DISTANCE TRANSFORMATION FOR NETWORK DESIGN PROBLEMS

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1 DISTANCE TRANSFORMATION FOR NETWORK DESIGN PROBLEMS A Ridha Mahjoub, Michae Poss, Luidi Simonetti, Eduardo Uchoa To cite this version: A Ridha Mahjoub, Michae Poss, Luidi Simonetti, Eduardo Uchoa. DISTANCE TRANSFORMA- TION FOR NETWORK DESIGN PROBLEMS. 7. <ha-697> HAL Id: ha Submitted on Nov 7 HAL is a muti-discipinary open access archive for the deposit and dissemination of scientific research documents, whether they are pubished or not. The documents may come from teaching and research institutions in France or abroad, or from pubic or private research centers. L archive ouverte puridiscipinaire HAL, est destinée au dépôt et à a diffusion de documents scientifiques de niveau recherche, pubiés ou non, émanant des étabissements d enseignement et de recherche français ou étrangers, des aboratoires pubics ou privés.

2 DISTANCE TRANSFORMATION FOR NETWORK DESIGN PROBLEMS A. RIDHA MAHJOUB, MICHAEL POSS, LUIDI SIMONETTI, AND EDUARDO UCHOA Abstract. We propose a new generic way to construct extended formuations for a arge cass of network design probems with given connectivity requirements. The approach is based on a graph transformation that maps any graph into a ayered graph according to a given distance function. The origina connectivity requirements are in turn transformed into equivaent connectivity requirements in the ayered graph. The mapping is extended to the graphs induced by fractiona vectors through an extended inear integer programming formuation. Whie graphs induced by binary vectors are mapped to isomorphic ayered graphs, those induced by fractiona vectors are mapped to a set of graphs having worse connectivity properties. Hence, the connectivity requirements in the ayered graph may cut off fractiona vectors that were feasibe for the probem formuated in the origina graph. Experiments over instances of the Steiner Forest and Hop-constrained Survivabe Network Design probems show that significant gap reductions over the state-of-the art formuations can be obtained. Key words. Extended Formuations, Network Design, Benders decomposition. AMS subject cassifications.. Introduction. Let G = (V, E) be an undirected graph with n vertices and m edges with positive costs c e, e E. Let D V V be a set of demands, each demand (u, v) D has its connectivity requirements: the existence of a certain number of (u, v)-paths, possiby those paths may need to satisfy some side constraints. The cass of Network Design Probems (NDPs) considered in this paper consists in finding a subgraph of G with minimum cost satisfying the connectivity requirements of a demands. For exampes: Steiner Tree Probem (STP): the input gives a set T V of terminas. D can be defined as {(u, v) : u, v T, u v}. The connectivity requirement of a demand (u, v) D is the existence of a path joining u and v. Steiner Forest Probem (SFP): An arbitrary set D is given as input. The connectivity requirement is the existence of a path joining the vertices in each demand. Hop-constrained Steiner Tree Probem (HSTP): the input gives a set T V of terminas, a root vertex r T and an integer H. The demand-set D is defined as {(r, v) : v T \{r}}. The connectivity requirement is the existence of a path with at most H edges (hops) joining the vertices in each demand. Hop-constrained Survivabe Network Design Probem (HSNDP): the set D and integers H and K are given as input. The connectivity requirement of a demand (u, v) D is the existence of K edge-disjoint (u, v)-paths, each path having at most H edges. Severa other NDPs found in the iterature aso beong to this cass, the connectivity Université Paris-Dauphine, Pace du Marécha De Lattre de Tassigny, 75775, Paris Cedex 6, France (mahjoub@amsade.dauphine.fr). UMR CNRS 556 LIRMM, Université de Montpeier, 6 rue Ada, 9 Montpeier Cedex 5, France. (michae.poss@irmm.fr). COPPE - PESC, Federa University of Rio de Janeiro, Cidade Universitria, Centro de Tecnoogia, Boco H, zip 9-97, Rio de Janeiro, Brazi. (uidi@ic.uff.br). Universidade Federa Fuminense, Rua Passo da Pátria, 56 CEP -, Niterói, Brazi (uchoa@producao.uff.br).

3 requirement may ask for a number of node-disjoint paths, side constraints may incude maximum deay, etc. On a those cases, the natura formuations over the design variabes x e, e E, take the foowing format: (a) Min c e x e (b) (c) e E S.t. x P (u, v) (u, v) D x e binary e E, where the binary points in poyhedra P (u, v) correspond to a subgraphs satisfying the connectivity requirements of demand (u, v). Even on cases when each poyhedron P (u, v) is integra, i.e., when the formuation aready contains the best possibe inequaities that are vaid for each individua demand, the overa formuation is often sti weak. For exampe, consider the cassic STP, where the natura Formuation () corresponds to the one proposed by Aneja []. In that case, constraints (b) are given by the so caed undirected cut constraints: for each cut separating two terminas u and v, there must be at east an edge in the soution. The inear reaxation of Aneja s formuation does not provide very good bounds. Even when the formuation is reinforced with sophisticated cuts discovered after poyhedra investigation [, 5], the duaity gaps may be sti significant. Fortunatey, Wong [] proposed a much better STP formuation. The main observation is that the set of demands D = {(u, v) : u, v T, u v} can be repaced by D = {(r, v) : v T \ {r}}, where r T is an arbitrariy chosen termina, without changing the optima soution. Then, G can be repaced by a bidirected graph G = (V, A) where A has two opposite arcs (i, j) and (j, i) for each edge e = {i, j} in E, with c((i, j)) = c((j, i)) = c(e). The probem now consists in ooking for a minimum cost subgraph of G having a directed path from r to every other termina. The proposed formuation uses directed cut constraints: for each directed cut separating r from a termina, there must be at east an arc in the soution. Wong s formuation is remarkaby strong. Its duaity gaps are ess than.% on amost a SteinLIB instances [5], except on the artificia instances created with the intent of being hard. In fact, some of the most effective STP codes of today, which are capabe of soving non-artificia instances (ike those from VLSI design) with tenths of thousands of vertices, usuay do not bother to separate cuts other than the simpe directed cut constraints. Instead, their agorithmic effort is focused on devising graph reductions and dua ascent procedures in order to speedup the soution of that inear reaxation [9, ]. Of course, one woud ike to have simiar reformuation schemes abe to produce strong bounds for other NDPs. An advance in that direction was the work by Gouveia et a. [] on hop/diameter constrained spanning/steiner Tree Probems. On a those cases it was possibe to show that the probem coud be transformed into a STP over a directed graph composed of up to H ayers, each ayer consisting of copies of the origina graph. Additiona arcs joining the ayers and a few extra vertices/arcs are required. By appying Wong s directed cut formuation over the transformed graphs, very sma gaps are obtained. Even though the ayered graph approach mutipies the size of the graph, it sti aowed the soution of instances with hundreds of vertices. Recenty, the ayered graph approach was successfuy appied on a number of other NDPs [, 7,, ], effective techniques were introduced the cope with the arge transformed graphs and to hande other kinds of constraints.

4 Fig.. Optima soution of a HSNDP instance with K = and H = ; n =, D = {(, v) : v =,..., }, compete graph, Eucidean costs. Nevertheess, we shoud remark that a those strong reformuation schemes are restricted to reativey simpe NDPs. As the reformuations depend on transforming the probem into a directed STP, they are restricted to probems where the optima soution topoogy shoud be a singe tree. NDPs with more compex topoogies do not seem to admit directed formuations. Sometimes this is reated to the fact that both orientations of an edge can be used in the paths connecting different demands. Figure shows the optima soution of a HSNDP instance with demand-set D = {(, v) : v =,..., }, K = and H =. Even tough a demands incude vertex, which makes this vertex a natura root for the probem, it can be seen that the same edge can be used in different directions by different demands. For exampe, demand (, 7) is connected by paths 5 7, 8 9 7, and 8 7; demand (, 5) is connected by paths 5, 6 5 and Suppose that G is transformed into a bidirected graph. The corresponding directed soution woud need to pay for both arcs (5, 7) and (7, 5), which is not correct. Even the seemingy simpe SFP, where the soution may be formed by a set of disconnected subtrees, does not seem to admit a strong reformuation by turning it into a directed probem over a transformed graph. Here, the difficuty ies in the fact that it is not possibe to know beforehand which demands wi be connected by the same subtree. In fact, the strongest known reformuation for the SPF, proposed in [6], is ony sighty better than the natura Formuation (). The Distance Transformation (DT) is an origina reformuation technique proposed to obtain stronger formuations for genera NDPs, incuding those where the currenty know reformuation techniques do not seem to work. Starting from Formu-

5 ation (), the resuting reformuation wi have the foowing format: (a) (b) (c) (d) Min c e x e e E S.t. Ax + Bw + Cy b (w, y) P (u, v) (u, v) D x e binary e E, where A, B, C and b are matrices of appropriated dimensions. Constraints (b) are probem-independent. They are devised to transform a soution over the variabes x into a distance expanded soution over the new variabes w and y. The origina connectivity constraints in each P (u, v), over the x variabes, are transformed into new connectivity constraints P (u, v), over the (w, y) variabes. The purpose of DT can be informay described as foows. Let x be a binary vector in {, } m and G(x) = (V, E(x)) be the subgraph induced by x. The distance transformation maps G(x) into a subgraph of the ayered graph that consists of n + copies of G pus edges between adjacent ayers. The mapping considers a chosen source subset S V and sends each vertex i V to a vertex in ayer, n, according to its distance from set S. Each edge {i, j} G(x) is sent to an edge in the ayered graph according to the ayers of i and j. The transformed graph is represented by variabes w and y, the mapping is encoded in Constraints (b). The interest of the transformation is the foowing. If x is integra, the transformed graph is isomorphic to G(x). However, its extension to graphs G(x) induced by fractiona vectors eads to transformed graphs that are ess connected than G(x). Hence, the corresponding (w, y) fractiona soution is much more ikey to be cut by inequaities (c). At first gance, this concept may seem simiar to the ayered formuations used in [,, 7,, ]. They are, however, competey different. In those previous works, the roe of the ayered graphs was to mode connectivity requirements with hop/distance/deay constraints. In contrast, the DT is not inked to any particuar kind of connectivity requirements. In fact, those techniques are orthogona: it is possibe to combine DT and ayered formuations... Contributions and structure of the paper. The idea behind DT has its origin in an extended formuation (denoted as DT-HOP MCF) introduced in Mahjoub et a. [8] for the HSNDP. The numerica resuts obtained were very positive, significant gap reductions with respect to previous known formuations ed to the soution of severa open instances from the iterature. There was however no theoretica justification for the gap reductions obtained by what seemed to be an ad-hoc reformuation for that specific network design probem. The purpose of this paper is to introduce the Distance Transformation, a generic graph transformation that works for the aforementioned cass of network design probems. We show in Section. how the distance transformation is naturay defined as a graph transformation. The transformation is extended to graphs defined by fractiona vectors by using convexity arguments. We expain then in Section. how this idea distance transformation reduces the connectivity of fractiona vectors by spitting nodes. Section studies inear programming formuations for the distance transformation. A natura formuation is presented in Section.. Section. studies reaxations of the formuation to make it more effective computationay. Section shows how basic connectivity requirements can be incorporated into the inear programming formuations through inequaities or fow formuations. The resuting formuations are

6 (a) G (b) DT S (G) Fig.. Origina graph G and DT S (G) with S = {, }. iustrated numericay in Section 5 on two network design probems: the Steiner Forest Probem and the Hop-constrained Survivabe Network Design Probem. The paper is finay concuded in Section 6... Notations. Let F (, ) = [, ] m \ {, } m be the set of vectors in the unit hypercube having at east one fractiona component. Undirected edges are denoted by {i, j}, and directed edges are denoted by (i, j). In addition, we use the foowing conventions in a figures in the paper. The set of source nodes S contains the vertices fied in back. Pain edges and nodes have weights equa to, whie dotted edges and nodes with dotted circe have weights equa to.5. The set of extreme points of a poytope P is denoted as ext(p ).. The unitary distance transformation. We introduce formay the distance transformation in Subsection. and study in Subsection. the connectivity of the ayered graphs obtained for fractiona vectors x. Linear programming formuations for DT sha be discussed in the next section... Definition. The distance transformation considers an arbitrary subset of root nodes S V and maps any subgraph G = (V, E ) of G = (V, E) to a ayered graph. The mapping is based on the distance in G between any node i V and S. Define the ayered and undirected graph G = (V, E), where V = V... V n, with V = {i : i V } for each =,..., n and i be the copy of vertex i in the -th eve of graph G. Then, the edge set is defined by E = {{i, j } {i, j} E, n, +} {{i n, j n } {i, j} E}. Let P(G) be the set of a subgraphs of G that contain a nodes of V and P(G) be the set of a subgraphs of G that contain exacty one copy of each node of V, formay: P(G) = {G = (V, E ) E E} and P(G) = {G = (V, E ) V V and V {i : n} =, i V, E E}. The distance transformation is a function DT S : P(G) P(G) that maps any subgraph G P(G) to ayered graph DT S (G ) P(G) defined as foows. Let dist G (i, S) be the ength of the shortest path in G connecting i to a node in S or if no such path exists. For each i V such that dist G (i, S) <, vertex i dist G (i,s) beongs to DT S (G ). Otherwise, i n DT S (G ). For any {i, j} G, {i, i } beongs to DT S (G ) if and ony if = dist G (i, S) and = dist G (j, S). Remark that for any {i, j} G, dist G (i, S) dist G (j, S). In other words, the images of adjacent vertices in G ie in adjacent eves in DT S (G ). Figure shows an exampe of distance transformation. A crucia property of the distance transformation is that G and DT S (G ) are isomorphic.

7 6 Any graph in P(G) can be characterized as G(x) = (V, E(x)) where x {, } m is a binary vector whose component {i, j} is equa to if and ony if {i, j} E(x); we say that G(x) is induced by x. We can aso describe a graph G = (V, E ) in P(G) using vectors w {, } n(n+) and y {, } (n )m, defined as foows: wi = iff i V ; y ij = iff {i, j } E. Conversey, given binary vectors w and y, G(w, y) denotes the subgraph of G that contains the vertices (resp. edges) associated to the components of w (resp. y) equa to. Hence, we aso define the function DT S : {, } m {, } n(n+)+(n )m as DT S (x) = (w, y) where (w, y) characterizes the graph DT S (G(x)). In this paper, we address network design probems formuated as inear programs with binary variabes. In this context, x is a vector of optimization variabes comprised between and. To use the distance transformation, we must describe DT S through a system of inear constraints such that for each x {, } m, (x, w, y) is feasibe if and ony if (w, y) = DT S (x). Certainy, the smaest poytope defined by such constraints is () P S = conv{(x, DT S (x)), x {, } m }. Using P S, we are abe to define DT S that is defined for any vector in x [, ] m, fractiona or not, as the foowing projection: () DT S (x) = {(w, y) (x, w, y) P S }. The more genera definition () is compatibe with the previous definition for integra vectors because whenever x is integra DT S (x) reduces to the singeton {DT S (x)}. Otherwise, set DT S (x) is a poytope with non-zero dimension. Therefore, DT S : [, ] m [, ] n(n+)+(n )m is a point-to-set mapping. We introduce next a straightforward characterization of DT S (x). Theorem. Let x q, q =,..., m, be the enumerated set of a vectors in {, } m. A vector (w, y) beongs to DT S (x) if and ony if there exists a vector of convex mutipiers λ such that x = m q= λq x q and (w, y) = m q= λq DT S (x q ). Proof. Direct from () and (). Extending the previous definitions to fractiona vectors, we can define G(x) and G(w, y) as the graphs induced by the positive components of x and (w, y), respectivey. We provide next an exampe of DT S (x ) for the fractiona vector x = (.5,.5,.5) associated to the graph G = (V, E ) with V = {,, } and E = {{, }, {, }, {, }}, and S = {}. Figures (a) (h) represent the graphs induced by the vectors of DT S (x) for each x {, }. In view of Theorem, any eement in DT S (x ) is obtained from expressing x as a specific convex combination of the vectors in {, }. Hence, one readiy verifies by examination that ext(dt S (x )) contains the vectors that induce the four graphs from Figure, which correspond to four different convex combinations describing x. One can see the spitting of nodes occurring in the graphs induced by a vectors of ext(dt S (x )). The spitting of nodes corresponds to fractiona vaues of w. As wi be seen in the next section, the spitting of nodes for a vectors of ext(dt S (x )) is a necessary condition for DT to be usefu. This does not aways happens. Consider a simiar exampe, except that S = {, }. The graphs for vectors in ext(dt S (x )) are shown in Figure 5. We see that the eft graph does not contain spit nodes. Hence, that choice of S woud not give a usefu transformation.

8 7 DT S( )= DT S( )= (a) G (x ) (b) G (x ) DT S( )= DT S( )= (c) G (x ) (d) G (x ) DT S( )= DT S( )= (e) G (x 5 ) (f) G (x 6 ) DT S( )= DT S( )= (g) G (x 7 ) (h) G (x 8 ) Fig.. DT S (x) for each x {, }... Connectivity of DT S. The distance transformation transates the origina network design probem defined in G into a network design probem defined in G, by importing to G the connectivity requirements described in G. In this section, we iustrate how this is done for the simper case of demands requiring the existence of K edge-disjoint paths between some pairs of vertices. To this end, we add up to D supervertices to G and G(w, y), obtaining G = (V, E) and G(w, y) = (V(w), E(w, y)), respectivey. Namey, for each demand (u, v) D, we create two supervertices s(u) and t(v) respectivey inked to vertices u and v by directed edges (s(u), u ) and (v, t(v)) for each n. For any x {, } m, the requirement of the existence in G(x) of K edge-disjoint (u, v)-paths becomes the requirement of the existence in G(w, y) of K edge-disjoint (s(u), t(v))-paths. To mode these connectivity requirements by inear constraints, we need to consider graphs G(x) and G(w, y) as weighted graphs. The weight on any edge in G(x) is equa to the vaue of the associated component of x. For G(w, y), we must distinguish between the undirected edges, and the directed ones that ink G(w, y) to the supervertices. The weight on any undirected edge is equa to the vaue of the associated component of y, whie the weight on any directed edge inking s(u) (resp. t(v)) and u (resp. v ) is equa to Kwu (resp. Kwv). We define next the connectivity of weighted graph G(x) as the vector C(x) R D + with C uv (x) equa to the maximum fow between u and v in G(x). Simiary, we define the connectivity of weighted

9 8 = ½ DT S( )+ ½ DT S( ) = ½ DT S( )+ ½ DT S( ) = ½ DT S( )+ ½ DT S( ) = ½ DT S( )+ ½ DT S( ) Fig.. Graphs in ext(dt S (G (x))) when S = {}. Fig. 5. Graphs in ext(dt S (G (x))) when S = {, }. graph G(w, y) as the vector C(w, y) R D + with C uv (w, y) equa to the maximum fow between the supervertices s(u) and t(v) in G(w, y). With these definitions, the network design probem defined in G with optimization variabes x and connectivity requirements (5) C uv (x) K, (u, v) D, can be reformuated as a network design probem defined in G with optimization variabes x, w, and y, and containing two groups of constraints:. Constraints specifying that (x, w, y) P S ;. Connectivity requirements constraints (6) C uv (w, y) K, (u, v) D. In the foowing we denote the feasibiity set of constraints (5) by C = {x [, ] m C uv (x) K, (u, v) D}

10 9 and the projected feasibiity set of constraints (6) by C P S = Proj x {(x, y, w) P S C uv (w, y) K, (u, v) D}, where Proj x (X ) denotes the projection of set X on variabes x. We aso denote the convex hu of the feasibe soutions of the network design probem defined by connectivity requirements (5) as C opt = conv{x {, } m C uv (x) K, (u, v) D}. The resut beow shows that the approximation of C opt provided by C P S is tighter than the one provided by C. Proposition. For any connectivity requirements constraints of the form (5) and any S, it aways hods that (7) C opt C P S C. Proof. Let x {, } m and DT S (x) = (w, y). The first incusion foows from the fact that G(x) and G(w, y) are isomorphic, and thus, C uv (x) = C uv (w, y) for each (u, v) D. To prove the second incusion, we first prove that C uv (x) C uv (w, y), (u, v) D. Let (u, v) D and et g be any vector defining a fow from s(u) to t(v) in G(w, y). Then, we can fatten G(w, y) to obtain a fow f from u to v. Namey, we define the fow f on edge {i, j} G(x) as the sums of the fows described by g on a {i, j } G(w, y). The fows on the directed edges inking the supervertices s(u) and t(v) to G do not matter since we are ony interested in a fow from u to v. It is easy to see that the resuting fow f satisfies the capacity constraints, the baance constraints and conveys the same amount of fow from u to v that g conveys from s(u) to s(v). Therefore, if x C P S, then we aso have that x C, proving the incusion. The power of distance transformation ies in its reduction of the connectivity of the graphs induced by (w, y) DT S (x) for fractiona vectors x (, ) m. Even when a fractiona soution x is not cut by connectivity requirement constraints (5), meaning that x C, it may we happen that a (w, y) DT S (x) are cut by connectivity requirement constraints (6), impying x / C P S. The next resut provides an exampe of x C \ C P S. Proposition. Consider the network design probem defined on graph G from Figure under the connectivity requirements C (x) and C (x). It hods that (8) C P S C. Proof. The incusion foows from Proposition. To see that the incusion is strict, we show that the fractiona soution x defined by x = x = x =.5 beongs to C \C P S. Ceary, x C since we can use the cyce to send haf a unit in each direction for both demands in D. To see that x / C P S, we must show that a (w, y) DT S (x) vioate C (w, y) or C (w, y). First, we remark that (9) n wi =, i V, =

11 because any (w, y) DT S (x) can be extended to (x, w, y) P S, which can be written as a convex combination of binary vectors (x, w, y) satisfying equation (9). Consider then the four extreme points of DT S (x), which are depicted in Figure where a edges have vaues equa to.5. We can see that for each of these graphs, vertex or vertex (or both) beongs to the eve, so that the corresponding vaue of w i =.5 for i = or. Since eve is not connected to,.5 units of fow cannot reach vertex and equation (9) impies that C (w, y) or C (w, y) is vioated. Since any (w, y) DT S (x) can be written as a convex combination of these graphs, there is aways a fraction of the fow that cannot reach vertex, proving the resut. The intuitive idea behind the distance transformation is that fractiona soutions are often mapped to ayered graphs where node spitting occurs, as in Figure. The node spitting then cuts some paths in the origina graph which resuts in a decrease of connectivity. This decrease is often enough to cut the fractiona soutions. Figure shows that the node spitting breaks the cyce in a cases but the upper one. Nevertheess, the connectivity of the upper-eft graph is aso reduced because of the imited weight avaiabe on the directed edge inking the supervertices to the ayered graph. When no node-spitting occurs, as in the eft graph of Figure 5, graphs G(x) and G(w, y) are isomorphic so that they satisfy the same connectivity requirements. It is therefore usefu to be abe to discover whether a given distance transformation eads to spitting of the fractiona vectors. The resut beow partiay answers this question by providing an approach to find out whether the graph induced by a given fractiona soution (x, w, y) P S contains spit nodes. Proposition. Let x F (, ) and (w, y) DT S (x). positive convex mutipiers λ such that () (x, w, y) = Q λ q (x q, w q, y q ), q= Consider a set of Q where x q {, } m and (w q, y q ) = DT S (x q ) for each q =,..., Q. Then, any node i V is spit Q times in G(w, y), where Q {,..., Q} corresponds to the number of different vaues in set () {dist G(x q )(i, S), q =,..., Q}. Proof. From equation (), we have that wi = Q q= λq w q i. By definition of DT S, we further have that w q dist G(x q )(i,s) i = for each q {,..., Q}, so that wi = λ q, q:dist G(x q ) (i,s)= which is positive for each {,..., n} corresponding to a vaue in the set defined in (). Proposition is an important resut in understanding the structure of fractiona vectors in P S, which is the key to efficienty appy distance transformation to network design probems. In particuar, the proposition shows that any node i V corresponding to (x, w, y) defined by equation () is spit if and ony if () dist G(xq )(i, S) dist G(x q )(i, S),

12 for some q q in {,..., Q}. The difficuty of using the node spitting to cut a particuar fractiona soution x is that we must ensure that the spitting occurs for a (w, y) DT S (x), which is a compex task in genera. Nevertheess, for some very specia cases it is possibe to predict that spitting aways occurs, see the resut beow. Proposition 5. Consider the distance transformation defined by a unique root S = {i} for some i V, et x F (, ), and consider a node j V \ {i}. If x ij (, ), node j is spit in G(w, y) for a (w, y) DT S (x). Proof. Let (x, w, y) = Q λ q (x q, w q, y q ) q= be any vector in P S. Because x ij (, ), there exists q and q in Q such that x q ij = and x q ij =. Hence, () hods, yieding the resut.. Linear programming formuations for the DT... Natura formuation for P S. So far we have been using abstract distance transformations based on the idea poytope P S. To use DT in practice, we shoud have a inear formuation for P S, providing its convex hu in the idea case. Unfortunatey, the compete description of P S is not easy to find. Beow, we provide a poynomia formuation for P S which, athough not competey describing P S, eads to very good improvements in the inear programming reaxation of some network design probems with connectivity requirements. Let δ E (S) = {{i, j} E, {i, j} S = } and E = E \ δ E (S). The formuation beow inks the three groups of variabes x,w, and y using the foowing constraints: (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) n = n yij + = y (n )(n ) ji y (n )(n ) ij (y (+) ij wi =, i S n wi =, i V \S = yij = x ij, {i, j} δ E (S), i S + y (+) ji ) = x ij, {i, j} E y ij yij + y(+) ij + y ( ) y ij + y(+) ji w j, {i, j} δ E (S), i S yij + y ij w i, yij + y ji {i, j} w E j, + y (n )(n ) ji + y (n )(n ) ij w i ji wi, + y ( ) ij wj, {i, j} E ; =,..., n w n w n {j,i} E i, j, x, w, y. {i, j} E y ( ) ji, i V \S; =,..., n Constraints (a b) state that each vertex shoud be in one of its possibe eves. Constraints (c h) state that each origina edge variabe x ij shoud be transated

13 (a) G(x) (b) G(w, y) (c) G(w, y) for some for the unique (w, y) ext(dts F (x)) (w, y) DT S (x) Fig. 6. (S = {}). Exampe of fractiona x, the singe point in DT S (x), and a point in ext(dt F S (x)) into a variabe y ij such that both w i and w j have vaue one. Constraints (i) state that a vertex i can ony be in eve < n if it is reached by at east one edge {j, i} from eve ( ). Let PS F be the poytope defined by constraints (). We prove beow that the above constraints yied a vaid formuation for poytope P S defined in (). Proposition 6. Linear constraints () yieds a vaid formuation for poytope P S. Hence, given x {, } m, (x, w, y) PS F if and ony if (w, y) = DT S(x). Proof. The vaidity of (a j) foows from the observation that (x, DT S (x)) satisfies the constraints for each x {, } m. To see that (a j) is aso a formuation for P S, we consider a binary vector x {, } m and see by induction on that PS F contains a unique soution where, for each node i connected to S in G(x), w i = if and ony if i is at distance from S. If i is not connected to S, the corresponding (b) constraint ensures that wi n =. The above formuation enabes us to extend DT S to fractiona vectors as done in Section.. The pendant of equation () for PS F is () DT F S (x) = {(w, y) (x, w, y) P F S }, and P PS F impies that DT S(x) DTS F (x) for any x [, ]n. When x is fractiona, the exampe from Figure 6 shows that the incusion can be strict. Namey, Figure 6(c) depicts the graph induced by a vector (w, y) DTS F (x) that does not beong to DT S (x) because it cannot be obtained as the convex combination of binary vectors. In fact, for that exampe, DT S (x) is reduced to the singeton (w, y) that induces the graph in Figure 6(b). For any x [, ] m, one can aso extend the connectivity requirements to the graphs induced by any (w, y) DTS F (x). We define C F S = Proj x {(x, y, w) P F S C H uv(w, y) K, (u, v) D}, and Proposition can be competed with the foowing resut. Proposition 7. For any connectivity requirements constraints, it aways hods that (5) C P S C F S C.

14 s() t() t() Fig. 7. Graph G(w, y) = (V(w), E(w, y)) associated to the second extreme point (w, y) depicted in Figure and considering demands (, ) and (, ). Thin edges (resp. nodes) correspond to the components of y (resp. w) equa to zero and do not beong to E(w, y) (resp. V(w)). Proof. The incusion C P S C F S foows from the fact that P PS F. Then, constraints (d) and (f) enabe us to prove incusion C F S C by using the same fattening argument as the one used in the proof of Proposition. We study next whether incusions in (5) can be strict. We see easiy that Proposition extends to C F S, providing an instance for which C F S C. We then turn to incusion C P S C F S. The exampe from Figure 6 does not ead to strict incusion because the graphs of Figures 6(b) and 6(c) satisfy the same connectivity requirements, namey C(w, y). However, we do have numerica evidence that the incusion can be indeed strict, which is eft out of the manuscript to simpify our exposure... Limiting the eves. The DT described in the previous subsections can ead to arge formuations. Due to the number of ayers in graph G, formuation () introduces O(nm) variabes and constraints. For some NDPs, we can eiminate most eves without affecting the transformation. For instance, if a demands in D have a common extremity, and the connectivity requirements asks for paths bounded by some number H, then we can restrict the number of ayers to H +. This exampe arises in the survivabe network design probems studied by [8]. However, it is possibe to define DTs that use ony a sma number of ayers L, regardess to the NDP under consideration. This decreases the size of the associated inear programming formuations, but may aso decrease the node spitting, and thus, the gains in terms of gap reduction. In fact, there is a trade-off between the chosen vaue of L and the quaity of the DT. We suppose that G(x) is the graph induced by some binary vector x and that G L is a graph that consists of L ayers. The truncated distance transformation DTS L sets the image of node i in G L to ayer min(dist G(x) (i, S), L ). This means that eves from L to n of the origina G are fattened into a singe eve L in G L ; nodes not connected to S are sti mapped to ayer L. Edges are mapped subsequenty according to the images of their extremities. This wi not affect much the quaity of the DT when few nodes i have dist G(x) (i, S) L in typica soutions x. Formuation () is adapted for this modification by changing constraints (g i), reducing the size of the formuation to O(Lm) variabes and constraints.. Formuating the connectivity requirements.

15 .. Simpe connectivity requirements. In order to provide a inear programming formuation for an NDP, one sti has to reformuate the connectivity requirements constraints with inear constraints. We start our approach with the simpe constraints (6) C uv (w, y) K, (u, v) D considered in the previous sections. Reca that constraint (6) impose that, for each (u, v) D, the vaue of the maximum fow between s(u) and t(v) in G(w, y) be not smaer than K, see Figure 7 for an exampe of graph G(w, y) = (V(w), E(w, y)). We describe next how to express the constraints for a singe demand {u, v} D using either fow variabes or cutset inequaities, and disregarding the eve reductions discussed in the previous section. The fow formuation compements Formuation () with two fow variabes f ij and f ji and two fow variabes fs(u)u and f vt(v) for each undirected edge {i, j } in G(w, y), for each demand (u, v) D and {,..., n} (notice that if u beongs to S, we ony introduce fow variabe fs(u)u, see Figure 7, and simiary if v S). Then, we impose that capacity be respected for a edges (7) f ij + f ji y ij, {i, j } E fs(u)i Kw i, i V fit(v) Kw i, i V, fow be conserved for nodes in V (8) f s(u)i f it(v) + {i,j } δ E (i ) ( f ji and that the fow exiting supervertex s(u) exceeds K (9) fs(u)i K, if u S, n fs(u)i K, otherwise. = ) f ij =, i V, Aternativey to constraints (7 9), cutset inequaities impose that () Kwu + Kwv + y ij K, U V. :u V\U :v U {i,j } δ E (U) We iustrate next cutset inequaities on an exampe based on the soution depicted in Figure 7 together with connectivity requirement C (w, y). If U = {,, }, then inequaity () becomes w + y + y, which is vioated by the soution depicted in Figure 7... Hop constraints. We study next the more compex connectivity requirements obtained by imiting the path ength (hops) used to transmit the fow by a given integer H. Namey, we consider matrix C(x) R D (n ) + with C H uv(x) equa to the maximum fow between u and v in G(x) using paths with at most H hops. Simiary, we define matrix C(w, y) R D (n ) + with C H uv(w, y) equa to the maximum fow between the supervertices s(u) and t(v) in G(w, y) using paths with at

16 5 (a) Graph G. (b) Graph G. s() t() (c) Graph G for H =. Fig. 8. Hop-eve graph. most H + hops. With these definitions, we see immediatey that C(x) = C n (x) and C(w, y) = C n (w, y). The counterparts of (5) and (6) for C are () C H uv(x) K, (u, v) D, and () C H uv(w, y) K, (u, v) D. Remind that connectivity constraints () can be expressed by a hop-indexed fow formuation first introduced in [9]. The formuation has been extended to hande () in [8], denoted DT-HOP indexed fow formuation therein, and we reca it beow. Given a demand (u, v) D and an integer H, the formuation considers a directed ayered graph G uv = (V uv, A uv ), where the definition of V uv = V uv VH+ uv depends on whether {u, v} intersects S. If {u, v} S =, then V uv = {s(u)}, V uv = {u n}, Vh uv = V \ {{u, v } n}, h =,..., H +, VH+ uv = {v n} and VH+ uv = {t(v)}. If u S or v S, we have instead V uv = {u } or VH+ uv = {v }, respectivey, see Figure 8 for an exampe where u S and v / S. Let i h be the copy of i V in the h-th ayer of graph G uv, that is, i = i

17 6 for some i V, n and i h = i h. The arcs set is defined by (see again Figure 8) () () (5) (6) (7) (8) (9) A uv = {(s(u), u ) u V uv } {(u, i ) {u, i} E, u V uv, i V uv } {(i h, j h+ ) {i, j} E, i h Vh uv, j h+ Vh+, uv h H} {(i h, i h+ ) h H, i h V uv h } {(i H+, v H+ ) {v, i} E, v H+ VH+, uv i H+ VH+} uv {(v H+, t(v)) v H+ VH+} uv {(u, v H+ ) {u, v} E, u V uv, v H+ VH+}. uv Given this auxiiary graph, the DT-HOP indexed fow formuation compements formuation () with a fow variabe g uv,h ij for each arc (i h, j h+ ) A uv. To simpify notations, we omit index uv from the fow variabes in what foows. Then, we impose that capacity be respected for a edges () () () () () (5) H h= g s(u)u Kw u, u V uv gui y ui, {u, i} E, u V uv, i V uv ( g h ij + g h ji) yij, {i, j} E, i h V uv h, j h+ V uv h+, h H g H+ iv y vi, {v, i} E, v H+ V uv H+, i H+ V uv H+ g H+ vt(v) Kw v, v H+ V uv H+ g uv y uv, {u, v} E, u V uv, v H+ V uv H+. Arcs in (6) have an infinite capacity so that no capacity constraints are written for these arcs. The counterpart of fow conservations constraints in graph G uv is readiy obtained from (8). Finay, we need to impose that the fow exiting supervertex s(u) V uv exceeds K: (6) gs(u)u K. u V uv Capacity constraints () prevent the above system to define a pure network fow probem, having the integraity property. Hence, in genera we need to impose integraity restrictions on g. However, in the cases H =,, one can readiy extend the resuts from [, ] to show that the DT-HOP indexed formuation is indeed integra. Actuay, for those cases it is possibe avoid incuding the DT-HOP indexed variabes and constraints in the Formuation () and repace them by cuts separated by the min-cut agorithm over G uv, as shown in the foowing exampe. Consider graph G = (V, E), defined by V = {,,,, 5} and E = {{, }, {, }, {, }, {, }, {, 5}, {, 5}}, et G = (V, E) be the associated ayered graph and Figure 9(a) represent a fractiona soution that satisfies C 5(x). Figure 9(b) represents the graph associated to some (w, y) DT S (x) that vioates C 5(w, y). We show next how to separate a vioated cut vioated by G(w, y). Without oss of generaity, we can restrict ourseves to the subsets of nodes and edges of G that beong to at east one graph of {G(DT (x )) : x {, } m }, which are represented in Figure 9(c) and can be obtained automaticay using preprocessing agorithms. The feasibiity of C 5(w, y) is tested by ooking for a feasibe fow of one unit between

18 (a) Fractiona soution G(x) that satisfies C 5 (x). 5 (b) G(w, y) for some (w, y) DT S (x) that vioates C 5 (w, y) (c) G(DT S (x )). x {,} m 5 5 s() 5 5 t(5) 5 5 (d) Transformation of G(w, y) foowing Figure from []. Thin nodes (resp. edges) correspond to the components of w (resp. y) equa to zero. Nodes in eve 5 are omitted because they are not connected to. Fig. 9. Separation of -path cut inequaities foowing []. s() and t(5) in the expanded graph depicted in Figure 9(d). Looking for a cut of minimum capacity that contains s(), we obtain either inequaity or inequaity y + y 5, w 5 + y 5, which are both vioated by the soution depicted in Figure 9(b). Those inequaities correspond to the counterparts of the -path-cut inequaities proposed in [, ]. Whenever H, the DT-HOP indexed fow formuation cannot be repaced by inequaities obtained by the min-cut agorithm. However, it can be numericay

19 8 efficient to avoid the incusion of the formuation and repace it by Benders inequaities for variabes y and w, simiary as []. 5. Numerica experiments for the DT. 5.. The Steiner Forest Probem Probem description. The SFP has the foowing natura formuation: (7a) (7b) (7c) Min S.t. c e x e e E e δ E (S) x e (u, v) D; S V, u S, v / S x e binary e E. This formuation is the basis for the cassica prima-dua -approximated agorithm for the SFP [8]. Nevertheess, (7) does not provide effective exact branch-andcut agorithms, duaity gaps of more than % are typica on practica instances. For exampe, consider an instance defined over a compete graph with vertices {,, } and D = {(, ), (, )}. The fractiona soution x = x = x =.5 mentioned previousy satisfies a constraints (7b) Numerica resuts. The foowing experiments were peformed in a singe core of a machine with processor i7 at.5 GHz and 6 GB of RAM. The branch-andcut agorithms were impemented over the XPRESS-Optimizer 7.. We performed tests with types of instances: Sma STF instances (pdh, di-yuan, dfn-gwin, poska and nobe-us) avaiabe in the SNDLib. Steiner instances C st,...,c st from the SteinLib. Those instances are defined over random graphs with 5 vertices. For an instance with terminaset T, we defined D as {(r, v) : v T, v r}, where r is the termina with smaer index. Those instances can be easiy soved by SPG codes using a directed formuation. Nevertheess, it is interesting to see how the DT can improve the undirected formuation. SPF instances C,...,C derived from the above instances as foows. The set D is obtained by pairing consecutive terminas in T. If T is odd, an extra demand from the first to the ast termina is incuded. To the best of our knowedge, some of those instances can be very hard for current soution methods. Tabe compares gaps (with respect to optima or best known UBs) and times to sove inear reaxations for: ()the natura formuation (7); () the ifted-cut formuation proposed in [6], the strongest SPF formuation avaiabe in the iterature; () DT reformuation over the natura formuation, for L =, L =, and L = 5, using unitary distances and with a singeton set S containing the first termina. Some comments on those resuts: Whie the ifted-cut formuation is not much better than the natura formuation, DT produces significanty smaer gaps. As expected, the average gaps decrease when L increase, but the improvement quicky becomes margina. The DT with L = seems to be the best compromise between gap and running times on most instances.

20 Whie the gap reductions are remarkabe for SNDLib and C st instances, they are ess impressive for the C instances. We verified that their distance transformed fractiona soutions were divided into a number of connected components. A those components, except the one that contained the vertex in S, are in eve L, where vertex spittings do not happen. In order to make the DT effective on that ast case, we devised an iterative scheme for choosing a arger set S. We start with a singe vertex in S and sove the inear reaxation of the corresponding DT. Whie the fractiona soution sti contains vertices in eve L, we introduce one vertex from the arger connected component in L and sove the new DT again. Tabe shows the resuts of this dynamic procedure for L =, L =, and L = 5. Whie the resuting gaps are quite better, they are sti arge when compared with those obtained in other types of instances. Finay, Tabe compares the resuts of the fu branch-and-cut over the origina formuation (7) with the branch-and-cut over the DT reformuation, for some chosen parameterization. We mark in bod the time of the method that coud sove the instance faster. If no method coud sove the instance, either because the time imit of 7 seconds was exceeded or because it went out of memory, we mark in bod the smaest fina gap obtained. We did not passed any externa upper bound to the branch-and-cut, those gaps are with respect to the best soution found by the method itsef. Athough the harder instances coud not be soved to optimaity, it is cear that the overa performance of the DT reformuation is much better. 5.. The Survivabe Network Design with Hop Constraints Probem. We consider in this section the HSNDP that has been defined in the introduction of the paper. We address the connectivity requirements by using the ayered fow formuation described in Section., which resuts in a arge extended formuation for the probem. It can be verified that using the fow formuations proposed in Section. to mode the connectivity requirements of HSNDP resuts in a formuation that is equivaent to the one originay proposed in [8]. Our objective in the section is to provide numerica evidence that the formuation is we-suited for Benders decomposition. Specificay, we compare the foowing two approaches for soving the extended formuation. On the one hand, we feed the formuation directy into CPLEX, eaving a parameters to their defaut vaues. On the other hand, we consider a Benders decomposition that decomposes the extended formuation into a master probem, that contains ony the design variabes x, w and y and Benders cuts, and one subprobem for each demand D that contains the fow variabes g associated to that demand together with the fow conservations constraints and capacity constraints described in Section.. Our agorithm foows modern impementations of Benders decomposition (e.g. [7]) and soves the master probem through a branch-and-cut agorithm. Every time an integer soution ( x, w, ȳ) is found in the branch-and-cut tree, a subprobems are soved to see if the soution ( x, w, ȳ) is feasibe for the origina probem. If this is not the case, the subprobems return one or more Benders cuts that are added to the master probem at a nodes of the branch-and-cut tree. In addition, we feed the sover with the best known soution (denoted BKS in Tabe ) and aow it to fathom any node worse than BKS. Tabe reports the resuts of our computationa experiments. The tabe aso contains the resuts obtained by using the cassica ayered formuations from [] (denoted HOP formuation in the tabe) where the connectivity requirements are imposed through fows on ayered graphs that are buit directy on the origina graph G. Hence, the HOP formuations contrasts with the the HOP-eve formuation that modes fows on ayered graphs that are buit on the top of the ayered graph G. 9

21 Natura Lifted-cut DT L= DT L= DT L=5 Instance V E D UB Gap (%) T (s) Gap (%) T (s) Gap (%) T (s) Gap (%) T (s) Gap (%) T (s) pdh di-yuan dfn-gwin poska nobe-us Avg c st c st c st c st c5 st c6 st c7 st c8 st c9 st c st Avg c c c c c c c c c c Avg Avg Tabe Natura [], Lifted-cut [6] and DT reformuation root gaps: S =. L= L= L=5 Inst Gap(%) T(s) Gap(%) T(s) Gap(%) T(s) c c c c c c c c c c Avg Tabe Root gaps for the dynamic choice of S in DT.

22 Inst Nodes Gap(%) T Nodes Gap(%) T(s) pdh L= S = di-yuan L= S = dfn-gwin L= S = poska..5.. L= S = nobe-us L= S = c st L= S = c st L= S = c st o.m..5 7 L=7 S = c st L=7 S = c5 st L=7 S = c6 st L= S = c7 st L= S = c8 st 7. o.m...7 L= S = c9 st o.m L= S = c st L= S = c L= S = c L= S = c L=7 S =8 c o.m L= S = c L= S =9 c L= S = c L= S = c8.6 o.m L= S = c o.m L= S =6 c L= S =7 Tabe Comparison of fu branch-and-cut over natura formuation and over DT reformuation

23 HOP formuation DT-HOP formuation CPLEX MIP Benders decomposition H K BKS root LB root gap(%) root T(s) nodes fina LB fina UB tota T (s) root T(s) root cuts nodes fina LB fina UB tota T (s) TC TE TC TE CPLEX MIP Benders decomposition H K BKS root root root nodes fina fina tota root root nodes fina fina tota LB gap(%) T(s) LB UB T (s) T(s) cuts LB UB T (s) TC TE TC TE Tabe Comparison of HOP and DT-HOP formuations: direct soution by CPLEX MIP sover or branch-and-cut after Benders decomposition.

24 Those experiments were carried out on a computer equipped with a processor Inte(R) Xeon(R) CPU X56 at.6ghz and GB of RAM memory, using Concert Technoogy for JAVA of CPLEX.6. [6]. The time imit was set to 6 seconds. The TC and TE instances used in tests are widey used the iterature. They correspond to compete graphs, vertices are associated to points in the pane, the costs are the Eucidean distances. TC- has vertex in the center and demands D = {(, v) : v =,..., }, TE- is simiar but has vertex in a corner, TC- and TE- are simiar but D =. We performed tests taking H {, } and K {, }. Tabe is divided into two parts. On the top, it shows resuts for HOP fomuation; on the bottom for DT-HOP formuation. Coumns root LB and root gap show the ower bounds and gap with respect to the Best Known Soution for HOP and DT- HOP formuations (those vaues do not depend whether Benders decomposition is used or not). The remaining coumns are statistics for the compete exact method, either by feeding the compete formuation to CPLEX MIP sover or by performing a branch-and-cut after the Benders decomposition. Therefore, we are comparing four methods on each instance. We mark in bod the time of the method that coud sove the instance faster. If no method coud sove the instance within the time imit we mark in bod the best fina ower bounds obtained. It can be seen that: DT works very we for that probem. Excepting TC- with H = and K =, a method based on DT-HOP was the winner. Benders decomposition can be a good aternative for mitigating the probems reated to the arge size of the reformuations obtained by DT. DT-HOP with Benders was the best method in 9 out the 6 instances. It aso aowed us to sove TC- with H = and K = to optimaity for the first time, and compute new upper bounds for instance TE- with H = and K =. 6. Concusions. A growing part of the Integer Programming research is devoted to finding new effective extended formuations for certain famiies of probems. This paper contributes in this direction, introducing a technique with the potentia of strengthening existing formuations for a arge cass of NDPs. The increase in formuation size is not necessariy exaggerated (actuay, the increase can be controed by the parameter L), making the overa approach computationay appeaing in a number of cases. We finish the paper by commenting, based on the presented experiments, on the factors that seem to make the DT more or ess suited to a particuar NDP. The DT seems to work better on NDPs with sparser soutions. For exampe, on HSND instances with K = the average gap was reduced by 7%, on HSND instances with K = the more modest average gap reduction of % was obtained. This is coherent with the theory presented in Section that asserts that DT works by spitting nodes according to the different distances (to the sources) induced by decomposing fractiona soutions. Denser fractiona soutions provide more ways of performing the decomposition, resuting in ess node spittings. The DT seems to work better on probems where the soutions are ikey to have a sma diameter. For exampe, this happens on HSND because the of the hop constraint. This aows using sma vaues of L without compromising the strength of the reformuation. Remark that SFP soutions are very sparse, but typicay have arge diameter. Athough the ast characteristic is not favorabe, the DT with imited vaues of L sti