Computers & Operations Research

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1 Computes & Opeations Reseach 37 (2010) Contents lists available at ScienceDiect Computes & Opeations Reseach jounal homepage: Min-degee constained minimum spanning tee poblem: New fomulation via Mille Tucke Zemlin constaints Ibahim Akgün, Babaos Ç. Tansel Depatment of Industial Engineeing, Bilkent Univesity, Bilkent 06800, Ankaa, Tukey A R T I C L E I N F O A B S T R A C T Available online 24 Mach 2009 Keywods: Mixed intege pogamming Degee-enfocing constaints Mille Tucke Zemlin constaints Minimum spanning tee Flow fomulation Rooted aboescence Given an undiected netwok with positive edge costs and a positive intege d > 2, the minimum-degee constained minimum spanning tee poblem is the poblem of finding a spanning tee with minimum total cost such that each non-leaf node in the tee has a degee of at least d. This poblem is new to the liteatue while the elated poblem with uppe bound constaints on degees is well studied. Mixedintege pogams poposed fo eithe type of poblem is composed, in geneal, of a tee-defining pat and a degee-enfocing pat. In ou fomulation of the minimum-degee constained minimum spanning tee poblem, the tee-defining pat is based on the Mille Tucke Zemlin constaints while the only ealie pape available in the liteatue on this poblem uses single and multi-commodity flow-based fomulations that ae well studied fo the case of uppe degee constaints. We popose a new set of constaints fo the degee-enfocing pat that lead to significantly bette solution times than ealie appoaches when used in conjunction with Mille Tucke Zemlin constaints Elsevie Ltd. All ights eseved. 1. Intoduction Minimum spanning tee (MST) poblems aise quite natually in tanspotation and communication netwok design when it is necessay to povide a minimum-cost connectivity among a numbe of geogaphically dispesed locations o system components. Vaious examples of minimum cost tee netwoks ae given by Ahuja et al. [1] fom netwok design in tanspotation, telecommunication, data stoage, and cluste analysis. We conside in this pape a topology constained vesion of the minimum spanning tee poblem in which a minimum cost spanning tee is sought fo while equiing that each node in the tee be eithe a leaf node o a cental (non-leaf) nodethatisadjacenttoatleastd nodes. The minimum-degee equiement fo cental nodes may aise in distibution netwoks when fixed chages associated with a facility may be lage enough to suggest that at least a cetain numbe of end-uses be seved by it to justify the opening and opeation costs associated with it. Minimum-degee constaints in tee netwoks ae also encounteed in telecommunication netwoks in the pocess of designing local access netwoks that feed taffic between a main netwok and a lage numbe of end-uses (teminals) (e.g., Geen Coesponding autho. Fax: addess: iakgun@bilkent.edu.t ( I. Akgün). [2]). Installation costs fo netwok components (e.g., concentatos and computes) can be bette justified when the numbe of nodes seved by them is not less than an acceptable theshold. The poblem is also of inteest fom a modeling and computational standpoint as appoaches that ae well studied fo uppe degee constained poblems do not necessaily wok well fo lowe degee constained poblems. To define the poblem of inteest, let G = (V, E) be an undiected connected netwok with node set V, edgesete, and positive edge costs c e (e E). A spanning tee of G is a connected sub-gaph of G that has no cycles and spans all nodes. Given a positive intege d, a spanning tee is a minimum-degee constained spanning tee if the degee of each node elative to the tee is eithe 1 o at least d. Fig. 1 gives two such tees fo d = 4. Fom now on, we efe to minimum-degee constained spanning tees as feasible tees. If d 2, all spanning tees ae feasible tees and the degee constaints can be ignoed. The distinction between feasible and infeasible tees becomes impotant fo d > 2. Note that if d n V, no feasible tee exists fo G. We assume fom this point on that 2 < d < n. We efe to the poblem of finding a minimum cost spanning tee of G as the minimum spanning tee poblem and that of finding a minimum cost feasible tee as the minimum-degee constained minimum spanning tee (MDC-MST) poblem. While MST is solvable in low ode polynomial time by the algoithms of Kuskal [3] and Pim [4], MDC-MST is poven to be NP-had fo d /$ - see font matte 2009 Elsevie Ltd. All ights eseved. doi: /j.co

2 I. Akgün, B. Tansel / Computes & Opeations Reseach 37 (2010) Fig. 1. Two feasible solutions fo MDC-MST with d = 4. by Almeida et al. [5]. The complexity status of this poblem is open fo d = 3. The MDC-MST is new to the liteatue. To ou knowledge, the fist and the only study in the liteatue on MDC-MST is that of Almeida et al. [5] (efeed to as AMS in the sequel). In thei study, AMS intoduced the poblem, discussed its popeties and complexity, pesented some popeties egading the numbe of leaf and non-leaf nodes, gave single- and multi-commodity flow-based fomulations fo the poblem, and pesent computational esults fo thei fomulations. Thei epoted esults show that many of the test poblems with up to 50 nodes can be solved within 3 h of CPU time, but thee ae also many test poblems in the same test bed that emain unsolved within the 3 h limit. We popose in this eseach new fomulations that give substantially impoved solution times fo the same set of test poblems. Additional impovement has been obtained by obseving that the linea pogamming (LP) elaxations of the poposed fomulations lead, in geneal, to tighte bounds if the oot node fo the tee is moe judiciously selected. Based on this, we give a methodology to select a oot node fo the tee that significantly impoves solution times fo poposed and pevious models. MDC-MST is closely elated to the degee-constained minimum spanning poblem (DC-MST) whee the degee equiement fo nonleaf nodes is an uppe bound athe than a lowe bound. DC-MST is poven to be NP-had by Gaey and Johnson [7]. Unlike MDC-MST, DC-MST is a well-studied poblem. Some of the notable contibutions on DC-MST ae Deo and Hakimi [8], Savelsbegh and Volgenant [9], Zhou and Gen [10], Knowles and Cone [11], Caccetta and Hill [12], Ribeio and Souza [13], Andade et al. [14], and Kishnamoothy et al. [15]. AMS compaed DC-MST and MDC-MST by using flow-based fomulations of both. They obseved that flow-based fomulations of MDC-MST show significantly poo pefomance with espect to LP bounds and solution times when compaed to flow-based fomulations of DC-MST. In this egad, MDC-MST appeas to be quite elusive, at least when compaed to DC-MST. The emainde of this pape is oganized as follows. Section 2 eviews flow-based models of AMS. Section 3 gives ou poposed fomulations. Section 4 gives ou methodology to select the oot node. Section 5 gives computational esults and compaes poposed and pevious models. Section 6 concludes this pape. 2. Review of the models of AMS Minimum spanning tee poblems with additional estictions on the stuctue of the tee (e.g., degee equiements) ae fomulated in geneal using two sets of constaints, one set ensuing that a spanning tee is obtained and the othe set ensuing that the esulting tee satisfies the stuctual equiements. The stuctual equiement in the poblem we study is the minimum-degee equiement on nonleaf nodes and will accodingly be efeed to as degee-enfocing constaints. The emaining potion of minimum spanning tee fomulations consists of constaints that ensue that the esulting set of acs is a spanning tee. We efe to this potion of the fomulations as tee-defining constaints. A numbe of diffeent appoaches ae available fo modeling spanning tee featues including fomulations based on packing, cut-sets, and flows (Magnanti and Wolsey [16]). Among diffeent fomulations, flow-based fomulations seem to be a most pefeed one because they ae compact in the numbe of vaiables and that they, especially the multi-commodity vesions, give a bette epesentation of the spanning tee polyhedon [16]. Following this fact, AMS use diected single- and multi-commodity flow-based fomulations to model spanning tee featues. Flow-based fomulations ae defined on a diected netwok G = (V, A) obtained fom G = (V, E) by eplacing each undiected edge {i, j} E, whee i j, by two diected acs (i, j)and(j, i) with symmetic costs c ij = c ji. A node is selected as the oot nodeandactsasa single souce fo the flow to be sent to the emaining n 1 nodes each of which acts as a sink node with a demand of one unit. In the single-commodity fomulation, the oot node has a supply of n 1 units of a commodity and sends them out into the netwok to satisfy the unit demand at each sink node. In the multi-commodity case, the n 1 demand nodes still have unit demands but each demand is fo a diffeent commodity and the oot node has a supply of one unit of each commodity (e.g., Magnanti and Wolsey [16]). In eithe case, the set of acs with positive flows in a feasible solution define an aboescence which is a diected tee such that evey node othe than the oot node has exactly one incoming ac while the oot node has no incoming ac. In a feasible aboescence, if the oot node is a leaf node, it has one outgoing ac but no enteing acs. Any demand node that is a leaf node has one incoming ac but no outgoing acs. If the oot node is a cental node, it has d o moe outgoing acs but no incoming acs while a demand node that is a cental node has one incoming ac and at least d 1 outgoing acs. Two example aboescences ae shown in Fig. 2 fo the case of d = 4. AMS use thee sets of decision vaiables to fomulate MDC-MST: (1) binay design vaiables x ij that take on the value of 1 if ac (i, j) is in the design and 0 othewise, (2) binay node vaiables w i that take on the value of 1 if node i is a cental node and 0 if node i is a leaf node, and (3) non-negative flow vaiables y ij specifying the amount of flow in ac (i, j) fo the single-commodity flow fomulation and the flow vaiables f k specifying the amount of flow sent fom the ij oot node to demand node k passing though ac (i, j) fo the multicommodity flow fomulation. In the flow-based fomulations that we give next, the degee enfocing constaints ae constaints (2) (5). We efe to constaints (2) (5) as DEF1 (the fist set of degee-enfocing constaints). The emaining set of constaints, othe than set estictions and nonnegativity, constitutes the tee-defining pat of the fomulation. The tee-defining pat is diffeent fo single and multi-commodity fomulations.

3 74 I. Akgün, B. Tansel / Computes & Opeations Reseach 37 (2010) Fig. 2. Two feasible solutions on a diected gaph with d = 4. SCF/DEF1: Single-commodity flow model with the fist set of degee-enfocing constaints z = min x,w c ij x ij (1) (i,j) A s.t. x ij 1 + (d 1)w i, i = (2) j x ij 1 + (n 2)w i, i = (3) j x ij (d 1)w i, i (V ) (4) j x ij (n 2)w i, i (V ) (5) j x ij = 1, j (V ) (6) i j y ij y ji = 1, i i j (V ) (7) x ij y ij, i V, j (V i ) (8) y ij (n 1)x ij, i V, j (V i ) (9) x ij {0, 1}, i V, j (V i ) (10) w i {0, 1}, i V (11) y ij 0, i V, j (V i ) (12) Objective function (1) minimizes the total cost of the acs in the solution. DEF1 constaints ae (2) (5) and tee-defining constaints ae (6) (9). Constaints (2) and (3) and constaints (4) and (5) define lowe and uppe bounds on the numbe of outgoing acs fom the oot node and non-oot nodes, espectively. Constaints (6) equie that the numbe of incoming acs to any non-oot node be equal to 1. Constaints (7) ae flow-consevation constaints. Constaints (8) and (9) ae coupling constaints equiing that any ac with a positive flow be in the design and that the amount of flow though an ac be bounded above by n 1. Even though this can be impoved to n 2 fo non-oot nodes, we etain n 1 in (9) to be consistent with the fom used by AMS. Constaints (10) (12) give the appopiate set estictions and non-negativity on the decision vaiables. MCF/DEF1: Multi-commodity flow model with the fist set of degee-enfocing constaints. In addition to (1) (6), (10), and (11), i k f k ij i f k ji = 0, j, k (V ), j k (13) f j = 1, ij j (V ) (14) i f k ij x ij, i V, j, k (V i ) (15) f k ij 0, i V, j, k (V i ) (16) Constaints (13) (15) togethe with constaints (6) ae multicommodity flow-based tee-defining constaints. Constaints (13) and (14) ae flow-balance constaints and constaints (15) ae coupling constaints. Note that constaints (13) and (14) and constaints (15) ae commodity-distinguished vesions of constaints (7) and constaints (8) and (9), espectively. Constaints (16) ae non-negativity estictions on flow vaiables. AMS define two valid inequalities which ae added to the models SCF/DEF1 and MCF/DEF1 to obtain a total of six diffeent fomulations (thee fo each). These valid inequalities ae x ij w i, i, j (V ), i j (17) n 2 w i (18) d 1 i V Valid inequality (17) equies that a node be a cental node if thee is an outgoing ac fom it while valid inequality (18) defines an uppe bound on the numbe of cental nodes in a solution. The validity of the uppe bound is poven by AMS. We use these valid inequalities in ou fomulations as well. We efe to the vesion of DEF1 that includes the valid inequalities (17) and (18) as DEF1. As to the numbe of constaints and vaiables, SCF/DEF1 has 3n 2 2n 1 constaints, n 2 n binay vaiables, and n 2 n continuous vaiables while MCF/DEF1 has n 3 + n 2 n 1 constaints, n 2 n binay vaiables, and n 3 2n 2 continuous vaiables. 3. Poposed fomulations fo MDC-MST In this section, we popose a new set of degee-enfocing constaints efeed to as DEF2. We also popose to use Mille Tucke Zemlin (MTZ) [6] constaints fo the tee-defining pat as an altenative to single o multi-commodity flow constaints DEF2: the poposed set of degee-enfocing constaints Let w ic and w il be a pai of binay vaiables associated with node i with w ic = 1(w il = 1) if node i is a cental (leaf) node and w ic = 0 (w il = 0) if not. DEF2 constaints ae as follows: w ic + w il = 1, i V (19) x ij 1, i = (20) j x ij dw ic + w il, i = (21) j x ij 1 + (n 2)w ic, i = (22) j x ij (d 1)w ic, i (V ) (23) j x ji + x ij d (d 1)w il, i (V ) (24) j j x ji + x ij 1 + (n 2)w ic, i (V ) (25) j j x ij w ic, (i, j) A, i, j (26) x ij + w il + w jl 2, (i, j) A, j (27) n 2 w ic (28) d 1 i V x ij 0, (i, j) A, j = (29) x ij + x ji 1, (i, j) A, i < j (30) x ij = n 1 (31) j i w ic, w il {0, 1}, i V (32)

4 I. Akgün, B. Tansel / Computes & Opeations Reseach 37 (2010) Constaints (19) equie that each node be eithe a cental node o a leaf node. Constaints (20) (31) expess some stuctual popeties of a feasible solution. Constaints (20) (22) define lowe and uppe bounds on the numbe of outgoing acs fom the oot node. Constaint (20) establishes that the numbe of outgoing acs at the oot node is at least 1. Constaints (21) and (22) equie that the numbe of outgoing acs at the oot node be equal to 1 when is a leaf node and be at least d and at most n 1 when is a cental node. Constaint (20) is actually edundant; howeve, it helps to impove the solution times. Constaints (21) and (22) ae equivalent to constaints (2) and (3), espectively, in tems of the new node vaiables. Constaints (23) (25) set uppe and lowe limits on the degee of non-oot nodes. Constaints (23) equie, as constaints (4), that the numbe of outgoing acs fom a non-oot node be at least if the node is a cental node. Constaints (24) state that the total numbe of outwad and inwad acs of each non-oot node is at least d when a non-oot node is a cental node and at least 1 when a non-oot node is a leaf node. Constaints (24) ae simila to constaints (23), and hence to constaints (4), except that constaints (24) take into account both inwad and outwad acs while constaints (23) take into account only outwad acs. Constaints (24) can be obtained by adding constaints (4) and (6) in tems of new vaiables, i.e., constaints (23) and (6). Because each non-oot node is equied to have exactly one incoming ac by constaints (6), constaints (24) ae actually nothing moe than adding the same tems to the leftand ight-hand sides of constaints (4). Constaints (25) estict the numbe of inwad and outwad acs of a non-oot node to be at most 1 when the node is a leaf node and at most n 1 when the node is a cental node. Constaints (25) can be obtained by adding constaints (5) and (6) in tems of new vaiables. Thus, the elationship between constaints (25) and (5) is simila to that between constaints (24) and (4). Constaints (26) ae exactly the valid inequalities (17) in tems of w ic. They equie that a non-oot node be a cental node if thee is an outgoing ac fom it. Note that constaints (5) can be obtained by summing both sides of constaints (26) ove all nodes j adjacent to node i, i.e., constaints (26) ae a disaggegated vesion of constaints (5). Constaints (27) pevent acs between pais of leaf nodes. Constaints (28) ae the valid inequalities (18) in tems of the new vaiables. Constaints (29) do not allow any acs incoming to the oot node. Constaints (30) state that a pai of acs of opposite diections between a pai of nodes is not possible. This set of constaints is actually a set of valid inequalities. Constaints (31) equie that the total numbe of acs in the solution be equal to n 1, which is a known fact fo a tee (e.g., [1]). Finally, constaints (32) give the zeo/one estictions on the decision vaiables w ic and w il. Note that constaints (26) and (27), and (29) (31) must be satisfied by any tee poblem and ae not paticula, in this sense, to MDC-MST. They ae not an essential pat of degee-enfocing constaints, but we keep them thee because thei pesence leads to bette computational pefomance than thei omission. Note that these constaints ae not an essential pat of tee-defining constaints, eithe. New flow-based fomulations fo MDC-MST can easily be obtained by eplacing DEF1 constaints (2) (5) with DEF2 constaints (19) (32). In fact, DEF2 (o any othe set of degee-enfocing constaints) can be coupled with any set of tee-defining constaints to obtain a new fomulation. Fo instance, MCF/DEF2, the multicommodity flow-based model with the poposed set of constaints, is composed of the objective function (1) and constaints (6), (10), (13) (16), and (19) (32). Poposition 1. Let DEF1P and DEF2P be two diffeent fomulations of MDC-MST whee DEF1 and DEF2 ae used as degee-enfocing constaints in the two fomulations, espectively, while all emaining constaints, including tee-defining constaints and intege estictions, ae common. Denoting by F(P LP ) the set of feasible solutions of the LP elaxation of any intege linea pogamming poblem P, we have F(DEF2P LP ) F(DEF1P LP ). Accodingly, DEF2 dominates DEF1. Poof. Let (x, y, w c, w l ) F(DEF2P LP ) whee x, y, w c,andw l ae the vectos of vaiables x ij, y ij, w ic,andw il, espectively. Put w = w c.we now pove (x, y, w) F(DEF1P LP ). It suffices to show that (x, y, w) satisfies constaints (2) (5) as the only constaints that ae in DEF1P LP that ae not included in DEF2P LP ae these constaints. The feasibility of (x, y, w c, w l )todef2p LP implies that (x, y, w c, w l ) satisfies the DEF2 constaints (19) (32) as well as the tee-defining constaints (6) (9). Constaint (2) is implied by (19), (21), and the fact that w = w c.constaint (3) is implied by (22) and w = w c. Constaints (4) ae implied by (23) and w=w c. Constaints (5) ae implied by (6), (25) and w=w c. Hence, (x, y, w) F(DEF1P LP ) and the poof is complete. We emak that the poof of the poposition is still valid if we change DEF1 to DEF1 in the poposition. This follows fom the fact that constaints (17) and (18) of DEF1 ae nothing but constaints (26) and (28) of DEF2, espectively, upon eplacing w with w c. Poposition 2 is an immediate consequence of Poposition 1 and the foegoing emaks when the poblem unde consideation is taken to be a flow-based fomulation, MCF o SCF. Poposition 2. (i) F(MCF/DEF2 LP ) F(MCF/DEF1 LP ) F(MCF/DEF1 LP). (ii) F(SCF/DEF2 LP ) F(SCF/DEF1 LP ) F(SCF/DEF1 LP). While it is possible to dop the vaiables w il fom DEF2 by eplacing w ic with w i and w il by 1 w i, the pesence of the vaiables w il in DEF2 poduces on the aveage bette solution times than when they ae absent. We attibute this to diffeent banch-and-bound stuctues and cuts that may be geneated by the solve when these vaiables ae pesent than when they ae absent. Computational studies indicate that flow-based models with DEF2 show bette pefomance with espect to both LP bounds and solution times than the ones with DEF1 and DEF1.Thesolution times fo poblems solved to optimality ae almost halved. In paticula, MCF/DEF2 gives consideably bette LP bounds than SCF/DEF1 o SCF/DEF1 (discussed in Section 5). Howeve, due to elatively high memoy stoage equiements of MCF/DEF2, it cannot solve most of the poblems with 50 nodes within the 3-h limit of CPU time. SCF/DEF2 can solve moe poblems than MCF/DEF2; howeve, thee still emain poblems not solved within the allotted time. Fo these easons, we use the Mille Tucke Zemlin constaints (Mille et al. [6]) as an altenative to flow-based fomulations Fomulations based on the Mille Tucke Zemlin sub-tou elimination constaints Fomulations based on MTZ constaints also ceate a ooted aboescence. In this egad, the diected netwok stuctue defined above is used. To fomulate MTZ constaints, in addition to the binay design vaiables x ij, non-negative node-labeling vaiables u i ae used. These labels ae assigned in such a way in any feasible solution that each diected ac included in the aboescence is diected fom a node with a lowe label into a node with a highe label. This ensues that the node labels fom an inceasing sequence on any diected path so that any node peviously visited on a diected path cannot be e-visited, theeby peventing fomation of sub-tous.

5 76 I. Akgün, B. Tansel / Computes & Opeations Reseach 37 (2010) Fig. 3. Two feasible spanning tees with labels assigned by MTZ constaints. The basic MTZ constaints [6] ae given below. The tem basic is used hee because these constaints ae changed late to obtain an impoved vesion of these constaints. BMTZ: Basic Mille Tucke Zemlin sub-tou elimination constaints u i u j + nx ij n 1, (i, j) A, j (33) u i n 1, i (V ) (34) u i 1, i (V ) (35) u i 0, i = (36) u i 0 i (37) MTZ constaints ae oiginally defined fo the taveling salesman poblem (TSP) (Lawle et al. [17], Padbeg and Sung [18], Nemhause and Wolsey [19]). In the context of TSP, MTZ constaints eliminate all sub-tous that do not contain the base (oot) node by assigning unique labels u i to nodes such that the label of a node epesents the ank-ode in which the node is visited in a taveling salesman tou. That is, base node is assigned a label of 0 while the i-th node visited afte node is assigned a label of i. In ou case, constaints (33) pevent sub-tous by ensuing that each ac included in the aboescence is diected fom a lowe labeled node to a highe labeled node. The uniqueness of node labels is not equied. Constaint (36) assigns a label of 0 to the oot node, while constaints (34) and (35) define uppe and lowe bounds on the labels that can be assigned to non-oot nodes, espectively. In the oiginal pape [6], the u i vaiables ae unesticted. Bounds (34) and (35) ae intoduced late on. Two new fomulations of MDC-MST whee the tee-defining pat consists of MTZ constaints while the degee-enfocing pat is eithe DEF1 o DEF2 ae given below: BMTZ/DEF1: Basic MTZ model with the fist set of degee-enfocing constaints. Objective function (1), constaints (2) (6), (10) and (11), and (33) (37). BMTZ/DEF2: Basic MTZ model with the poposed set of degeeenfocing constaints. Objective function (1), constaints (6), (10), and (19) (37). By specializing Poposition 1 to the Basic MTZ-based fomulations, we have the following. Poposition 3. F(BMTZ/DEF2 LP ) F(BMTZ/DEF1 LP ) F(BMTZ/DEF1 LP). In the context of TSP, u j = u i + 1 wheneve x ij = 1 given that j and hence the whole ange of label values is used. In ou fomulation of MDC-MST, the fact that the same label value may be assigned to moe than one node esults in not using the whole ange of label values. This actually allows feasible solutions with diffeent labeling stuctues. Fo example, a feasible solution whee the same label is not assigned to all nodes at the same distance fom the oot is possible. Specifically, in the assignment of labels to nodes, thee ae thee possible cases fo an edge {i, j}: eithe x ij = 1, o x ji = 1, o both x ij = 0andx ji = 0. If x ij = 1, then u j u i + 1. Similaly, if x ji = 1, then u i u j + 1. If both x ij = 0andx ji = 0, then u i u j n 1and u j u i n 1. In this espect, any assignment of labels satisfying the afoementioned conditions gives a feasible solution. Two example feasible solutions ae given in Fig. 3. The special stuctue of MDC-MST allows us to make some impovements in the MTZ constaints that impove both the LP bounds and the solution times. These impovements ae obtained in two steps. In the fist step, constaints ae added to allow feasible solutions with a cetain labeling stuctue. In a feasible solution of MDC-MST, each node is eithe a cental node o a leaf node. Because a non-oot leaf node has one incoming ac whose oigin is necessaily a cental node, then a feasible solution can be obtained by equiing that the labels of all non-oot leaf nodes be geate than the highest possible label of cental nodes. This condition is easily fulfilled if we assign the label value n 1 to each non-oot leaf node while pemitting cental nodes to take label values of at most n 2. If all nodes othe than the oot node ae leaf nodes, then the oot node eceives the node label 0 and all othe nodes eceive node labels of n 1. If thee is a nonoot cental node, then its label will be between 1 and n 2. Thus, in finding feasible solutions fo MDC-MST, looking only fo solutions in which the label values of non-oot leaf nodes ae esticted to n 1 and the label values of cental nodes ae esticted to be less than o equal to n 2 is sufficient. In the second step, the ange of labels is esticted to a cetain inteval so that the feasible egion of the linea pogamming elaxation is futhe deceased. Recalling that the numbe of cental nodes is bounded above by n 2, it is diect to conclude that the labels of non-oot cental nodes may be esticted to the inteval fom n 1 n 2 to n 2. The label of the oot node is not included in this inteval because it is not known a pioi if the oot node will be a cental node o not. Thus, the label of the oot node is set to n 1 n 2 1. Two such feasible solutions with d = 4 ae given in Fig. 4. Note that non-oot cental node in the gaph on the ight can also take on the value of 6. We now give IMTZ, the Impoved MTZ constaints. IMTZ: Impoved Mille Tucke Zemlin sub-tou elimination constaints In addition to (33) and (34), and (37), u i (n 1)w il, i (V ) (38) u i (n 1) w ic, i (V ) (39) n 2 u i (n 1) 1, i = (40) d 1 n 2 u i (n 1), i (V ) (41) d 1 Constaints (38), togethe with constaints (34) that give an uppe bound on the node labels u i, establish that the labels of all non-oot leaf nodes ae equal to n 1. Constaints (39) estict the labels of cental nodes to be at most n 2. Constaint (40) sets the label of the oot node to n 1 n 2 1. Constaints (41) equie that the n 2. label values of non-oot nodes be at least n 1

6 I. Akgün, B. Tansel / Computes & Opeations Reseach 37 (2010) Fig. 4. Two feasible spanning tees with labels assigned by MTZ constaints with d = 4. Table 1 Chaacteistics of test poblems. P. ID P. type V d Instance = m 1 SYM SYM SYM SYM SYM SYM SYM SYM SYM SYM SYM SYM SYM SYM SYM CRD CRD CRD CRD CRD CRD CRD CRD CRD CRD CRD CRD CRD CRD CRD Poposition 4. Let BMTZP and IMTZP be two diffeent fomulations fo MDC-MST whee BMTZ and IMTZ ae used as tee-defining constaints, espectively, togethe with a set of degee-enfocing constaints, e.g., DEF1 o DEF2. Then, F(IMTZP LP ) F(BMTZP LP ), i.e., IMTZP dominates BMTZP. Poof. We note fist that all constaints of IMTZP and BMTZP ae alike except that (35) and (36) in BMTZP ae eplaced by (38) (41) in IMTZP. Conside now any feasible solution (x, u, w c, w l )toimtzp LP whee u includes all node vaiables except u which is just a constant defined by (40). This constant is eplaced by anothe constant defined by (36) in BMTZP. The solution (x, u, w c, w l ) satisfies all constaints of BMTZP LP since the[ ange [1, n 1] fo node] labels u i (i ) inbmtzp LP includes the ange (n 1) n 2, n 2 imposed on cental nodes by constaints (39) and (41) in IMTZP LP as well as the ange [n 1, n 1] imposed on leaf nodes by constaints (34) and (38) in IMTZP LP. This implies (x, u, w c, w l ) F(BMTZP LP ) and completes the poof. Due to Poposition 4, the LP polytope of IMTZP is a subset of the LP polytope of BMTZP. Computational studies in Section 5 (Table 1) veify this fact empiically. We give below two new fomulations of MDC-MST whee the tee-defining pat consists of IMTZ: IMTZ/DEF1: Impoved MTZ model with the fist set of degeeenfocing constaints. Objective function (1), constaints (2) (6), (10) and (11), (33) and (34), and (37) (41). IMTZ/DEF2: Impoved MTZ model with the poposed set of degeeenfocing constaints. Objective function (1), constaints (6), (10), (19) (32), (33) and (34), and (37) (41). As a coollay to Popositions 1 and 3, we can state the following poposition. Poposition 5. (i) F(IMTZ/DEF2 LP ) F(IMTZ/DEF1 LP ). (ii) F(IMTZ/DEF1 LP ) F(BMTZ/DEF1 LP ). (iii) F(IMTZ/DEF2 LP ) F(BMTZ/DEF2 LP ). IMTZ/DEF2 has 3.5n n 4.5 constaints, n 2 + n binay vaiables, and n continuous vaiables and is much moe compact than MCF/DEF1 with espect to the numbe of vaiables and constaints. On the othe hand, IMTZ/DEF2 has moe constaints but fewe vaiables than SCF/DEF1. A feasible solution equies that each non-oot node has exactly one inwad ac, which is povided by constaints (6). Howeve, computational studies show that bette solution times ae obtained by using them in fom, i.e., i x ij 1. In this egad, all solution times fo models using DEF2 ae obtained by using this fom. The inequality fom of (6) is well justified by the pesence of constaint (31) that limits the numbe of acs in the solution to n 1. Without (31), a solution esulting fom the inequality fom of constaints (6) may violate the tee stuctue if the ac costs do not satisfy the tiangle inequality and if the whole ange of labels is not used, but this will not occu with (31). MTZ constaints ae attactive due to thei compactness. Howeve, they ae well known fo poducing weak LP elaxation bounds. Oman and Williams [20] compaed the stengths of seveal fomulations of TSP by thei LP elaxation bounds. They found that the LP elaxation polytope obtained by MTZ constaints contains some of the seven existing fomulations. Specifically, the fomulation with MTZ constaints gives weake LP bounds than the ones based on single- and multi-commodity flow fomulations. This has led to vaious studies that augment the MTZ constaints to stengthen the LP bounds (e.g., Desoches and Lapote [21];Gouveia[22]; Gouveia and Pies [23]; Sheali and Discoll [24]). Although most studies focus on the TSP o TSP-elated poblems, the fomulations o liftings in those studies can be adapted to othe poblems whee sub-tous ae not allowed. Fo instance, Gouveia [22] used MTZ constaints in the context of hop-constained MST (HMST) whee each path stating fom the oot is equied to have at most a fixed numbe of hops (acs) and offes liftings to constaints (33) (35). We ty those liftings and the ones by Desoches and Lapote [21] in ou study as well. The

7 78 I. Akgün, B. Tansel / Computes & Opeations Reseach 37 (2010) Fig. 5. Two feasible spanning tees with u j = u i + 1 wheneve x ij = 1. liftings contibute to incease the LP elaxation bounds significantly fo BMTZ/DEF1 and IMTZ/DEF1 but do not incease o slightly incease (unde 1%) the LP bounds fo BMTZ/DEF2 and IMTZ/DEF2. Howeve, the liftings do not help to impove the solution times, which is in compliance with what Gouveia [22] has obtained fo HMST. Because the contibutions of the liftings to LP bounds and computation times ae vey maginal fo DEF2, we decide not to use the liftings in ou models to bette assess the computational effects of ou poposed fomulations. Note also that by augmenting MTZ constaints (e.g., [21]), it is actually possible to have u j = u i + 1 when x ij = 1. In this case, solutions such as the ones given in Fig. 5 ae obtained. Howeve, this pevents us fom using constaints (38), i.e., the fist impovement suggested above. 4. Root node selection Clealy, the optimal solution values of MDC-MST instances do not change depending on the oot node. Howeve, the LP elaxation bounds o how the solve poceeds may change, affecting the solution times significantly. Of the models of AMS, only MCF/DEF1 is symmetic elative to the oot and hence the LP elaxation bounds ae the same fo all oots [5]. All othe models given in the pape ae not symmetic and hence the LP bounds may change. AMS empiically showed that solution times may change significantly even when the fomulation is symmetic and suggested that the selection of the oot node may be of impotance. Howeve, they do not popose any methodology to select the oot node. They test the pefomance of thei models by selecting the fist node as the oot node. In ou studies, we obtain esults fo two diffeent oot nodes, namely the fist node and the node selected by a new methodology poposed heein. The methodology consists of (1) finding the smallest thee values in each ow of the cost matix, (2) finding the sum of the thee smallest values in each ow, and (3) selecting the node coesponding to the ow with the smallest sum found in step (2). The methodology is based on the idea that acs with lowest costs ae likely to be in the solution. Empiical esults show that the solution times ae impoved significantly fo the models of AMS and IMTZ/DEF2 when the oot node is selected with the poposed methodology. The oot nodes m calculated by using the methodology ae given in the last column in Table Computational studies Computational studies ae pefomed by using specially stuctued, had CRD and SYM instances which ae complete gaphs with Euclidean costs set to intege units (e.g., [15]). CRD instances ae 2-dimensional Euclidean poblems whee the points ae geneated andomly with a unifom distibution in a squae. SYM instances ae analogous to CRD instances but with points geneated in highe dimensional Euclidean space. These poblems have been widely used in the liteatue to test DC-MST (e.g., [9,25,26]). Following AMS, CRD and SYM instances defined on netwoks with 30 and 50 nodes ae used in ou computational studies. Fo each netwok size, thee diffeent instances ae tested fo diffeent values of d. Table 1 summaizes the chaacteistics of test poblems. Computational tests ae pefomed on a PC with a 3.0 GHz Intel Coe 2 duo pocesso and 3 GB of RAM by using ILOG CPLEX 9.0. The models ae un until optimality is attained o fo 3 h (10,800 CPU s) at maximum and by using default settings of CPLEX (e.g., moving the best bound stategy fo banching is used, cuts ae allowed, see [27]) except that file stoage is set to 3, which allows tee file to be stoed on the had disk when it eaches the default limit in ode not to un out of memoy. To compae ou esults to those of AMS, we have modeled and solved the models of AMS on the same PC. In the tables pesenting computational studies, LP elaxation bounds, un times, optimal objective function values, and elative optimality gaps ae given. Relative optimality gap is defined as BP BF /( BP ), whee BP is the objective function value of the best intege solution and BF is the best emaining objective function value of any unexploed node (see [27]). Undelined values in the tables show that the poblem is not solved to optimality within the allotted time of 10,800 s Compaison of MTZ-based models among themselves Table 2 gives computational esults fo BMTZ/DEF1, BMTZ/DEF2, IMTZ/DEF1, and IMTZ/DEF2 fo = 1. In tems of LP bounds, the esults show that the weakest LP bounds ae obtained fo BMTZ/DEF1. The bounds fo IMTZ/DEF1 ae bette than those fo BMTZ/DEF1 implying that IMTZ constaints ae stonge than BMTZ constaints. The diffeence between BMTZ and IMTZ when they ae used with DEF1 is not obseved when they ae used with DEF2. Both BMTZ/DEF2 and IMTZ/DEF2 give the same LP bounds which ae much bette than those obtained fom IMTZ/DEF1. The fact that both basic and impoved vesions of MTZ give the same bounds when used with DEF2 indicates that DEF2 dominates and oveshadows any contibutions that might have been coming fom the impoved stuctue of IMTZ ove BMTZ. The fact that IMTZ/DEF2 (as well as BMTZ/DEF2) poduces much bette LP bounds than IMTZ/DEF1 indicates that the majo contibution to the impovement in the LP bounds comes fom DEF2. This shows that DEF2 is significantly stonge in poducing LP bounds than DEF1. In tems of solution times, IMTZ/DEF2 gives the best pefomance with espect to CPU times. Even though the pefomance of BMTZ/DEF2 is close to that of IMTZ/DEF2 fo SYM instances, the supeio pefomance of IMTZ/DEF2 in CPU time becomes moe ponounced fo CRD instances. In this egad, we take IMTZ/DEF2 as the main model to compae with the flow-based models Compaison of degee-enfocing constaints based on LP bounds Table 3 gives LP elaxation bounds fo MCF/DEF1, MCF/DEF2, BMTZ/DEF1, and BMTZ/DEF2 (IMTZ/DEF2). The last two columns clealy indicate that, as explained in Section 5.1, DEF2 is stongly bette than DEF1 when used with MTZ constaints. The columns fo MCF/DEF1 and MCF/DEF2 give futhe evidence fo the dominance of DEF2 ove DEF1. Its dominance becomes moe appaent as the degee equiement d inceases.

8 I. Akgün, B. Tansel / Computes & Opeations Reseach 37 (2010) Table 2 Solution times, integality gaps, and LP elaxation bounds fo BMTZ/DEF1, BMTZ/DEF2, IMTZ/DEF1, and IMTZ/DEF2 with = 1. P. ID BMTZ/DEF1 IMTZ/DEF1 BP Gap (%) Time (s) z LP BP Gap (%) Time (s) z LP , , P. ID BMTZ/DEF2 IMTZ/DEF2 BP Gap (%) Time (s) z LP BP Gap (%) Time (s) z LP , , , , , , , , , , Undelined values show that the poblem is not solved to optimality. The dominance of DEF2 ove DEF1 can be bette assessed by compaing the columns fo MCF/DEF2 and BMTZ/DEF2. Even though MCF constaints ae known to give much stonge epesentation of the spanning tee polytope than that of the MTZ constaints (e.g., Oman and Williams [20]), BMTZ/DEF2 becomes competitive with MCF/DEF2 especially fo SYM instances. Fo CRD instances, MCF/DEF2 gives bette LP elaxation bounds than BMTZ/DEF2. In this egad, the dominance of MCF constaints ove BMTZ is not compensated fo by DEF2. Howeve, DEF2 consideably deceases the gap between the LP elaxation bounds of MCF- and MTZ-based fomulations implying its stength ove DEF1. That is, the methodology does not guaantee a node with the best LP bound; computational studies show that LP bounds ae on the aveage bette (sometimes the best) at least fo the poblems studied. Howeve, Table 5 indicates that significant impovements in solution times of IMTZ/DEF2 ae ealized fo = m, fo which moe details ae given in Section 5.4. As Table 6 demonstates, the solution times of MCF/DEF1 ae also significantly impoved, implying that using =m impoves the solution times of the flow-based models as well, especially fo hade CRD instances. Fo example, the solution times of , , and s fo P. 16, 20, and 21, espectively, ae impoved to 88.72, and s Compaison of esults fo diffeent oot nodes Table 4 gives LP elaxation bounds of MCF/DEF1, MCF/DEF2, and IMTZ/DEF2 fo = 1and = m, i.e., the methodology-selected oot node. The table demonstates that LP elaxation bounds obtained with = m ae bette in geneal than the ones obtained with = Compaison of IMTZ/DEF2 and flow-based models with espect to solution times Table 5 gives computational esults fo IMTZ/DEF2 and the flowbased models of AMS. In epoting the computational esults fo the flow-based models, esults ae not given fo each model sepaately.

9 80 I. Akgün, B. Tansel / Computes & Opeations Reseach 37 (2010) Table 3 Compaison of degee-enfocing constaints by LP elaxation bounds with = 1. P. ID MCF BMTZ DEF1 DEF2 DEF1 DEF2 (IMTZ/DEF2) Table 4 LP elaxation bounds of diffeent models fo diffeent oot nodes. P. ID MCF/DEF1 MCF/DEF2 IMTZ/DEF2 ( = 1) ( = m ) ( = 1) ( = m ) ( = 1) ( = m ) Fo = m,seetable 1. In all cases, the best objective function value (BP), the un time, and the optimality gap of the model with the best esults (the smallest solution time fo the poblems solved to optimality and the smallest optimality gap fo the poblems not solved to optimality) ae epoted. Computational esults show that the flow-based models of AMS can optimally solve all 12 poblem instances with 30 nodes by at least one of thei fomulations. The solution times change fom 0.91 to s fo SYM instances and fom to s fo CRD instances. Regading poblems with 50 nodes, flow-based models of AMS cannot optimally solve 6 instances, all of which ae CRD instances, out of 18 within the allotted time. Fo 12 solved poblems (of which 9 ae SYM and 3 ae CRD instances), the solution times change fom to s fo SYM instances and fom to s fo CRD instances. These esults show that the solution times of SYM instances ae much bette than those of CRD instances. Computational esults fo IMTZ/DEF2 show that, when the oot node is the fist node, IMTZ/DEF2 can solve all instances with 30 nodes except P. 17. Fo all solved poblems except P. 16, the solution times ae incompaably bette than those of AMS. The solution times change fom 0.05 to 1.14 s fo SYM instances and fom to s fo CRD instances. As to the poblems with 50 nodes, IMTZ/DEF2 can solve all instances solved by AMS with incompaably much bette solution times. The solution times change fom 0.92 to s fo SYM instances and fom to s fo CRD instances. IMTZ/DEF2 cannot solve fou instances, which ae also not

10 I. Akgün, B. Tansel / Computes & Opeations Reseach 37 (2010) Table 5 Solution times and integality gaps. P. ID SCF/MCF with DEF1 o DEF1 ( = 1) IMTZ/DEF2 ( = 1) IMTZ/DEF2 ( = m ) BP Gap (%) Time (s) BP Gap (%) Time (s) BP Gap (%) Time (s) , , , , , , , , , , , , , Undelined values show that the poblem is not solved to optimality. Fo = m,seetable 1. Table 6 Solution times and integality gaps of flow-based models fo diffeent oot nodes. P. ID MCF/DEF1 ( = 1) MCF/DEF1 ( = m ) BP Gap (%) Time (s) BP Gap (%) Time (s) Fo = m,seetable 1. solved by AMS. Howeve, IMTZ/DEF2 can solve two CRD instances unsolved by AMS. The solution times fo those poblems ae and s, which ae also incompaably bette. Computational esults fo IMTZ/DEF2 show that, when the oot node is selected by using the poposed methodology, the solution times obtained with the fist node being oot node ae impoved significantly. Fo example, the solution time of s fo P. 16 is impoved to s and P. 17 not solved in 10,800 s is solved in s. In this case, fou poblems out of the six that ae not solved by AMS ae now solved to optimality with solution times changing fom to s. Fo unsolved poblems, IMTZ/DEF2 with = m has smalle optimality gaps than flow-based models and IMTZ/DEF2 with = 1. Moeove, it has bette objective function values and lowe bounds. Specifically, the lowe bounds fo P. 22 ae 5306, , and fo flow-based model, IMTZ/DEF2 with = 1, and IMTZ/DEF2 with = m, espectively. Fo P.23, the lowe bounds ae , , and In this egad, because inceasing the lowe bounds constitutes most of the solution time, it is highly likely that optimality will be eached in shote times when = m. When the esults ae consideed as a whole, it is obseved that the solution times of IMTZ/DEF2 ae incompaably bette than those of the flow-based models in geneal. This combined with the fact that the test poblems ae specially stuctued leads us to conclude