Ideal, non-extended formulations for disjunctive constraints admitting a network representation

In this paper we reconsider a known technique for constructing strong MIP formulations for disjunctive constraints of the form x∈⋃i=1mPi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x \in \bigcup _{i=1}^m P_i$$\end{document}, where the Pi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_i$$\end{document} are polytopes. The formulation is based on the Cayley Embedding of the union of polytopes, namely, Q:=conv(⋃i=1mPi×{ϵi})\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q := \mathrm {conv}(\bigcup _{i=1}^m P_i\times \{\epsilon ^i\})$$\end{document}, where ϵi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon ^i$$\end{document} is the ith unit vector in Rm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^m$$\end{document}. Our main contribution is a full characterization of the facets of Q, provided it has a certain network representation. In the second half of the paper, we work-out a number of applications from the literature, e.g., special ordered sets of type 2, logical constraints, the cardinality indicating polytope, union of simplicies, etc., along with a more complex recent example. Furthermore, we describe a new formulation for piecewise linear functions defined on a grid triangulation of a rectangular region D⊂Rd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D \subset {\mathbb {R}}^d$$\end{document} using a logarithmic number of auxilirary variables in the number of gridpoints in D for any fixed d. The series of applications demonstrates the richness of the class of disjunctive constraints for which our method can be applied.


Introduction
Disjunctive programming was introduced by Egon Balas [2,3] in the 1970s as an extension of linear programming with disjunctive constraints. Disjunctive constraints can express logical conditions that the feasible solutions must satisfy. A disjunctive constraint can be represented as where each P i is a polyhedron. Suppose P i = {x ∈ R n : is a rational matrix and b (i) is a rational vector of matching dimension. Balas [4], and Jeroslow and Lowe [32] proved that if the P i are bounded and nonempty, thenx satisfies (1) if and only if the following set of constraints admits a solution: More generally, a MIP formulation for (1) is a set of constraints such thatx satisfies (1) if and only if (3) admits a feasible solution when x =x (see e.g. [26,32,[44][45][46]). The Linear Programming (LP) relaxation of (3) is the polyhedron Q determined by Ax + By + Cz ≤ b. For simplicity we assume that Q has at least one extreme point. The formulation is ideal , if z is integral in all extreme points of Q, and non-extended if it contains no y variables, and extended otherwise. Balas' formulation (2) is extended and ideal [32,45]. A lot of work has been done for devising (ideal) formulations for disjunctive constraints that do only involve the original x variables (in particular, no copies of them) as well as some additional integral z-variables, see Sect. 2 for a brief overview, and it is the main topic of this paper. For illustrating the benefits of such formulations in terms of the size of the formulation, consider the n-dimensional cross-polytope In fact, P equals the convex hull of the union of polytopes P i = {x ∈ R : −1 ≤ x i ≤ 1, x j = 0 ∀ j = i}, i = 1, . . . , n. A minimal linear representation of P consists of 2 n linear inequalities (see e.g. Balas [9], page 146), that is, −x 1 − · · · − x n ≤ 1 x 1 − · · · − x n ≤ 1 . . .
Now, an ideal, non-extended formulation for x ∈ n i=1 P i is This system has 2n continuous variables, and 3n + 1 linear inequalities (including the nonnegativity of the variables).
A further advantage of ideal, non-extended formulations for (1) is that they may be more efficient computationally than (2), since they do not contain copies of the original problem variables and constraints. This claim is supported by a recent study of Anderson et al. [1], who compared ideal non-extended, and extended formulations for expressing the maximum of affine functions on polyhedral domains, and showed that the former formulations result much shorter computation times.
To derive ideal, non-extended formulations for (1), we focus on a particular technique, pioneered by Vielma [45], which is based on the Cayley Embedding of a finite union of polyhedra. The Cayley Embedding of polyhedra was proposed by Huber et al. [25] for studying the Minkowski sum of point configurations in R n , see also [34,49], and for the union of polyhedra in (1) it would be of the form m i=1 P i × { i }, where i is the ith unit vector in R m . Vielma generalized the Cayley Embedding by replacing the unit vectors with distinct binary vectors h i ∈ {0, 1} d for some d ≥ log 2 m . Under the conditions that all the P i are non-empty, and pointed, Vielma proved that a non-extended, ideal MIP formulation for (1) is given by where P = (P i ) m i=1 , and H = (h i ) m i=1 is a family of distinct binary vectors in {0, 1} d . This formulation is implicit in the sense that it does not provide a description of Q(P, H) in terms of linear inequalities. Neverthless, Vielma determined the nontrivial facets 1 of the convex hull Q(P, H) for the SOS2 constraints of [13], and also for piecewise linear functions of two variables on grid triangulations of a square. In These ideas were generalized to combinatorial disjunctive constraints by Huchette and Vielma [27], which take the form where the S i are subsets of V := {1, . . . , n} such that m i=1 S i = V , P(S) := {x ∈ Δ V : x j = 0 ∀ j ∈ V \S}, and Δ V := {x ∈ R n ≥0 : n j=1 x j = 1} is the ndimensional standard simplex. Huchette and Vielma gave an explicit description of Q(P, H) by linear inequalities, where P := (P(S i )) m i=1 , and H is a set of m distinct binary vectors, under some mild technical assumptions. However, the construction of the inequalities may be computationally heavy in general, and the authors left open the characterization of those inequalities which induce facets of Q(P, H).
In this paper we propose a new systematic way of constructing ideal, non-extended formulations for disjunctive constraints (1), when the P i are polytopes, and a certain network representation exists. We will use the MIP formulation of Vielma [45] restricted to the unit vectors i . Nevertheless, if P emb admits a network representation of a given structure, to be defined next, then we can characterize the facets of P emb . Our method works not only for combinatorial disjunctions, but for a much wider class of polytopes, which will be illustrated in Sect. 5. A network representation for P emb consists of a network N = (V , A, c) from the following family. The set of nodes V comprises a source node s, and a sink node t, the nodes V s := {v 1 , . . . , v m } representing m alternatives, the nodes V t := {1, . . . , n} corresponding to the variables x 1 , . . . , x n in (6), and possibly some other nodes. We assume that V s ∩ V t = ∅. The set of arcs A contains, among others, the arcs (s, v i ) for i ∈ {1, . . . , m}, and also the arcs ( j, t) for j ∈ {1, . . . , n}, and neither s, nor t has other nodes adjacent to them. It is required that N \{t} decomposes into m directed trees rooted at s, G 1 , . . . , G m , where v i is a node of G i , the leafs of each G i lie in V t , and Fig. 1). For each i and arc a of G i , the capacity of a is c(a) := k a λ i for some rational number k a > 0. Moreover, c( j, t) = x j for each j ∈ V t .
We say that N represents P emb , if for each i, when setting λ i = 1, and the other coordinates of λ to 0, and for any x ≥ 0, (x, λ) ∈ P emb if and only if the network N parametrized by x and λ as above, admits a feasible s − t flow of value n j=1 x j , and (x, λ) satisfies the valid equations for P emb . Deciding whether a particular P emb admits a network representation from the above family is out of scope of this paper. However, in several practical applications, the construction of N is easy, see Sect. 5.
In the sequel we concentrate on two special cases only. Either the only valid equation We distinguish these two cases by writing P emb ≤ , and P emb = , respectively, for P emb . To simplify the presentation, let P emb * denote any of P emb ≤ and P emb = . We assume that each j ∈ V t is reachable from s by a directed path. This implies that x j = 0 is not a valid equation for P emb * . The following assumption is without loss of generality.
Main results of the paper. In the first part of the paper, we fully characterize the nontrivial facets of P emb * for any choice of the P i , provided it has a network representation N . At first, we prove that any s − t cut of the network N yields a valid inequality for P emb , and moreover, along with the non-negativity of the variables, and the valid equation(s), they describe P emb . Then, we fully characterize those s − t cuts of N , that induce facets of P emb . Briefly, some s − t cut (S,S) of N induces a facet of P emb ≤ if and only if the subgraphs of N spanned byS\{t}, and by S, respectively, admit some connectivity properties. In case of P emb = , slightly stronger conditions must hold. These results, along with (6) provide an explicit, ideal, non-extended MIP formulation for (1).
In the second part of the paper, we demonstrate the richness of the class of problems for which our machinery can be applied by deriving well-known formulations for problems in the literature, and working out some new ones. The list of examples include reformulation of SOS2 and SOSk sets, the Cardinality Indicating Polytope, the E V E N n polytope, the polytope of small cliques, the union of simplices, and some more complicated ones. In addition, based on the ideas of [28], we obtain a new formulation for piecewise linear functions defined on a grid-triangulation of a rectangular domain in R d using O(d log 2 n + 2 d d!) auxiliary variables.
Our results complement those of Vielma [45] and Huchette and Vielma [26][27][28]. On the one hand, Vielma [45] was able to derive the facets of the generalized Cayley embedding of SOS2 sets for any choice of the (distinct) binary vectors h i , but our method works for a much wider class of disjunctive sets than SOS2, and we have a full characterization of the non-trivial facets. On the other hand, in [26] a linear representation is obtained for the convex hull Q(P, H) of the MIP formulation for combinatorial disjunctive constraints (5) and any choice of distinct binary vectors , while our characterization of facets is valid for any choice of the P i provided that P emb in (6) admits a network representation.
In Sect. 2 we review some previous work. After some preliminaries in Sect. 3, we state and prove our main results in Sect. 4. In Sect. 5 we describe a series of applications, and conclude the paper in Sect. 6.

Terminology
Let P = {x ∈ R n | Bx ≤ b} be any convex polyhedron. A face F of P is a subset of P obtained by setting some inequalities to equations in the system of inequalities Bx ≤ b defining P. In particular, P is a face of itself, and also the empty set is a face of P. A facet F is a face of P such that F = P, and there exists no face F of P such that F ⊂ F ⊂ P (the containments are proper). A vertex of P is a face which consists of exactly one point of P.
The dimension of P, dim(P), is the maximum number of affinely independent points in P minus one. Equivalently, dim(P) = n − m = , where m = is the maximum number of linearly independent equations which are satisfied by all the points of P.
Let i ∈ R m be the i-th unit vector in R m , i.e., i i = 1 and i k = 0 for all k ∈ {1, . . . , m}\{i}.
A network N = (V , A, c) is a directed graph with vertex set V , arc set A, and arc capacities c. Let V 1 , V 2 be disjoint subsets of the set of vertices V of the network N . Then [V 1 , V 2 ] ⊆ A is the set of those arcs (u, v) ∈ A such that u ∈ V 1 and v ∈ V 2 . Let s and t be distinct vertices of N . An s − t cut (S,S) is a partitioning of V into subsets S andS = V \S such that s ∈ S and t ∈S. The associated cut-set is [S,S]. For any node u, let δ out u := [{u}, V \{u}], and Γ out u = {v ∈ V : (u, v) ∈ δ out u }. Recall the general network N defined above. For fixed x and λ, let N x,λ denote the network with arc capacities determined by x and λ.

Previous work
The ultimate reference on disjuntive programming is the recent book of Balas [9]. The first results were summarized in a research report which got published only some 24 years later [6], but the first papers appeared in print already in the 1970's, see [2] and [3]. In these early works, extended formulations were obtained for general disjunctive programs, the concept of duality was developed and a sequential convexification procedure was devised. Questions related to facets of general disjunctive programs were raised and partially answered in [11].
Balas [2,3,6] investigated the convex hull of feasible points of a disjunctive program by giving two distinct linear programming formulations: one involving only the original problem variables, and another using additional continuous variables. We focus on the latter one. Let . . , m, and cl conv(·) is the closed convex hull operator. Balas has shown thatx ∈ P if and only if the LP-relaxation of (2) admits a feasible solution when x =x, provided that the P i are nonempty and bounded, or some additional technical assumptions hold.
Jeroslow and Lowe [32] introduced the concept of MIP-representable sets, namely, S ⊆ Q n is MIP-representable if there are rational matrices A, B, C, and a vector b with the property that: x ∈ S if and only if, for some y, z ≥ 0 with z integer, Ax + By +Cz ≤ b. They also pointed out the connection to disjunctive programming, and in particular they showed that (2) is ideal. Basu et al. [12] gave an alternative definition, and the concept was generalized to convex sets by Lubin et al. [37]. Vielma [44] provided a broad overview of MIP formulations in mixed-integer programming, including MIP representability of sets and functions.
Conforti and Wolsey [19] generalized the idea of Balas [6] for expressing the union of polyhedra in a higher dimensional space by first lifting each polyhedron in some space where it admits an easy extended formulation. They applied this results to mixed integer sets, like the continuous mixing set with upper bound, a mixing set with two divisible capacities, and a divisible capacity single node flow model.
In the formulation (2), the original x variables are copied m times, which is computationally unattractive. However, Conforti et al. [18] has recently shown that Balas' extended formulation for conv(P 1 ∪ P 2 ) is optimal in the following sense. For any polynomial σ , there exist polytopes P 1 and P 2 of size 2 f 1 and f 2 , respectively, such that any formulation of conv(P 1 ∪ P 2 ) of size bounded by σ ( f 1 + f 2 ) must have Ω(n) additional variables.
In the best case, neither variable copies, nor new λ i variables are needed to get an explicit linear description of the convex hull of the union of some polyhedra. For instance, Balas et al. [10] considered upper monotone polytopes in [0, 1] n . A polytope P = {x ∈ [0, 1] n | Ax ≥ 1} is upper monotone if A is a non-negative matrix. Among other results, Balas and his co-authors derived the convex hull of the union of two upper monotone polytopes P 1 ⊆ [0, 1] n 1 × {0 n 2 } and P 2 ⊆ {0 n 1 } × [0, 1] n 2 in disjoint spaces, and in partially overlapping spaces as well without using any extra variables. They also obtained the facets of conv(P 1 ∪ P 2 ), and applied their characterizations to the union of matroid polytopes, and to logical inference. The latter result extends that of [50], see also [7]. These results were generalized to nonlinear and possibly unbounded orthogonal disjunctive sets in [42].
The second best option is that no variable copies are used in a formulation, only some new (binary) variables. Jeroslow [31] studied a special class of disjunctive constraints, where P i := {x ∈ R n | Ax ≤ b (i) } for i ∈ {1, . . . , m}, that is, only the right-handsides change in the definitions of the P i . Let P := cl conv( m i=1 P i ) and Q the set of those x ∈ R n that have an extension (x, λ) ∈ R n+m satisfying While P ⊆ Q always holds, the converse inclusion is not true in general. Jeroslow gave two sufficient conditions for Q = P. Blair [14] devised new necessary and sufficient conditions for Q = P, and proved that deciding whether equality holds is an NP-hard decision problem. Blair's condition was later generalized by Vielma [46] (Corollary 2). Balas [5] gave yet another sufficient condition for Q = P, and applied it to multiple network polyhedra. Vielma [45] developed ideal, non-extended MIP formulations for modeling disjunctive constraints based on the concept of Cayley Embedding, and applied it to SOS2 constraints, and to piecewise linear functions on the grid. In [27], Huchette and Vielma provided a very general ideal formulation for combinatorial disjunctive constraints based on the generalized Cayley Embedding of [45], and applied it to univariate and bivariate piecewise linear functions. Lee and Wilson [36] studied the modeling of piecewise linear functions on arbitrary triangulations of their domains. Their model can be reinterpreted as the Cayley Embeddig of a union of polyhedra (one polytope for each triangle) using the vectors i . Vielma [46] generalized the Cayley Trick for polyhedra to the union of convex sets. He obtained ideal, non-extended formulations for m i=1 C i using the corresponding gauge functions, and applied them to a wide range of disjunctive constraints. Vielma also characterized the boundary structure of the Cayley Embedding of the union of convex sets, which in the polyhedral case is as follows. Each face of P emb is of the form conv(∪ m i=1 (F i × { i })), where the F i are faces of the P i with intersecting normal vectors, see also [51].
Branching schemes constitute another approach for deriving ideal, non-extended formulations for (combinatorial) disjunctive constraints. Tomlin [43] proposed a new modeling of SOS2 sets with binary variables, while Martin et al. [38] introduced SOSk constraints and a branching scheme for modeling two-variable piecewise linear functions. Vielma and Nemhauser introduced the concept of independent branching schemes for modeling a constraint very similar to the combinatorial disjunctive constraint (5) of [26], the only difference being that An independent branching scheme of depth d for (5) is given by pairs of disjoint sets L k , ). This yields the formulation This formulation is ideal. Of course, the existence of a branching scheme of depth d depends on the sets S i . For SOS1 and SOS2 constraints, the authors obtained formu-lations with log 2 n binary variables and twice as many additional constraints. They also devised MIP formulations for 2-variable piecewise linear functions on a grid triangulation of a square using a logarithmic number of new variables and constraints in the number of triangles, see also [47]. When Δ V is replaced by [0, 1] V in the above definitions, they obtained an independent branching scheme of depth d = m. In [26], Huchette and Vielma gave a necessary and sufficient condition for the existence of an independent branching scheme of depth d for a combinatorial disjunctive constraint. The authors described a number of techniques for constructing independent branching schemes of depth log 2 m , and applied them to several well-known disjunctive constraints, including SOS2, SOSk, and Grid Triangulations. In [28], new methods are described for constructing independent branching schemes for univariate and bivariate piecewise linear functions (on grid triangulations), and the computational merits of the various approaches are evaluated. Ceria and Soares [16] generalized the formulation (2) to closed convex sets C i using the perspective mappings of the convex functions defining the C i . The authors also described a primal procedure which converges to the optimum under mild conditions. On/off constraints can be used to model disjunctive constraints and the description of the set of feasible solutions in linear and nonlinear mathematical programs is the topic of e.g., [15,[22][23][24]42,46]. In particular, Hijazi et al. [24] considered a linear program with a disjunctive constraint x ∈ P 0 ∪ P 1 , where P 0 is given by lower and upper bounds for the variables, while P 1 is specified by one linear inequality and another set of lower and upper bounds on the variables. The authors provided a complete description of conv(P 0 ∪ P 1 ) using one new binary variable only. Note that their description needs exponentially many new constraints in the number of the variables, but the authors left open which inequalities induce facets.
For more results on extended formulations in combinatorial optimization see the review paper [17], and also [35,40].

Preliminaries
In this section we derive a (not necessarily minimal) linear representation of P emb * .
Here and below, we always assume that P emb * admits a network representation N = (V , A, c). As we will see, the linear representation is closely related to the s − t cuts of N . We will define a dominance relation between s − t cuts, and prove that dominated s − t cuts are redundant in the linear representation.
To start with, we describe some reduction rules for the arc capacities of N without affecting any s − t flow of value n j=1 x j . Recall that for any i the arc capacities of the subnetwork The capacity c(a) = k a λ i of the unique arc a = (u, v) ∈ E(G i ) can be reduced without affecting the set of feasible s − t flows of value n j=1 x j in N x,λ for any (x, λ) ∈ P emb * if one of the following two transformations can be applied to it: and k a > k a for the unique arc a := (u , u) ∈ E(G i ), then k a can be reduced to k a , thus c(a) becomes k a λ i .
Throughout the paper we assume that N is reduced , i.e., the above transformations cannot be applied to it. Now, we express the capacity of the s − t cuts of N x,λ in terms of x and λ.
In order to describe P emb ≤ by linear inequalities, we define the polytope while for P emb Let Q * denote one of Q ≤ and Q = .
and (x, i ) satisfies the valid equations for P emb * . By the MAX-FLOW MIN-CUT Theorem of Ford and Fulkerson [20], N (x, i ) admits an s − t flow of value n j=1 x j if and only if all the s − t cuts have a capacity of at least n j=1 x j . Now, observe that these are precisely the inequalities defining Q * , whence (x, i ) ∈ Q * . To finish the proof, take any point (x, λ) ∈ P emb * , and notice that it is a convex combination of some points with λ integral. Since Q * is a convex polytope, the statement follows.
Our next goal is to prove the converse inclusion. Proposition 2 Q * has vertices, and in each vertex (x, λ) of Q * , the λ is a 0/1 vector.
Proof By Assumption 1, and since P emb * ⊆ Q * , Q * is not empty. Since all variables are non-negative in the definition of Q * , it resides in the non-negative orthant of the (n + m)-dimensional real vector space, and thus it has vertices. Let and Q * is convex. Conversely, consider any point (x, λ) ∈ Q * , and consider any s−t flow of N x,λ of value n j=1 x j . Clearly, such a flow must exist, since the minimum capacity of an s −t cut of N x,λ is n j=1 x j by the definition of Q * . Let f i e be the flow on arc e of G i . Then we have since a flow of value n j=1 x j must saturate all the ( j, t) arcs of N . If λ i = 0, then we pick the vector x i provided by Assumption 1, and we have

Observation 2
The vertices of F S are those vertices of P emb * that are in F S .
Consequently, in all the vertices (x,λ) of F S ,λ = i for some i.

Proposition 3 No facet F S of P emb *
induced by an s − t cut (S,S) of N equals the face induced by x j = 0 for any j.
Now suppose j ∈ V t \S. Then x j = 0 is satisfied by all points (x, λ) ∈ F S if and only if the source s is not connected to j ∈ V t by a directed path, which we excluded.
In order to characterize those s − t cuts (S,S) of N that induce facets of P emb * , as we will see, the single most important parameter is the intersection of S with V t .
A cut (S,S) ∈ C(U ) induces an inequality of the following general form (cf. Observation 1): We are only interested in the minimum capacity s − t cuts in C(U ). We can identify these cuts without fixing λ and x as follows. Since N \{t} is the union of the directed graphs G i , where the G i share only the nodes {s} ∪ V t , we can determine separately for each G i which of its nodes belong to S in order to minimize the coefficients k i .
This observation motivates the following definition.
The members of C min (U ) are the minimum capacity s − t cuts of N with respect to U .

Remark 1 For any
, which proves the statement. Recall that the set function which assigns to S the capacity of the s − t cut (S,S) is submodular, see e.g., Frank [21]. Using this, we can derive where the first inequality follows from the submodularity of the cut capacity function, and the second from the assumption that both of (Z 1 , Remark 1 implies that in the following definition, the face F S does not depend on which non-dominated s − t cut in C min (U ) is chosen. Proof Fix S and i, and we set λ = i . Thus, the arcs of G i have some positive capacities, while all the arcs in the other G k , k = i, subgraphs of N have zero capacity. We construct a vector x such that (x , i ) ∈ P emb * , and (x , i ) satisfies (11) at equality.

Observation 4 If F S is a facet of P emb *
First we consider P emb ≤ . We determine x by constructing an s −t flow f i saturating all the arcs in E(G i ) ∩ [S,S], and which has zero flow on any arc for all j, and it is easy to verify that (11) is satisfied by (x , i ) at equality.
As for P emb = , if the flow f i constructed above saturates the arc (s, v i ) (which is of capacity α i ), then we are done, since then n j=1 x j = f i (s,v i ) = α i , by flow conservation at the nodes. Otherwise, we have to augment f i with additional flow restricted to the subgraph G i (S) of G i spanned by the nodes S ∩ V (G i ). Consider any s − j path in G i (S) for some j ∈ U which consist of non-saturated arcs only. We increase the flow along this path until some arc becomes saturated. We repeat this until no more flow can be sent from s to some node in U in G i (S). Letf i be the resulting flow. If the arc (s, v i ) is saturated byf i , then we are done, since we can set Otherwise, a subset of the arcs saturated bŷ f i constitutes a cut-set separating v i from all the nodes in V t having capacity smaller than α i . But this contradicts Assumption 1.

Proposition 6 For any U
Proof Let (S, S) and (Z , Z ) be a dominating and a dominated s − t cut in C(U ), respectively. We claim that F Z ⊂ F S , and for all i, and at least one of these inequalities is strict. Hence, any Hence (x, i * ) ∈ F S \F Z .

Theorem 1 The set U ⊂ V t induces a facet of P emb
Proof Necessity : suppose that U induces a facet F of P emb . We claim that F ⊂ F Z and F = F Z , and thus F cannot be a facet, which contradicts our initial assumption.
Let (x, λ) ∈ F, and suppose that the capacity of (S, S) is j∈U x j + m i=1 k i λ i . We can decompose the second term of this sum along the components of N \(S ∪ {t}) as follows: However, this equation is equivalent to , there exist some x and i such that (x, i ) ∈ F, and x j > 0, otherwise all points in F satisfy x j = 0, and thus F cannot be a facet by Proposition 3. For each such j, we define a vectorx as follows:x j := 0, andx := x for all = j. Then (x, i ) ∈ F Z \F.
As for (ii), suppose the condition does not hold for some j ∈ U . Then x j = 0 holds for all (x, λ) ∈ F, which leads to a contradiction by Proposition 3.
Sufficiency : Assume that conditions (i) and (ii) of the Theorem hold. Let F := F S be the face of P emb ≤ induced by the dominating s − t cut (S, S) ∈ C min (U ). It suffices to prove that P emb ≤ does not admit any facet F such that F is a proper face of F . By Consider any s − t cut (Z ,Z ) of N which induces a facet F Z = F of P emb ≤ . By Proposition 6, we may also assume that (Z , Z ) is a dominating member of C min (Z ∩V t ). We distinguish two cases: U \Z = ∅, and U ⊂ Z .
First suppose U \Z = ∅, and we construct a point in If none of the sets R i satisfies these conditions, then S violates condition (ii) of the theorem.
Consider some j ∈ R i * ∩ U such that there is a directed path from s to j ∈ V t in the subnetwork G i (S) of G i (such a node exists by the choice of R i * ). Then there exists a point (x, i * ) ∈ F such thatx j > 0, since α i > k i . Consider the set of points X j = {x | 0 ≤ x j ≤x j , x j =x j for all j = j}. On the one hand, (x, i * ) ∈ F for all x ∈ X j . On the other hand, j ∈R i * ∩V t x j is not a constant on X j . Hence, there exists x ∈ X j such that ( Proof Suppose S\Z is not empty. We define the quantities A λ , B λ , C λ , and D λ as follows. Let A λ be the total capacity of those arcs (u, v) such that u ∈ S ∩ Z and v ∈ S\Z . We have , and therefore its capacity is at least that of (S,S). Consequently, A λ ≥ C λ + D λ for all (x, λ) ∈ P (where we compare the coefficients of the λ i ). On the other hand, (Z ,Z ) is a dominating member of C min (Z ∩ V t ), whence A λ + B λ < D λ for some (x, λ) ∈ P. Combining the two inequalities, we obtain B λ + C λ + D λ < D λ for some (x, λ) ∈ P, which is impossible, since B λ , C λ ≥ 0 always hold.
By condition (i), there is some i, and x such that (x, i ) ∈ F, j∈V t ∩Z \U x j > 0, j∈V t \Z x j > 0, and G i \S has a connected component with non-empty intersection with V t ∩ Z \U and also with V t \Z . However, Proof Suppose indirectly that any S ⊂ Z such that S ⊆ S and S \S ⊂ V (G i )\V t , violates condition (a), see Fig. 3 ∈ F Z , a contradiction. Let S be a subset of nodes that satisfies condition (a). Indirectly, suppose it violates (b). Then the cut-set ] must be split, and less than (k i − k Z i ) of it reaches the nodes j ∈ Z ∩ V t \U for any (x, i ) ∈ F. But then j∈V t ∩Z \U x j < k i − k Z i , again a contradiction. Using this claim, we can replace S by S 2 := S ∪ (Z ∩ V (G i )\S ). Then F S 2 = F. Therefore, if S 2 = S, then S is not dominating, a contradiction. Otherwise, G i \S has no connected component which contains some nodes j ∈ V (G i ) ∩ V t ∩ Z \U and j ∈ V (G i ) ∩ V t \Z simultaneously, again a contradiction. Now we turn to P emb = . By definition, m i=1 λ i = 1, and m i=1 α i λ i = n j=1 x j are valid for P emb = . However, adding the latter equation to Q ≤ may render some of the inequalities induced by s − t cuts to implicit equations for P emb = (for the same problem data).
But then x 1 = 1 3 λ 1 is a valid equation for Q = and it is linearly independent of x 1 + x 2 + x 3 = 1 and λ 1 + λ 2 = 1. Thus, a minimal linear representation of Q = is P emb After these preliminaries, we determine the maximum number of linearly independent valid equations for P emb = . Hence, equality holds throughout, and in particular (S ∩ S, S ∩ S) induces a valid equation for P emb = . Now, if V t ∩ S ∩ S = ∅, and V t ∩ S \(V t ∩ S) = ∅, then E can be further split, whence κ is not maximal, a contradiction. On the other hand, if S ∩ V t is the union of some of the E , then the equation induced by (S,S) is implied by (12). The same argument shows that the above partitioning is unique. Finally, by substituting i to λ i , we immediately derive κ =1 β i = α i .

Corollary 2
Let κ be defined as in Proposition 7. Then dim(P emb To facilitate the presentation, we assume that the rooted trees G i that constitute the network N have the following additional structure. For each i ∈ m , let A i ⊆ κ be the subset of those indices such that β i > 0. If |A i | ≥ 2, then G i has exactly Moreover, the subtree of G i rooted at w i has all leafs in E . If A i = { } for some ∈ κ , then all leafs of G i are in E , and no further assumptions are needed. Let N be the subnetwork of N spanned by the nodes {s, t} ∪ E along with those nodes of G i , i ∈ κ , which are on a directed path from v i to E .
Due to the valid equations, the facets of P emb = may have several equivalent forms. We argue that there is a normal form, which is easy to construct.

Proof The facet inducing inequality induced by (S,S) is
If U is the union of some of the E , then F S is not a proper face of P emb = , a contradiction. Hence, there exists some E such that E \U = ∅ and U ∩ E = ∅. If U ∩ E μ = ∅ for all μ = , then U ⊂ E and we are done. Otherwise, there are two cases. First suppose there exists E μ ⊂ U . If we add the valid equation (12) for E to the inequality induced by (S,S) we obtain This inequality corresponds to the dominating s − t cut (S 1 ,S 1 ) ∈ C min (U \E ), and F S = F S 1 . Finally, suppose U ∩ E μ = ∅ for some E μ = E . Since E ∩ E μ = ∅ by definition, it means that a proper subset of E μ is contained in U . We will show that then F S is not a facet of P emb = , a contradiction. For any (x, λ) ∈ F S , consider a feasible flow of value n j=1 x j . This flow saturates the arcs of G i in the cut-set [S,S] for any i ∈ m , and it is split between the nodes E μ \U and V t \(U ∪ E μ ). Hence, there exists β 1 iμ ∈ [0, β iμ ], independent of (x, λ), such that the amount of flow toward E μ \U is β 1 iμ λ i , since the total flow toward the nodes of E μ is β iμ for any (x, λ) ∈ P emb = . Therefore, the dominating s − t cut (Z ,Z ) ∈ C min (U ∪ E μ ) is such that F S ⊆ F Z . Moreover, F S = F Z , otherwise E μ could be further split, and F Z = P emb = , since Z ∩ E = S ∩ E = ∅, and V t ∩ Z \E = V t ∩ S\E = ∅. Then, F S is not a facet of P emb = .

= if and only if (i) for the dominating s − t cut (S, S) ∈ C min (U ), the subgraph N \(S ∪ {t}) is connected, and (ii) if |U | ≥ 2, then for any partitioning of U into non-empty subsets U 1 and U 2 , there exist i ∈ m , and a connected component C of (V (G i )∩ S, E(V (G i )∩ S))
which contains v i , and at least one point from each of U 1 and U 2 , such that for λ = i , the minimum capacity of a cut in C separating v i from V (G i ) ∩ U is larger than α i − k i , where k i is the total capacity of those arcs of G i which are in [S,S], and α i − k i > 0.
Proof Necessity : Suppose that the dominating s − t cut (S,S) ∈ C min (U ) induces a facet of P emb = . Observe that condition (i) coincides with Theorem 1(i). Since P emb = ⊂ P emb ≤ (for the same data), it is enough to verify that U satisfies condition (ii). Indirectly, suppose this is not the case, and there exists a partitioning U 1 ∪U 2 of U into nonempty subsets along with non-negative numbers β i Therefore, F S cannot be a facet of P emb = . Sufficiency: Suppose conditions (i) and (ii) are satisfied by U , and we have to prove that F := F S is a facet of P emb = for the dominating s − t cut (S,S) ∈ C min (U ). As in the proof of Theorem 1, we may assume that F is not contained in the faces of P emb = induced by x j = 0 for any j, or by λ i = 0 for any i. So, suppose (Z ,Z ) induces a facet F Z of P emb = and F ⊂ F Z . Since F Z is a facet, by Proposition 6, (Z ,Z ) is a dominating s − t cut in C min (Z ∩ V t ). We distinguish two cases: U \Z = ∅, and U ⊂ Z .
First suppose U \Z = ∅, and U ∩ Z = ∅. We construct a point (x, λ) in F\F Z to show that F F Z . By condition (ii), there exist i and a connected component C of (V (G i ) ∩ S, A(V (G i ) ∩ S)) such that v i ∈ C, C has non-empty intersection with U \Z and also with U ∩ Z , and for λ = i , the minimum capacity of a cut separating v i from V (G i ) ∩ U is larger than α i − k i , where α i − k i is the remaining quantity to be covered by a flow from v i . Since this minimum capacity is larger than the amount of flow to be sent from v i to the nodes V (G i ) ∩ U , the flows on the arcs of component C are not fixed. Therefore, there exist points (x 1 Let k Z i be the coefficient of λ i in the linear expression for the capacity of (Z ,Z ) as in Observation 1. Consider the partitioning . We argue that for any i, and any (x, i ) ∈ F, the sum of the x j in the three subsets equals α i − k i , α i − k Z i , and k i + k Z i − α i , respectively. For any (x, i ) ∈ F, j∈V t \U x j = k i by definition, and thus j∈U Then, by condition (i), there exists v i ∈ V s \S, such that v i is connected to some j 1 ∈ V t ∩ Z , and also to some node j 2 ∈ V t \(U ∪ Z ). Moreover, the capacity of the cut (in G i ) separating v i from these two sets must be greater than k i , since (S,S) is a dominating s − t cut. Clearly, there must exist (x, i ) ∈ F such that x j 2 > 0, say. Let δ > 0 be small enough such that increasing x j 1 by δ, while decreasing x j 2 by δ yields a vector x such that (x , i ) ∈ F. But x does not satisfy the equation j∈V t ∩Z x j = α i − k Z i , hence (x , i ) / ∈ F Z , a contradiction. Finally, if U ⊂ Z , then we can proceed as in the proof of Theorem 1.

Applications
In this section we present a number of applications where we apply Theorem 1 or Theorem 2 in order to derive the non-trivial facets (different to the non-negativity of the variables) of the corresponding formulations. For each application we will present a MIP formulation, then a network representation, and finally derive the dominating and facet defining inequalities, but we abandon straightforward technical details such as the network correctly represents the feasible solutions of the MIP formulation in a higher dimensional space. If not stated otherwise, the formulations contain no implicit equations.

Special ordered sets of type 2
A special ordered set of type 2 (SOS2) consists of vectors x ∈ R n ≥0 such that n j=1 x j = 1, and x has either one nonzero coordinate, or two consecutive nonzero coordinates, see [13].

≥0
defined as the convex hull of points Observe that in this formulation, if λ j = 1, then x j and x j+1 may be positive, but all other coordinates of x must be 0. Hence, if λ is integral, then proj x (P SO S2 n ) emb is indeed a SOS2. However, relaxing the integrality of the λ i variables in (13), we obtain a polytopeP SO S2 n which has a number of vertices with fractional λ i variables. For instance, for n = 5, x = (1/2, 1/2, 0, 0, 0), and λ = (1/2, 0, 1/2, 0), {1, . . . , n}. For all i = 1, . . . , n−1 we add arcs (s, v i ), (v i , i) and (v i , i + 1) to the network of capacity λ i each, and the arcs ( j, t) of capacity x j for j ∈ V t , see Fig. 4. By construction, we have Clearly, if i, k / ∈ U and j ∈ U for some 1 ≤ i < j < k ≤ n, then N SO S2 n \(S ∪ {t}) is disconnected, and thus the set U does not induce a facet of (P SO S2 n ) emb . We distinguish three cases.
Observe that (14) and (15) along with the non-negativity of the variables and the two valid equations, is precisely the reformulation of Padberg [39] for SOS2 (obtained by complety different means), see also [26,28,45].

Special ordered sets of type k
A special ordered set of type k (SOSk) consists of vectors x ∈ R n ≥0 such that n j=1 x j = 1, and x has at most k consecutive non-zero components, see [26,38]. Clearly, it generalizes SOS2 discussed in the previous section. Let (P SOSk n ) emb be the convex hull of those (x, λ) that satisfy the constraints The corresponding network N = (V , A, c) has a set of nodes V comprising a source s, a sink t, the nodes V s = {v 1 , . . . , v n−k+1 }, and also V t := {1, . . . , n}. s is connected to each of the v i by a directed arc of capacity λ i , and each j ∈ V t to t by a directed arc of capacity x j . Finally, v i is connected to the nodes i, . . . , i + k − 1 in V t by directed arcs of capacity λ i each. By applying Theorem 2 to the network N , one can prove analogously to the SOS2 special case the following:

Logical constraints
Consider a set of binary variables x j indexed by j ∈ J = {1, . . . , n}. For each be the convext hull of those binary vectors x ∈ {0, 1} J that satisfy the following disjunctive constraints: In order to obtain a MIP formulation for P L OG ≤ , we introduce a binary variable λ i for i = 1, . . . , m. Let (P L OG ≤ ) emb be the convex hull of points (x, λ) ∈ {0, 1} n × {0, 1} m that satisfy the constraints Firstly, we identify the dominating s − t cut (S,S) w.r.t. some U ⊂ V t . Clearly, We can apply Theorem 1 to determine those dominating s −t cuts of N with respect to some U ⊂ V t that induce facets of (P L OG ≤ ) emb , however, the condition we get is not much more specific than that of the theorem. All we can say is that the dominating s − t cut w.r.t. any U ⊂ V t satisfies condition (ii) of Theorem 1.
As a special case we reconsider the logical constraints of [7], which we summarized in Sect. 2. By splitting J into two subsets J 1 and J 2 , and after complementing all the variables, i.e.,x j = 1 − x j for all j, we can express P 1 and P 2 as x 1 , . . . , x r ; y 1 , . . . , y s ∈ [0, 1] |M 1 |+···+|M r |+|N 1 |+···+|N r | : Since P 1 and P 2 are in disjoint subspaces, the characterization of Theorem 1 becomes more apparent. Let E 1i = M i , and E 2 j = N j .

Proposition 11
The subset U ⊂ V t induces a facet of conv(P 1 ∪ P 2 ) if |V t \U | = 1, or there exists w 1i or w 2 j such that for the dominating s − t cut (S,S) ∈ C min (U ), w 1i ∈S and M i \U = V t \U , or w 2 j ∈S and N j \U = V t \U .
Proof By Theorem 1, N \(S ∪{t}) must be connected. However, the M i and the N j are disjoint by assumption, so, condition (i) holds only if the conditions of this statement are satisfied. On the other hand, V s ⊂ S, and thus condition (ii) of the same theorem is met as well.

Cardinality indicating polytope
The cardinality indicating polytope P C AR D n ⊆ R 2n+1 ≥0 is defined as the convex hull of the points  For k = 0, . . . , n and j = 1, . . . , n,  the arcs (v k , j) ∈ A of capacity λ k . For j = 1, . . . , n, the arcs ( j, t) ∈ A of capacity x j (Fig. 5). By construction, we have Proof By Proposition 6 it suffices to consider dominating s − t cuts to identify the facets of P C AR D n . For any nonempty subset I ⊆ {1, . . . , n}, let U := V t \I , and (S,S) the unique dominating s − t cut w.r.t. U . Since (S,S) is of minimum capacity w.r.t. U , v k ∈ S for all k > |I |, and v k ∈S for all k < |I |. Since, (S,S) is dominating w.r.t. U , v |I | ∈ S. Clearly, (S,S) satisfies the conditions of Theorem 2, thus the corresponding face F S is a facet of P C AR D n , and takes the form (19).
Notice that Theorem 10 of [35] gives precisely the same description of P C AR D n .

All even subsets
The parity polytope P E V E N n is defined as the convex hull of points x ∈ {0, 1} n : x has even an number of 1's .
The minimal description of P E V E N n in terms of linear inequalities in the space of original variables is attributed to Jeroslow [30]. We apply our technique to get one in a higher dimensional space. Let (P E V E N n ) emb be the convex hull of the set of points
Proof By Proposition 6 only dominating s − t cuts of N E V E N n can define facets. For any nonempty subset I ⊆ {1, . . . , n}, let U := V t \I , and (S,S) the unique dominating s − t cut w.r.t. U . Since (S,S) is of minimum capacity w.r.t. U , v k ∈ S for all 2k ≥ |I |, and v k ∈S for all 2k < |I |. In contrast to the cardinality indicating polytope, not all I determine facets of (P E V E N n ) emb , namely, |I | induces a facet if and only if |I | = 2, and there exists an integer k such that n > 2k and 2k > |I |. If this condition fails, then the s − t cut (S,S) corresponding to I does not satisfy condition (i) or (ii) of Theorem 2. The statement follows.

Polytopes of small cliques
Given a simple graph G = (V, E), and a constant c, let P C L Q ≤c be the polytope whose vertices correspond to cliques of G of size at most c, and It is easy to see that In fact, this polytope has a very simple structure.

Proposition 14
Proof The validity of the equations is easy to verify. They correspond to the dominating s − t cuts for U ⊂ V t such that |U | = n − 1, but they hold at equality for all (x, λ) ∈ (P C L Q =c ) emb and together imply j∈V x j = c. Moreover, the polyhedron on the right is integral, which can be verified by ad-hoc methods.
When c = 2, we can easily project out the λ i variables, by considering the stables sets of G. That is, using a result of [8], we apply the extreme rays of the polyhedron y B ≥ 0, where Bλ = x is the subsystem defining the connection between the x and λ in the definition of (P C L Q =c ) emb . For any stable set I of G, let y j = −1 for j ∈ I , and y j = 1 for j ∈ Γ (I ), the neighbors of I in G. These extreme rays y yield the valid inequalities x(I ) − x(Γ (I )) ≤ 0, for all stable set I of G.
These inequalities were derived in [33] by starting out from a different extended formulation, and Kaibel and Loos also show that along with x j ≥ 0, j ∈ V, and j∈V x j = 2, they suffice to describe P C L Q =c , and they give conditions for these inequalities to define facets. Now let us turn to (P C L Q ≤c ) emb . Clearly,

Proposition 15
The non-trivial facets of (P Proof By Theorem 1, the only sets U ⊂ V t that induce facets of (P C L Q ≤c ) emb are those, where U contains all but one of the nodes j ∈ V t , and the statement follows.
The special case with c = 2 is extensively studied in [29] and [33], where all facets of P C L Q ≤c are determined in the space of the original problem variables.

Union of simplicies
This application is from Jeroslow [31].
We assume that n j=1 L B i j ≤ U B i for all i, and all bounds are finite. Let (P Δ ) emb = conv(∪ m i=1 (P Δ i × i )). Clearly, (P Δ ) emb is the convex hull of those (x, λ) that satisfy the constraints In order to describe (P Δ ) emb , we form a network N = (V , A, c, lb), where V comprises a source s, sink t, the nodes V s = {v 1 , . . . , v m } as well as and ( j, t) for all j ∈ V t . There are both capacities c, and lower bounds lb on the arcs.
We have the following relation between the feasible flows of N and N .
Proof Part i) is obvious. As for ii), first suppose N x,λ admits a feasible flow f of value n j=1 x j , as claimed. The opposite direction can be proved similarly.
Let Q be convex hull of those (x , λ) such that x , λ ≥ 0, and N x ,λ admits a feasible s − t flow of value n j=1 x j . Now we use Theorem 1 to identify those facets of Q that corresponds to some s − t cuts of N and then we lift the description of Q to that of (P Δ ) emb . So, we have the following description for Q :

Observation 5 Q is of dimension n
Notice that this description is valid even if U B i = n j=1 L B i j for all i. Now we transform this description to one for (P Δ ) emb . But this is easy. By Proposition 16 we know that (x, λ) ∈ (P Δ ) emb if and only if Consequently, we have the following: This is precisely the description of Jeroslow obtained by completely different means.

Multi-variate piecewise linear functions
This application is from [26], see also [28]. The objective is to model the graph of a continuous piecewise linear function f : It is easy to see that (P pwl ) emb admits a network representation if and only if w j ∈ C i . Furthermore, A contains all the arcs (s, v i ) for i = 1, . . . , m, and ( j, t) for j = 1, . . . , n. As usual, c( j, t) = x j for j ∈ V t , while c(s, v i ) = λ i , and c(v i , j) = λ i for all (v i , j) ∈ A. The facets of (P pwl ) emb have a particularly nice geometrical characterization. Let κ(T ) denote the number of connected components 3 of some T ⊂ R d . Clearly, the dimension of (P pwl ) emb is n + m − 1 − κ(D).
Proof We construct a dominating s − t cut (S,S) from M and show that it determines an inequality equivalent to (24). Let U := ∪ i∈M Γ out v i , and S := {s} ∪ {v i ∈ V s : Since m i=1 λ i = n j=1 x j = 1 by assumption, we can rewrite this as Rearranging terms yields (24). We still have to verify that S meets the conditions of Theorem 2. In fact, if either of the conditions does not hold, then either the nodes in U are disconnected in the subnetwork spanned by S\{s}, or those in V t \U are disconnected in N \(S ∪ {t}), which implies κ(D\( i∈M C i )) + κ( i∈M C i ) > κ(D) + 1.
In fact, this characterization is the same as that of Lee and Wilson [36].
constitute a grid triangulation of D with m = 2S 1 S 2 triangles, is extensively studied in [26][27][28]45,48], and ideal formulations with O(log 2 m) new (binary) variables and constraints are provided. Different triangulation of the grid may define different piecewise linear functions, even if they agree on the grid points, for an example we refer the reader to [28].
The naive modelling (23) of the grid triangulations leads to m auxiliray λ i variables, which is far more than O(log 2 m) needed by the methods of [26][27][28]45,48]. On the other hand, the techniques of the above papers are difficult to generalize for 3 or more dimensions. Nevertheless, Huchette and Vielma [28] proved that intersecting ideal formulations for combinatorial disjunctions yields an ideal formulation: Theorem 3 [28] Fix s ∈ Z ≥1 , and consider s distinct combinatorial disjunctive con- for each t ∈ s . Then, an ideal formulation for s t=1 B t is  6 Covering the grid [0, 4] 2 ∩ Z 2 by 4 sets of node disjoint unit squares. The squares with the same number constitute one set of the partitioning selecting a cube, and then some additional combinatorial disjunction(s) to select a single simplex in the selected cube (this is how the compact formulation of [28] works in 2 dimensions). So, for t ∈ d , let m t = S t , and T i,t = {w ∈ W : w t ∈ {i − 1, i}} for i ∈ m t , namely, in T i,t we select all the grid points w ∈ D such that w t ∈ {i − 1, i}.
, and observe that this is the SOS2 constraint along the t th axis. Let Π t ⊂ R n × R r t be any ideal formulation for B t with r t = O(log S t ) variables and O(log S t ) constraints, see e.g., [28], and Sect. 2. By Theorem 3, joining the Π t , t ∈ d , yields an ideal formulation for d t=1 B t , and in fact it selects a single cube from the grid (by setting the λ t to unit vectors). However, we still have to select a simplex. All we need is one more combinatorial disjunction, which covers the vertices of all the d! d t=1 S t simplices such that each term in this disjunction covers simplices of node disjoint cubes. In fact, 2 d d! terms suffice, as the following statement shows.

Proposition 20 There is a combinatorial disjunction
-for any unit cube in the grid D ∩ Z d , and any simplex C ∈ C from the cube, there is a unique T i,d+1 containing vert(C), and -Any T i,d+1 consists of the vertices of simplices from vertex disjoint unit cubes of the grid.
Proof To cover a grid in Z d by sets of node disjoint unit cubes, we can partition the grid into 2 d subsets, see Fig. 6. Each of the 2 d sets of cubes yields d! disjunctive terms, namely, we identify each simplex in each unit cube uniquely by a number from 1, . . . , d!, and then we collect all vertices of those simplices having the same id. in a term of the disjunction. More formally, let R k for k ∈ 2 d be the sets of node disjoint unit cubes from the grid which together cover D. Each R k gives rise to d! sets T i,d+1 for i ∈ {d!(k − 1) + 1, . . . , d!k}, where T i,d+1 contains vert(C) for those simplices C from the cubes of R k having id (i − d!(k − 1)). It is easy to see that the sets T i,d+1 satisfy the conditions of the statement.
Moreover, joining any ideal formulations Π t for B t , t ∈ d + 1 , as in Theorem 3, yields an ideal formulation for d+1 t=1 B t .
Finally, using our techniques, it is straightforward to obtain an ideal formulation for P d+1 Node s is connected to each v i ∈ V s by an arc of capacity λ i , node v i is connected to each node j ∈ V t such that w j ∈ T i,d+1 by an arc of capacity λ i , and finally, each j ∈ V t is connected to t by an arc of capacity x j . We also have the equations m i=1 λ i = n j=1 x j = 1. However, there are no implicit equations for (P d+1 = ) emb , because if a partitioning E 1 , . . . , E κ of W , for some κ ≥ 2, induces linearly independent valid equations for (P d+1 = ) emb , then each T i,d+1 must be a subset of some E . It is easy to verify that this is impossible.
We can apply Theorem 2 to obtain the non-trivial facets of (P d+1 Note that there are only 2 d d! auxiliary variables in the resulting formulation, which is just 8 for d = 2, 48 for d = 3, and 384 for d = 4, and it is independent of the size of n. As for the number of facets of (P d+1 = ) emb , it can be exponential in n even if d = 2. However, to our best knowledge, for d ≥ 3, no ideal formulation is known with a logarithmic number of auxiliary variables and linear inequalities in n.

Proposition 21
Let U ⊂ V t , and (S,S) ∈ C min (U ). Then F S is a facet of (P d+1 = ) emb if and only if (a) N \(S ∪ {t}) is connected, and (b) for any partitioning of U into nonempty subsets U 1 and U 2 , there is some i such that T i,d+1 ∩ U = ∅ for = 1, 2, and T i,d+1 ⊆ U .

Modeling of variable intensity activities in a resource loading problem
This application stems from the paper [41], where a resource loading problem is studied. Here we give a new derivation of the polyhedral description of feasible intensity assignments to an activity.
Suppose there is a time horizon of n consecutive time periods of unit length each, and an activity which has to be scheduled in an interval of time periods uninterruptedly. In those time periods when it is processed, a fraction between given lower and upper bounds, L B and U B, must be assigned such that the sum of fractions is 1, where 0 < L B ≤ U B ≤ 1. Let E consist of all the possible execution intervals for the activity, i.e., [k, ] ∈ E if and only if 1 ≤ k ≤ ≤ n, and ( − k + 1)L B ≤ 1 ≤ ( − k + 1)U B. For ease of notation, [k, ] also represents the set of integers {t ∈ Z : k ≤ t ≤ }. If we select [k, ] ∈ E, then the intensity assignment x ∈ R n of the activity must satisfy the conditions j=k x j = 1, L B ≤ x j ≤ U B for j ∈ [k, ], and x j = 0 otherwise.
For each interval [k, ] ∈ E, we can strengthen the lower and upper bounds as follows: To avoid implicit equations, we assume that for each pair of consecutive time periods j, and j + 1, there exists some execution interval [k, ] ∈ E such that j, j + 1 ∈ [k, ], We can model the feasible intensity assignments by means of the following integer program: x j = 1 (28) Let P RL P be the convex hull of those (x, λ) vectors that satisfy these constraint. In order to get the (non-trivial) facets of P RL P , we model the above integer program by a network N = (V , A, c, lb), where V comprises a source node s, a sink node t, the nodes V s := {v k | [k, ] ∈ E}, and the nodes V t := {1, . . . , n}. We have one arc from s to each v k with lower bound lb(s, v k ) = ( − k + 1)L B k λ k , and capacity c(s, v k ) = λ k . Further on, for each v k ∈ V s and j ∈ V t , we have one arc (v k , j) of lower bound λ k L B k , and capacity λ k U k . Finally, we have one arc from each j ∈ V t to t of lower bound 0, and capacity x j . Observe that the positive lower bounds stem from the left hand side of (26). Let Let P RL P be the convex hull of those (x , λ) such that x, λ ≥ 0, [k, ]∈E λ k = 1, and N x ,λ admits a feasible s − t flow of value [k, ]∈E (1 − ( − k + 1)L B k )λ k . We apply the MAX FLOW-MIN CUT theorem to N to obtain a linear representation of P RL P : Let (S,S) be any s − t cut of N , the corresponding inequality is If we increase both sides by [k, ]∈E |[k, ]\U | · L B k λ k , we obtain where we used x j = x j + [k, ]∈E: j∈[k, ] L B k λ k . These are precisely the inequalities (27) obtained in [41] by a different argument. The following statement is analogous to Proposition 18.

Proposition 23
A minimal linear description of P RL P consists λ ≥ 0, the inequalities x j ≥ [k, ]∈E L B k λ k , (27)- (29), and those inequalities from (31) for which the corresponding inequality from (30) induces a facet of P RL P .
It remains to characterize those U ⊂ V t that induce facets of P RL P . First, for a fixed U ⊂ V t , we determine the non-dominated s − t cuts of N w.r.t. U . So we build a set S containing {s} ∪ U , and some of the nodes v k . We can decide for each v k whether to include in S or not in order to minimize the capacity of (S,S).
-If v k ∈ S, then the arcs (v k , j) for j ∈ [k, ]\U are adjacent to v k and leave S. The total capacity of these arcs is -If v k / ∈ S, then the arc (s, v k ) leaves S and its capacity is Comparing (32) and (33), we can derive the following. Proof We derive the two conditions of the theorem from those of Theorem 2. First note that if v k / ∈ E + , then v k ∈ S, as (S,S) is the dominating s − t cut of N w.r.t. U . Consider the first condition of Theorem 2. N \(S ∪ {t}) is not connected if and only if there exists a partitioning of V t \U into two subsetsŪ 1 = ∅ andŪ 2 = ∅, such that each v k ∈S ∩ V s is adjacent to some nodes ofŪ 1 only, or to some nodes ofŪ 2 only, that is, [k, ]\U ⊆Ū 1 or [k, ]\U ⊆Ū 2 . Since [k, ] is an interval, [k, ]\U consists of consecutive members of V t \U , and thus the latter condition is equivalent to the existence of some j ∈ V t \U (but not the last one) such that [k, ]\U ⊆ [1, j] or [k, ]\U ⊆ [ j + 1, n] for each v k ∈S. But this is precisely the negation of the first condition of this theorem.
The equivalence of the second condition of Theorem 2 to that of the present theorem can be shown analogously.
The above characterization of the facets of P RL P is equivalent to that obtained in [41] for the non-trivial facets of P RL P . However, in that paper a more problem specific approach is used.

Final remarks
In this paper we have proposed a new systematic way of constructing ideal, nonextended MIP formulations for disjunctive constraints of the form x ∈ m i=1 P i , where the P i are polytopes. Our method works if the Cayley Embedding of this union of polytopes admits a certain network representation. Then we can characterize the facets of the LP-relaxation of the MIP formulation. We have illustrated the richness of the class of problems that fits this framework by several examples from the literature, and we have also worked out a new one. We emphasize that our characterization of facets is valid for any union of polytopes such that the corresponding Cayley Embedding admits a network representation.
These results may also pave the way for modeling problems using building blocks representing the convex hull of polytopes with the above properties. Then, the solver could generate cuts for such building blocks as needed in the course of solving the problem by branch-and-cut.
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