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A Catalog of Steiner Tree Formulations

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A Catalog of Steiner Tree Formulations A Catalog of Steiner Tree Formulations Michel X. Goemansf Department of Mathematics, Room 2-372,Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 Young-soo Myungt Department of Business Administration, Dankook University, Cheonan, Chungnam 330, South Korea We present some existing and some new formulations for the Steiner tree and Steiner arborescence problems. We show the equivalence of many of these formulations. In particular, we establish the equivalence between the classical bidirected dicut relaxation and two vertex weighted undirected relaxations. The motivation behind this study is a characterization of the feasible region of the dicut relaxation in the natural space corresponding to the Steiner tree problem. 0 7993 by John Wiley & Sons, Inc. INTRODUCTION cost, where the cost of a tree is the sum of the costs of its edges. Given a digraph D = (V,A) and a root vertex We refer to undirected graphs as graphs and to di- r, a set of arcs is called an r-arborescence of D if it rected graphs as digraphs. In a graph G = (V, E), the forms a (not necessarily spanning) tree directed away elements of E are called edges and the edge e between from the root r. For a set T of terminals and a specified the vertices i and j is denoted by {i, j} or {j, i}. In a root vertex r E T, we define a Steiner arborescence as digraph D = (V, A), the elements of A are called arcs an r-arborescence spanning T. The Steiner arbores- and the arc a between i and j is denoted by (i, j).(i, j) cence problem is the problem of finding a minimum- and (j, i) do not represent the same arc. From any cost Steiner arborescence. Let T, = T\{r} and V, = graph G = (V, E), we can obtain a bidirected graph DG V\{r}, = (V, A) by replacing every edge of E by two arcs in The Steiner tree and Steiner arborescence problems opposite direction, i.e., A = {(i, j): {i, j}E E}. have extensively been studied in the literature. Two Given a graph G = (V, E) and a set T C V of termi- recent surveys on these Steiner problems have sum- nals, a Steiner tree is a tree spanning T. We do not marized formulations and solution methods [20, 291. require its leaves to be terminals. Let 1 be a cost func- Maculan [20] emphasizes exact algorithms and integer tion defined on the edge set E. The Steiner tree prob- programming formulations, whereas Winter [29] con- lem is the problem of finding a Steiner tree of minimum siders exact algorithms, heuristics, and polynomially solvable special cases. We associate to any Steiner tree an incidence vec- tor x such that xe = 1 if edge e E E is part of the Steiner *Supported by Air Force contract AFOSR-89-0271 and DARPA contract DARPA-89-J- 1988. tree and 0 otherwise. Let 9, denote the convex hull of ton leave at the Operations Research Center, MIT. Partial Sup- incidence vectors of Steiner trees in a graph G. 9,is port from the Yonam Foundation. called the Steiner tree polytope. Similarly, the inci- NETWORKS, Vol. 23 (1993) 19-28 0 1993 by John Wiley 8 Sons, Inc. CCC 0028-3045/93/01019-10 19 20 GOEMANSANDMYUNG dence vector w of a Steiner arborescence B is defined lem. In Section 2, we introduce two simple extended by w, = 1 if a E B and 0 otherwise. The Steiner arbo- undirected relaxations involving vertex variables and rescence polytope, denoted by 9,,,,is the convex hull we prove their equivalence to the bidirected dicut re- of incidence vectors of Steiner arborescences. laxation. Bounded analogs to these relaxations are We shall describe linear programming (LP) relaxa- presented in Section 3. Finally, in Section 4, we show tions of the Steiner tree and Steiner arborescence that the polyhedra defined in Section 2 are the domi- problems. An LP relaxation for the Steiner tree prob- nants of their bounded analogs of Section 4. This im- lem is a linear program of the form plies that all relaxations defined in this paper are equivalent for all nonnegative cost functions. In Sec- Minimize( z = lex,: x E R,) , tion 4, we also prove that the choice of the root vertex is unimportant when bidirecting an undirected in- stance. where R, is a polyhedral region with 9, c R,. More generally, we allow this definition to include extended relaxations of the form 1. A REVIEW OF CLASSICAL INTEGER PROGRAMMING FORMULATIONS Minimize[ z = 2,lex,: (x, s) E R-1, Given a graph G = (V,E) and a set S of vertices, 6(S) where 9,is contained within the projection proj.(R,,) represents the set of edges in E with exactly one end- of R,., onto the x variables defined as proj,(R,,) = point in S, whereas E(S)represents the set of edges in {x : (x, s) E R,, for some s}. We regard two relaxations E with both endpoints in S. The corresponding notions as equivalent for a class 3 of cost functions 1 : E + R if for a digraph D = (V,A)are as follows: For a set S c their optimal values are equal for any 1 E Y.Of course, V, 6-G) denotes the set of arcs {(i,j) E A : i $Z S, j E this does not necessarily imply that their feasible re- S}, 6+(S) = 6-(V\S) and A(S) = {(i,j) : i E S, j E S}. gions are equal. However, if two relaxations, defined For simplicity, we write 6-G) [resp., 6+(i)or S(i)] in- by R,,s and R:, , are equivalent for all cost functions of stead of 6-({i})[resp., S+({i})or S({i})].If x is defined x, then proj.(R,,) = proj,(R:,) and vice versa. In this on the elements of a set M (typically M is an edge set case, we say that R,,s and R:, are extended descriptions E, an arc set A, or a vertex set V),then we denote of R = proj,(R,,s) = proj,(R;,). These concepts are xi for N M by x(N). The only exceptions are defined analogously for the Steiner arborescence prob- a(.), 6-(.), a+(.), E(,),and A(.), which were defined lem. previously. The most classical relaxation for the Steiner arbo- rescence problem is the dicut relaxation introduced by 1.l. Classical Formulations for the Steiner Wong [30]. This relaxation can also be used for the Steiner tree problem since any instance of the problem Tree Problem can be equivalently formulated as a bidirected Steiner A Steiner tree can be seen as a minimall subgraph arborescence problem. Chopra and Rao [6] showed having a path between any pair of terminals. In fact, that this approach leads to better relaxations than do we can even restrict our attention to pairs containing a simple undirected relaxations for the Steiner tree prob- specified vertex r E T. This vertex r plays the role of lem. As a result, there has been little emphasis on root for the Steiner tree. This definition of Steiner undirected relaxations in recent years. In this paper, trees in terms of minimal subgraphs can be used to we show that undirected relaxations can be as tight as formulate the Steiner tree problem as an integer pro- bidirected ones, provided that we introduce some aux- gram when all cost coefficients are nonnegative. For iliary variables. In particular, by considering vertex this purpose, we introduce some flow variables and variables that either keep track of the vertices spanned consider the following program [3]: or the degrees of the vertices in the Steiner tree, we obtain two undirected relaxations that are equivalent to the bidirected dicut relaxation. These relaxations Minimize PEE lrxe are valid only for nonnegative cost functions. We also introduce tighter bounded analogs to these relaxations that appear to be equivalent. (1P:f)subject to: The paper is organized as follows: In Section 1, we review classical formulations for the Steiner tree and Steiner arborescence problems and we consider the use of bidirected relaxations for the Steiner tree prob- *With respect to inclusion. CATALOG OF STEINER TREE FORMULATIONS 21 where 1 i=r S,f = {(x, f): fk(6+(i))- fk(s-(i))= 1and k E T, [ o i E V\{k, r}J k fu Ix, e = {i, j} E E and k E T, ff 2 0 a E A and k E T,}, and D = (V, A) is the bidirected graph obtained from value of (LP;) can be computed a la Held and Karp G = (V, E) by bidirecting every edge of E. The con- [I51 by solving a sequence of minimum spanning tree straints (I) imply the existence of a unit flow from r to problems with Lagrangean costs [ 141. k, and if x, is integral, this means that there exists a In some cases, S, is integral, i.e., it is equal to its path from r to k in the subgraph {e E E: x, 2 I}. integer hull inr-hull(S,) defined as the convex hull Using the max-flow min-cut theorem, the projection conu(S, n ZIEl)of its integer points. For example, this S, of S,f onto the x variables can be characterized as happens when IT1 = 2 or when G is acyclic. The case JTI = 2 corresponds to the shortest path problem in an S, = {x: x(S(S)) 2 1 r 4 S and S fl T # 0 (3) undirected graph with no negative cycles.
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