Chvátal–Gomory cuts for the Steiner tree problem
Introduction
The Steiner tree problem is one of the most fundamental network design problems: Given a weighted, undirected graph and a set of terminal nodes , find a minimum-weight subtree of that spans . The problem can be formulated as an integer linear program (ILP) in many different ways, see for instance the surveys by Goemans and Myung [22], and Polzin and Vahdati Daneshmand [37] for different formulations and their relationship, or the findings in [9], [29] and [38]. In this work, however, we are interested in arguably the simplest possible formulation, namely the undirected cut formulation (UC) as stated by Aneja [5]. The linear programming (LP) relaxation of UC is weak; its worst-case integrality gap3 is known to be two even if restricted to seemingly easy instance classes and even for the easy spanning tree case where all nodes are terminals. Nonetheless, many classes of valid combinatorial inequalities that strengthen the linear programming (LP) relaxation of UC [11], [12], [14], [21] have been found and it is with these inequalities that this work is concerned. More precisely, we analyze how the partition and the odd-hole inequalities discovered by Chopra and Rao [11] can be obtained through Chvátal–Gomory (CG) cuts as used by general purpose integer linear programming solvers.
We remark that the undirected cut formulation is inferior to other formulations for most practical considerations. For instance, the implementations in [28], [31] rely on the (bi-)directed cut formulation (DC) as stated by Wong [40]. To the best of the authors’ knowledge, no example with an integrality gap of more than is known for DC (see [7] for the example) and in stark contrast to UC, the formulation is integral in the spanning tree case [19]. Conversely, we can strengthen the LP relaxation of UC such that all basic optimum solutions correspond to minimum spanning trees by adding all partition inequalities [10]. Chopra and Rao [11] prove that DC is at least as strong as UC plus all the partition and odd-hole inequalities. Goemans [21], and Margot, Prodon and Liebling [33], and Lucena [32] independently develop a tree formulation (TR) based on Edmond’s characterization of the spanning tree polytope [15]. The LP-relaxation of TR is at least as strong as the LP-relaxation of DC [22] and at least as strong as the LP-relaxation of UC plus all partition, odd-hole, and other rich classes of inequalities [21]. Polzin and Vahdati Daneshmand [37] review several tree formulations from the literature and analyze their relationship to each other and to a new strong common flow formulation. Several equivalent hypergraphic relaxations [7], [9], [29], [38], [39] are strictly stronger than DC, but incomparable to the common flow formulation [38]. The worst-case integrality gap of these formulations is at most , see [23]. While it is NP-hard to solve the hypergraphic relaxations exactly [23], their optimum can be approximated within a factor of in polynomial time (with a doubly exponential dependence on ) for any fixed using a result by Borchers and Du [6]. Filipecki and Van Vyve [16] present a hierarchy of extended path-based formulations in a recent work.
Independently of formulation strength, however, an interesting question is if the known classes of valid combinatorial inequalities can be recovered through general mixed integer cuts. For instance, Mirchandani [35] observes that 3-partition inequalities are in fact CG cuts of a special kind. Agarwal [1], [2] as well as Agarwal and Aneja [3] derive partition based facets for the related network loading problem4 through a similar procedure. Still, the network loading problem is not a generalization of the Steiner tree problem and its more complicated connectivity requirements make it so that the correct right-hand side of partition based inequalities is difficult to find. Nonetheless, Agarwal’s [2] experimental study demonstrates that the partition based inequalities obtained through CG cuts are highly effective in practical computations.
In this work, we generalize Mirchandani’s [35] observations and analyze how the known facial structure of the convex hull of the integer solutions to UC can be retrieved through CG cuts in the Steiner tree case. It will turn out that partition and odd-hole inequalities can be recovered with homogeneous CG cuts, i.e., CG cuts with multipliers of the form with . We show that any partition inequality induced by partitions is a CG cut derived from partition inequalities induced by partitions and that consequently, the Chvátal rank (the “recursion depth” needed to obtain the inequality with the CG cut generation procedure) as defined below is at most . Likewise, we show that for any , the Chvátal rank of any odd-hole inequality induced by partitions is at most . Additionally, we show what happens if the choice of cuts is restricted and analyze which valid inequalities can be obtained by systematically combining partition inequalities.
Despite the known superior formulations, it makes sense for our theoretical analysis to start with the simplest ILP formulation. Moreover, a generalization of our results to the Steiner forest problem would be interesting since the state-of-the-art approximation algorithms [4], [24], [27] for the Steiner forest problem are limited by the integrality gap of the canonical generalization of UC, and no formulation with an integrality gap of any constant less than two is known. Indeed, a recent work by Fiorini, Groß, Könemann and Sanità [17] uses CG cuts to develop a better approximation algorithm for the related tree augmentation problem.
Given an undirected graph , a non-negative edge cost for each edge and a set of terminal nodes , we call a (connected) subtree of a feasible Steiner tree or simply a Steiner tree if spans , i.e. if contains all terminal nodes in . We define the cost of as , see Fig. 1 for an example. The Steiner tree problem asks for a feasible Steiner tree with minimum cost. One way to cast the Steiner tree problem into the form of an ILP is the undirected cut formulation [5] which we want to study in more detail below.
We first need some notation. A cut-set of is a subset of the nodes and we define the corresponding cut as . We say that is cut by if . Furthermore, if has the property that and , i.e., if separates a terminal from a terminal , we say that is a Steiner cut-set and call a Steiner cut. For two disjoint sets , let be the set of edges with one end point in and one endpoint in . Again, we say that cuts any edge .
To derive the undirected cut formulation, consider any Steiner cut-set . Since any subtree of spanning must contain a path between and , it also must contain at least one edge of the corresponding Steiner cut . Moreover, if is a subgraph of containing at least one edge of each Steiner cut , then is a -spanning subtree by virtue of Menger’s Theorem [34]. Fig. 2 shows an example. We obtain the undirected cut ILP by introducing a binary variable for each edge and use the common abbreviation for some . where with being the set of all Steiner cut-sets. The constraints of type (1a) are called Steiner cut-set inequalities and we write to denote the Steiner cut-set inequality defined by ; they ensure that, indeed, at least one edge of each Steiner cut is selected in a feasible solution of (IPuc). Hence, any feasible solution to (IPuc) defines a feasible Steiner tree by our above observation. Conversely, any feasible Steiner tree defines a feasible solution for (IPuc) with if and otherwise. It follows that (IPuc) models the Steiner tree problem correctly: Any optimum solution to (IPuc) induces an optimum Steiner tree. We remark that – following the usual convention – we have relaxed to to ensure that has the non-negative orthant as its recession cone without affecting any optimum solution, thus removing irrelevant faces from the feasible region. This step is necessary for the validity of Lemma 1 and Theorem 12 which are due to Chopra and Rao [11] and which shall be discussed in Sections 2 Partitionings and partition inequalities, 4 Odd-hole inequalities . In the sequel, let be the Steiner tree polytope. We say that an inequality over with , is valid for a set if holds for all .
Given an integer linear program with a constraint matrix , a right-hand side , and an objective function , the most common solution approach is to first solve the LP relaxation x . Unfortunately, the optimum of the LP relaxation can be arbitrarily far away from the optimum of the ILP and in consequence, the LP relaxation by itself is not necessarily a good approximation of the ILP. Therefore, we are interested in additional constraints (i.e., inequalities) that are valid for the feasible region {} of the ILP, but are (ideally) not valid for the feasible region of the LP relaxation: Adding such an inequality to the LP relaxation will cut off a part of its feasible region and potentially improve the approximation. Consequentially, such an inequality is often called a cutting plane or short, a cut.
A systematic way to find cutting planes is through the Chvátal–Gomory (CG) procedure. This procedure is particularly well suited for combinatorial problems since it requires that the constraint matrix and the right-hand side are integer. Then, given any system of linear inequalities with an integer matrix and an integer right-hand side over , any multiplier such that is integer defines a CG cut [13], [25] The cut (2) is valid for {}: Clearly, as a non-negative linear combination of the constraints, is valid for , and since for any integer vector its left-hand side is integer the right-hand side of can be rounded up to obtain (2). If is of the form with we say that the resulting cut is homogeneous. The entirety of all CG cuts derivable from induces the first Chvátal closure Chvátal [13] showed that out of the infinitely many possible number of cuts only a finite number is required to describe . Thus, the procedure can be applied recursively and we obtain the th Chvátal closure as the first Chvátal closure of the th closure . The Chvátal rank of a cut is the smallest such that the cut can be obtained from . We refer to [36] and [18] for a more detailed introduction into the topic. In particular, Fischetti and Lodi [18] propose a general algorithm for separating CG cuts via mixed integer linear programming.
Section snippets
Partitionings and partition inequalities
We are now interested in the Chvátal closure of . As an introductory example, consider a partitioning of the node set into three disjoint Steiner cut-sets . As seen in Fig. 3, adding yields the valid inequality with integer left-hand side and thus is a (homogeneous) CG cut; a well-known fact that was for instance observed by Mirchandani [35]. The inequalities of type (4) are called -partition
A more finely grained analysis of the Chvátal closure
So far, we have derived CG cuts from partitionings of by fixing a partitioning and then combining all partition inequalities corresponding to coarsenings of with a given number of partitions. We now want to analyze which cuts are attainable if we only use a subset of all possible coarsenings. To address this question, we divide the coarsenings into different classes.
Let be a partitioning of and let be a -coarsening of . For all , let be the
Odd holes
Let us now consider the case where has a contraction minor of a special structure: Given , a -odd-hole is a graph with a node set with the convention that . We choose as the Steiner nodes and as the terminals. The edge set consists of a simple cycle connecting the Steiner nodes and two edges for each connecting to . Fig. 6 shows a picture of the graph. Chopra and Rao show that
Outlook
We have analyzed the relationship between the Chvátal closure of the undirected cut formulation (IPuc) and the facial structure of the Steiner tree polytope . Yet, there are parts of the facial structure that we have not tackled: Chopra and Rao [12], Goemans [21] as well as Didi Biha, Kerivin and Mahjoub [14] give further classes of valid and facet-defining inequalities for ; in particular extending the -partition and -odd-hole inequalities studied here. While the techniques presented
Acknowledgments
We thank the anonymous reviewers for their helpful comments. Part of this work has been conducted while the second author was a researcher at the University of Cologne.
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Both authors have equally contributed to this work.