even, and by Corollary 2.3,whenever D is odd.

In order to adapt formulation (1)–(8) to DCMSTP, under an even D, one must restrict variables y to assume values in theinterval [0,D/2]. This is accomplished by appending inequalities

ye ≤ D/2, ∀e ∈ E (11)

to that formulation. In association with this, recall that ye = 0 results for an edge e that is not part of the spanning tree.Additionally notice that if edge e belongs to T and one of its end nodes implies a leaf for graph G, the integer assigned to emay be 0, even though this edge is part of T . To prevent that from happening, inequalities

ye ≥ xe, ∀e ∈ E (12)

must also be appended to (1)–(8). Finally notice that

e∈E wexe is the objective function to be minimized over formulation(1)–(8), (11) and (12).

If D is odd, a new set of variables, {ze ∈ {0, 1} : e ∈ E}, must be introduced into the formulation. Variables z are used toidentify, in accordance with Corollary 2.3, those edges of T that are allowed to be assigned an α value of ⌈D/2⌉. In this case,each variable ye is restricted to assume values in the range [0, ⌊D/2⌋ + ze] and constraints

e∈E

ze = 1 (13)

and

ye ≤ ⌊D/2⌋ + ze, ∀e ∈ E, (14)

are used to ensure that only one y variable may take a value of ⌈D/2⌉. Finally, notice that

e∈E wexe should be minimizedover formulation (1)–(8), (12)–(14) of the odd constrained DCMSTP.

3.2. The capacitated minimum spanning tree problem

Assume that a connected graph G is given with edge weights {we ≥ 0 : e ∈ E}, a designated root node r and aninteger capacity limit L. CMSTP [11] is then defined as the problem of finding a minimumweight spanning tree such that allsubtrees, i.e., maximal subgraphs connected to the root by a single edge, have at most L nodes. CMSTP has been proven tobe NP-hard in [11]. In this subsection, formulation (1)–(2), (5)–(10), is adapted to CMSTP. This is accomplished by enforcingthe conditions imposed by Corollary 2.7.

6 S. Urrutia, A. Lucena / Discrete Applied Mathematics ( ) –

CMSTP calls for a spanning tree whosemaximum edge betweenness centrality is at most L. This is enforced by appendinginequalities

ye ≤ L, ∀e ∈ E (15)to our formulation of the problem.

These new inequalities do not complete the formulation of CMSTP. In fact, if one of the subtrees hasmore than n/2 nodesthen the tree may have a maximum betweenness centrality at most L while that subtree may have more than L nodes.

This observation leads us to append the following equations to the formulation:

s2e = 0, ∀e = {r, j} ∈ E (16)and

s3e = 0, ∀e = {i, r} ∈ E. (17)

If edge e is part of the spanning tree (i.e., if xe=1), s1e = 0 holds through constraint (2). Therefore, Eq. (5) implies that s2e = 1or s3e = 1. Consequently, in this case, Eq. (16) (resp. (17)) indicates that Theorem 2.4 conditions are to be enforced via theend node j (resp. i) of e implicitly limiting the number of nodes in the subtree not containing r . Finally notice that restrictingvariables y to assume values in the range [0, L] (observe that a value of 0 can only be taken by variables associated withunused edges of G), leads to CMSTP formulation (1)–(2), (5)–(10), (15)–(17). Accordingly, the objective function tominimizeover it is given by

e∈E wexe.

3.3. The homogeneous probabilistic minimum spanning tree problem

The Probabilistic Minimum Spanning Tree Problem (PMSTP) was first introduced by Bertsimas [4]. Given a spanning treeT of a graphG = (V , E) and a subset A ⊆ V of active nodes, the active tree AT (T , A) is theminimum subtree of T connecting allnodes of A. For PMSTP one is given edge weights {we ≥ 0 : e ∈ E} and independent node activation probabilities {pi : i ∈ V },where pi is the probability of node i belonging to A. One then aims at finding a spanning tree T of G where the expectedweight of the active tree AT (T , A) is minimum. We refer the reader to [4] for combinatorial properties of the problem andfor a NP-hardness proof.

In this subsection we address the homogeneous version of PMSTP, denoted here HPMSTP. In this case, every vertex has asame constant probability of activation, i.e., pi = p for every i ∈ V . As such, the objective function contribution of every treeedge does not only depend on its weight. It also depends on the number of nodes in each subtree resulting from the edge’sremoval from T , i.e., on the edge’s betweenness centrality. The probability of an edge being active is equal to the probabilitythat at least one node is active in each of these subtrees. Setting q = 1− p, the expected value of the active cost of a givenspanning tree T may be computed as [4]

E[LT ] =e∈T

we(1− qbe)(1− qn−be), (18)

where be is the betweenness centrality of edge e. If one defines constants cek = we(1−qk)(1−q(n−k)), the objective functionof HPMSTP is then given by

E[LT ] =e∈T

cebe . (19)

To tackle the problem, let us consider formulation (1)–(2), (5)–(10), with variables y limited to take values in the range[0, ⌊n/2⌋], as enforced by the inequalities

ye ≤ ⌊n/2⌋, ∀e ∈ E. (20)We also use additional variables {ℓek ∈ {0, 1} : e ∈ E, k ∈ [0, . . . , ⌊n/2⌋]}, defined as follows: ℓek = 1 when edge e has be-tweenness centrality k in the spanning tree solutionwhile ℓek = 0 applies, otherwise. These variables appear in the problemformulation via constraints

n/2k=0

ℓek = 1, ∀e ∈ E, (21)

n/2k=0

k · ℓek = ye, ∀e ∈ E, (22)

andℓek ∈ {0, 1}, ∀e ∈ E, k ∈ [0, . . . , ⌊n/2⌋], (23)

where Eq. (21) states that the betweenness centrality of every edge is unique while Eq. (22) couples the new variables tovariables y. A formulation for HPMSTP is then given by (1)–(2), (5)–(10) and (20)–(23). Accordingly, the objective functionto minimize over it is

e∈E

n/2k=1 cekℓek.

S. Urrutia, A. Lucena / Discrete Applied Mathematics ( ) – 7

Table 1Computational results.

p |V | Solution cost Time (s) B&C nodes Gap (%)

0.3 10 126.0491 22.48 124,158 –0.3 11 128.9377 20.38 70,077 –0.3 12 149.2856 462.75 1,399,920 –0.3 13 163.8729 3342.95 15,654,202 –0.3 14 188.5673 3600 12,499,400 31.39

0.5 10 194.6484 7.79 47,195 –0.5 11 195.1494 8.60 26,095 –0.5 12 219.0867 124.54 702,649 –0.5 13 237.0359 1013.06 5,009,153 –0.5 14 259.9077 3600 13,371,400 16.65

0.8 10 259.1613 3.00 10,627 –0.8 11 259.7752 2.97 11,352 –0.8 12 285.7642 11.96 29,577 –0.8 13 303.6594 35.07 176,101 –0.8 14 319.8154 864.35 3,772,112 –

4. Computational results

All formulations introduced in the previous section involve a polynomial number of variables and constraints. As such,for the problems they address, theymay be combinedwith an existingMILP solver to produce readily available off-the-shelfexact solution algorithms. However, since all of these formulations use the big-M strategy, one should not expect resultsthat are competitive with those found in the literature. A single exception applies to PMSTP. For that problem, to the best ofour knowledge, only two heuristics [1,2], based on genetic algorithms, could be found. Apparently, so far, no exact solutionalgorithm has been suggested for that problem or for any of its variants. We thus combine our formulation of HPMSTP withMILP solver CPLEX 12.1 to obtain, for the first time, proven optimal solutions for that variant of the problem.

Computational experiments were conducted on a 2 GHz Intel Xeon machine with 8 GB of RAM memory, running underthe Linux operational system. Test instances used were defined over complete graphs where the number of vertices rangedfrom 10 to 14. These graphs originate from the underlying graph corresponding to DCMSTP instance c_v15_a105_d4, arandom instance generated with 15 nodes, 105 arcs and diameter 4 (see [13], for further details). More specifically, theycorrespond to the subgraphs defined by the first n vertices of that graph, where 10 ≤ n ≤ 14.

Table 1 shows the computational results obtained. Every line in that table corresponds to a different test instance.Additionally, first column entries indicate node activation probabilities. This is followed, in column two, by the numberof vertices for the underlying graphs. Best feasible solutions found in less than one CPU hour are indicated in column three.Columns four and five respectively give theCPU times spent byCPLEX to findprovenoptimal solutions and the correspondingnumber of enumeration tree nodes. Finally, column six shows the ratios attained, after one CPU time hour, for the percentagegaps between best primal and best dual bounds, computed over best primal bounds.

From the results obtained, one may observe that, in less than one CPU hour, CPLEX was capable of solving to provenoptimality all test instances involving up to 13 nodes. As such, one may conclude that, under acceptable CPU times, ourformulation may be used to solve small HPMSTP instances. A second observation is that HPMSTP appears to become moredifficult to solve when the node activation probabilities decrease. Indeed, that had already been predicted by Bertsimas [4],who observed that, for the special case when node activation probabilities are all equal to one, HPMSTP becomes equivalentto the classical Minimum Spanning Tree Problem, which is polynomially solvable. On the other hand, differently from that,when all probabilities approach 0, HPMSTP is an NP-hard network design problem that appears, in practice, very difficult tosolve to proven optimality.

Closing this section, we should point out that although we were not able to find formulations that do not rely on thebig-M strategy, this is not a conclusive indication that they do not exist. Additionally, there should exist applications where,for small dimension instances, formulations based on the characterizations proposed here, even when subjected to the big-M restriction, may run faster than alternative formulations. Indeed, this is the frequently observed case for another type ofbig-M based spanning tree formulation, i.e., that implied by the Miller–Tucker–Zemlin (MTZ) inequalities [10]. Among thenumerous existing examples of that one may refer the reader to [13]. There, a lifting of the MTZ inequalities leads to thesituation hinted above. Another observation that could be made is that our characterizations open up additional modelingpossibilities. DCMSTP and HPMSTP are problems that are not very straightforward to model. However, the acyclic graphcharacterizations we suggested allowed them to be modeled in a quite natural way. One may then think about combiningour characterizations with more effective subtour breaking inequalities that do not involve big-Ms.

5. Conclusions

In this paper two new characterizations of acyclic graphs were introduced. Additionally, they were specialized to someimportant special cases. In doing so, new formulations were suggested for three Combinatorial Optimization problems that

8 S. Urrutia, A. Lucena / Discrete Applied Mathematics ( ) –

have restricted acyclic graphs as feasible solutions. For one of these problems, i.e., the Homogeneous Probabilistic MinimumSpanning Tree Problem, our proposed formulation was used, under MILP solver CPLEX 12.1 to obtain, for the first time,proven optimal solutions to the problem.

The characterizations introduced here are likely to be relevant in the theoretical study of structural properties for thegraphs they address. Additionally, they may also be specialized to other particular cases of interest.

As they stand, the formulations investigated in this paper are, generally speaking, uncompetitivewith existing alternativeformulations. However, drawing from what is observed for MTZ based formulations, that also involve big-M restrictions,liftings of our subtour breaking inequalities may possibly be attained, thus leading to stronger formulations. These, as isthe case for reinforced MTZ based formulations, may eventually turn out to be competitive for small dimension instances.Additionally, disjunctive cuts [3] that result from our characterizations may also offer the possibility of strengthening ourformulations. Finally, due to the disjunctions implied in our formulations, it also appears attractive to investigate the use ofConstraint Programming based solution approaches [12] as a possible way of avoiding the big-M trap.

Acknowledgments

The authors wish to thank the three anonymous referees for their detailed reading of our paper and also for their varioussuggestions that helped to improve it.

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