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Theorem 1.8. Let \( a, b \) be integers not both 0, and let \( g \mathrel{\text{:=}} \gcd \left( {a, b}\right) \) . Equation \( {ax} + {by} = c \) admits an integral solution if and only if \( c \) is an integer and \( g \) divides \( c \) . Furthermore, there exists a \( 2 \times 2 \) integral matrix \( T \) whose inv... | Proof. We may assume without loss of generality that \( a \geq b \geq 0 \) and \( a > 0 \) . We already established the existence of a matrix \( T \) as claimed in the theorem. Equation \( \left( \begin{array}{ll} a & b \end{array}\right) \left( \begin{array}{l} x \\ y \end{array}\right) = c \) can be rewritten as \( \... | Yes |
Corollary 1.9. Given \( a \in {\mathbb{Z}}^{n} \smallsetminus \{ 0\} \), let \( g \mathrel{\text{:=}} \gcd \left( {{a}_{1},\ldots ,{a}_{n}}\right) \) . Equation \( {ax} = c \) admits an integral solution if and only if \( c \) is an integer and \( g \) divides \( c \) . | Furthermore, there exists an \( n \times n \) integral matrix \( T \) whose inverse is also integral such that \( {aT} = \left( {g,0,\ldots ,0}\right) \) . All integral solutions of \( {ax} = c \) are of the form\n\n\[ T\left( \begin{array}{l} \frac{c}{g} \\ z \end{array}\right) ,\;z \in {\mathbb{Z}}^{n - 1} \]\n\nProo... | Yes |
Theorem 1.10. Given \( a \in {\mathbb{Z}}^{n} \smallsetminus \{ 0\} \) and \( c \in \mathbb{R} \), let \( S \mathrel{\text{:=}} \left\{ {x \in {\mathbb{Z}}^{n} : {ax} \leq c}\right\} \) , and let \( g \mathrel{\text{:=}} \gcd \left( {{a}_{1},\ldots ,{a}_{n}}\right) \) . Then\n\n\[ \operatorname{conv}\left( S\right) = \... | Proof. By Corollary 1.9 there exists an integral \( n \times n \) matrix \( T \) whose inverse is integral such that \( {aT} = \left( {g,0,\ldots ,0}\right) \) . Let \( {S}^{\prime } \mathrel{\text{:=}} \left\{ {y \in {\mathbb{R}}^{n} : {Ty} \in S}\right\} \) . Note that \( \operatorname{conv}\left( S\right) = \left\{ ... | Yes |
Theorem 1.12. Every rational matrix with full row rank can be brought into Hermite normal form by a finite sequence of unimodular operations. | Proof. Let \( A \in {\mathbb{Q}}^{m \times n} \) be a rational matrix with full row rank. Let \( M \) be a positive integer such that \( {MA} \) is an integral matrix. We prove the result by induction on the number of rows of \( A \) . Assume that \( A \) has been transformed with unimodular operations into the form \(... | Yes |
Lemma 1.16. Let \( U \) be an \( n \times n \) nonsingular matrix. The following are equivalent.\n\n(i) \( U \) is unimodular,\n\n(ii) \( U \) and \( {U}^{-1} \) are both integral,\n\n(iii) \( {U}^{-1} \) is unimodular,\n\n(iv) For all \( x \in {\mathbb{R}}^{n},{Ux} \) is integral if and only if \( x \) is integral,\n\... | Proof. (i) \( \Rightarrow \) (ii) Assume \( U \) is unimodular. By standard linear algebra, \( {U}^{-1} \) equals the adjugate matrix of \( U \) divided by \( \det \left( U\right) \) . Since \( U \) is integral, its adjugate is integral as well, thus \( {U}^{-1} \) is integral because \( \det \left( U\right) = \pm 1 \)... | Yes |
Theorem 1.17. Let \( A \) be a rational \( m \times n \) matrix with full row-rank, and let \( b \in {\mathbb{R}}^{m} \) . Let \( H = \left( \begin{array}{ll} D & 0 \end{array}\right) \) be the Hermite normal form of \( A \), and \( U \) be a unimodular matrix such that \( H = {AU} \) . Then \( {Ax} = b, x \in {\mathbb... | Proof. By Lemma 1.16, for all \( y \in {\mathbb{R}}^{n} \) we have that \( y \in {\mathbb{Z}}^{n} \) if and only if \( {Uy} \in {\mathbb{Z}}^{n} \), therefore \( y \) is an integral solution of \( {Hy} = {AUy} = b \) if and only if \( x \mathrel{\text{:=}} {Uy} \) is an integral solution of \( {Ax} = b \) . By Remark 1... | Yes |
Theorem 1.19 (Fredholm Alternative). A system of linear equations \( {Ax} = \) \( b \) is infeasible if and only if there exists a vector \( u \in {\mathbb{R}}^{m} \) such that \( {uA} = 0 \) , \( {ub} \neq 0 \) . | A constructive proof of Theorem 1.19 is straightforward using Gaussian elimination on the system \( {Ax} = b \) . | No |
Theorem 1.20 (Integer Farkas Lemma or Kronecker Approximation Theorem). Let \( A \) be a rational matrix and \( b \) a rational vector. The system \( {Ax} = b \) admits no integral solution if and only if there exists a vector \( u \in {\mathbb{R}}^{m} \) such that \( {uA} \in {\mathbb{Z}}^{n},{ub} \notin \mathbb{Z} \)... | Proof. Assume that \( {Ax} = b \) admits an integral solution \( x \) . Then, for any vector \( u \) such that \( {uA} \) is integral, \( {ub} = {uAx} \) is an integer.\n\nSuppose now that \( {Ax} = b \) does not have an integral solution. If system \( {Ax} = b \) is infeasible, Theorem 1.19 shows that there exists \( ... | Yes |
Proposition 2.1. The sets \( K \) and \( {K}^{C} \) coincide. | Proof. It suffices to show that (i) if \( C \) is a minimal cover of \( K \), the inequality \( \mathop{\sum }\limits_{{i \in C}}{x}_{i} \leq \left| C\right| - 1 \) is valid for \( K \) and (ii) the inequality \( \mathop{\sum }\limits_{{i = 1}}^{n}{a}_{i}{x}_{i} \leq b \) is valid for \( {K}^{C} \) . The first statemen... | Yes |
In the example of Fig. 2.1, the inequalities \( {x}_{2} + {x}_{3} + {x}_{4} \leq 1 \) and \( {x}_{2} + {x}_{4} + {x}_{5} \leq 1 \) are clique inequalities relative to the cliques \( \{ 2,3,4\} \) and \( \{ 2,4,5\} \) in \( {G}_{A} \) . Note that the point \( \left( {0,1/2,1/2,1/2,0}\right) \) satisfies \( {Ax} \leq 1 \... | A better formulation of \( {Ax} \leq 1 \) , \( x \in \{ 0,1{\} }^{n} \) is obtained by replacing the constraint matrix \( A \) by the maximal clique versus node incidence matrix \( {A}_{c} \) of the intersection graph of \( A \) . For the example of Fig. 2.1, \( {A}_{c} \mathrel{\text{:=}} \left( \begin{array}{lllll} 1... | Yes |
Proposition 2.4. Let \( \mathcal{S} \) be a clutter and \( \mathcal{T} \) its blocker. Then \( \mathcal{S} \) is the blocker of \( \mathcal{T} \) . | Proof. Let \( \mathcal{Q} \) be the blocker of \( \mathcal{T} \) . We need to show that \( \mathcal{Q} = \mathcal{S} \) . By definition of clutter, it suffices to show that every member of \( \mathcal{S} \) contains some member of \( \mathcal{Q} \) and every member of \( \mathcal{Q} \) contains some member of \( \mathc... | Yes |
Proposition 2.6. The convex hull of solutions to (2.21) is\n\n\\[ \n\\mathop{\\sum }\\limits_{{i = 1}}^{k}{y}_{i} = y \n\\]\n\n\\[ \n{A}_{i}{y}_{i} \\leq {b}_{i}{x}_{i}\\;i = 1,\\ldots, k \n\\]\n\n\\[ \n0 \\leq {y}_{i} \\leq {u}_{i}{x}_{i}\\;i = 1,\\ldots, k \n\\]\n\n\\[ \n\\mathop{\\sum }\\limits_{{i = 1}}^{k}{x}_{i} ... | Proof. Let \( P \\subset {\\mathbb{R}}^{n} \\times {\\mathbb{R}}^{kn} \\times {\\mathbb{R}}^{k} \) be the polytope given in the statement of the proposition. It suffices to show that any point \( \\bar{z} \\mathrel{\\text{:=}} \\left( {\\bar{y},{\\bar{y}}_{1},\\ldots ,{\\bar{y}}_{k},{\\bar{x}}_{1},\\ldots ,{\\bar{x}}_{... | Yes |
Proposition 2.7. Any 0,1 polynomial program (2.22) can be formulated as a pure 0,1 linear program by introducing additional variables. | Proof. Note that, for any integer exponent \( k \geq 1 \), the 0,1 variable \( {x}_{j} \) satisfies \( {x}_{j}^{k} = {x}_{j} \) . Therefore we can replace each expression of the from \( {x}_{j}^{k} \) with \( {x}_{j} \) , so that no variable appears in \( f \) or \( {g}_{i} \) with exponent greater than 1 .\n\nThe prod... | Yes |
Theorem 3.1. A vector \( \left( {{\bar{x}}_{1},\ldots ,{\bar{x}}_{n - 1}}\right) \) satisfies the system (3.2) if and only if there exists \( {\bar{x}}_{n} \) such that \( \left( {{\bar{x}}_{1},\ldots ,{\bar{x}}_{n - 1},{\bar{x}}_{n}}\right) \) satisfies \( {Ax} \leq b \) . | Proof. We already remarked the \ | No |
Example 3.3. Consider the system \( {A}^{3}x \leq {b}^{3} \) of linear inequalities in three variables | Applying Fourier’s procedure to eliminate variable \( {x}_{3} \), we obtain the system \( {A}^{2}x \leq {b}^{2} \) :\n\n\[ \text{-}{x}_{1} \leq - 1 \]\n\n\[ \text{-}{x}_{2} \leq - 1 \]\n\n\[ \begin{matrix} & - & {x}_{1} & - & {x}_{2} & & & \leq & - 3 \end{matrix} \]\n\n\[ {x}_{1} + {x}_{2}\; \leq \]\n\nwhere the last t... | Yes |
Theorem 3.4 (Farkas’ Lemma). A system of linear inequalities \( {Ax} \leq b \) is infeasible if and only if the system \( {uA} = 0,{ub} < 0, u \geq 0 \) is feasible. | Proof. Assume \( {uA} = 0,{ub} < 0, u \geq 0 \) is feasible. Then \( 0 = {uAx} \leq {ub} < 0 \) for any \( x \) satisfying \( {Ax} \leq b \) . It follows that \( {Ax} \leq b \) is infeasible and this proves the \ | No |
Theorem 3.6. The system \( {Ax} + {By} \leq f,{Cx} + {Dy} = g, x \geq 0 \) is feasible if and only if \( {uf} + {vg} \geq 0 \) for every \( \left( {u, v}\right) \) satisfying \( {uA} + {vC} \geq 0 \) , \( {uB} + {vD} = 0, u \geq 0 \) . | Theorem 3.6 can be derived from Theorem 3.4. We leave this proof as an exercise. | No |
Theorem 3.7 (Linear Programming Duality). Given a matrix \( A \in {\mathbb{R}}^{m \times n} \) and vectors \( c \in {\mathbb{R}}^{n}, b \in {\mathbb{R}}^{m} \), let \( P \mathrel{\text{:=}} \{ x : {Ax} \leq b\} \) and \( D \mathrel{\text{:=}} \{ u : {uA} = \) \( c, u \geq 0\} \) . If \( P \) and \( D \) are both nonemp... | Proof. For every \( x \in P \) and \( u \in D \), we have \( {cx} = {uAx} \leq {ub} \), where the equality follows from \( {uA} = c \) and the inequality follows from \( u \geq 0,{Ax} \leq b \) . Hence \( \max \{ {cx} : x \in P\} \leq \min \{ {ub} : u \in D\} \) . Since \( D \neq \varnothing \), this also implies that ... | Yes |
Theorem 3.8 (Complementary Slackness). Given a matrix \( A \in {\mathbb{R}}^{m \times n} \) and vectors \( c \in {\mathbb{R}}^{n}, b \in {\mathbb{R}}^{m} \), let \( P \mathrel{\text{:=}} \{ x : {Ax} \leq b\} \) and \( D \mathrel{\text{:=}} \{ u : {uA} = c, u \geq 0\} \) . Given \( {x}^{ * } \in P \) and \( {u}^{ * } \i... | Proof. We have that \( c{x}^{ * } = {u}^{ * }A{x}^{ * } \leq {u}^{ * }b \), and by Theorem 3.7 equality holds if and only if \( {x}^{ * } \) and \( {u}^{ * } \) are optimal solutions for \( \max \{ {cx} : x \in P\} \) and \( \min \{ {ub} : u \in D\} \) . Since \( {a}^{i}{x}^{ * } \leq {b}_{i} \) and \( {u}_{i}^{ * } \g... | Yes |
Proposition 3.9. Let \( P \mathrel{\text{:=}} \{ x : {Ax} \leq b\} \) and \( D \mathrel{\text{:=}} \{ u : {uA} = c, u \geq 0\} \) , and suppose \( P \neq \varnothing \) . Then \( \max \{ {cx} : x \in P\} \) is unbounded if and only if \( D = \varnothing \) . Equivalently, \( \max \{ {cx} : x \in P\} \) is unbounded if ... | Proof. By Farkas’ lemma (Theorem 3.5), \( D = \varnothing \) if and only if there exists a vector \( \bar{y} \) such that \( A\bar{y} \leq 0 \) and \( c\bar{y} > 0 \) . If \( D \neq \varnothing \), then by Theorem 3.7 \( \max \{ {cx} : x \in P\} = \min \{ {ub} : u \in D\} \), therefore \( \max \{ {cx} : x \in P\} \) is... | Yes |
Theorem 3.11 (Minkowski-Weyl Theorem for Cones). A subset of \( {\mathbb{R}}^{n} \) is a finitely generated cone if and only if it is a polyhedral cone. | Proof. We first show that if \( C \subseteq {\mathbb{R}}^{n} \) is a finitely generated cone, then \( C \) is polyhedral. Let \( R \) be an \( n \times k \) matrix such that \( C = \left\{ {x \in {\mathbb{R}}^{n} : \exists \mu \geq }\right. \) \( 0 \) s.t. \( x = {R\mu }\} \) . We need to show that there exists a matri... | Yes |
Proposition 3.12. Given a rational matrix \( A \in {\mathbb{R}}^{m \times n} \), there exist rational vectors \( {r}^{1},\ldots ,{r}^{k} \in {\mathbb{R}}^{n} \) such that \( \{ x : {Ax} \leq 0\} = \operatorname{cone}\left( {{r}^{1},\ldots ,{r}^{k}}\right) \) . Conversely, given rational vectors \( {r}^{1},\ldots ,{r}^{... | Proof. The statement follows from the proof of Theorem 3.11 and Remark 3.2(ii). | No |
Theorem 3.13 (Minkowski-Weyl Theorem [279,348]). A subset \( P \) of \( {\mathbb{R}}^{n} \) is a polyhedron if and only if \( P = Q + C \) for some polytope \( Q \subset {\mathbb{R}}^{n} \) and finitely generated cone \( C \subseteq {\mathbb{R}}^{n} \) . | Proof. Let \( P \) be a subset of \( {\mathbb{R}}^{n} \) . We need to show that the following two conditions are equivalent.\n\n1. There exist a matrix \( A \) and a vector \( b \) such that \( P = \left\{ {x \in {\mathbb{R}}^{n}}\right. \) : \( {Ax} \leq b\} \) .\n\n2. There exist \( {v}^{1},\ldots ,{v}^{p} \in {\math... | Yes |
Proposition 3.15. Let \( P \mathrel{\text{:=}} \left\{ {x \in {\mathbb{R}}^{n} : {Ax} \leq b}\right\} = \operatorname{conv}\left( {{v}^{1},\ldots ,{v}^{p}}\right) + \operatorname{cone}\left( {{r}^{1},\ldots ,{r}^{q}}\right) \) be a nonempty polyhedron. Then\n\n\[ \operatorname{rec}\left( P\right) = \left\{ {r \in {\mat... | Proof. If \( \bar{r} \) satisfies \( {Ar} \leq 0 \), then \( A\left( {x + \lambda \bar{r}}\right) \leq b + {\lambda A}\bar{r} \leq b \), for every \( x \in P \) and \( \lambda \in {\mathbb{R}}_{ + } \), so \( x + \lambda \bar{r} \in P \). It follows that \( \operatorname{rec}\left( P\right) \supseteq \left\{ {r \in {\m... | Yes |
Theorem 3.17. Let \( P \mathrel{\text{:=}} \left\{ {x \in {\mathbb{R}}^{n} : {Ax} \leq b}\right\} \) be a nonempty polyhedron. Then\n\n\[ \operatorname{aff}\left( P\right) = \left\{ {x \in {\mathbb{R}}^{n} : {A}^{ = }x = {b}^{ = }}\right\} = \left\{ {x \in {\mathbb{R}}^{n} : {A}^{ = }x \leq {b}^{ = }}\right\} . \]\n\nF... | Proof. Since \( P \) is contained in the affine space \( \left\{ {x \in {\mathbb{R}}^{n} : {A}^{ = }x = {b}^{ = }}\right\} \), then \( \operatorname{aff}\left( P\right) \subseteq \left\{ {x \in {\mathbb{R}}^{n} : {A}^{ = }x = {b}^{ = }}\right\} \) . Also, trivially \( \left\{ {x \in {\mathbb{R}}^{n} : {A}^{ = }x = {b}^... | Yes |
The assignment polytope is the following\n\n\[ P \mathrel{\text{:=}} \left\{ {x \in {\mathbb{R}}^{{n}^{2}} : \;\begin{matrix} \mathop{\sum }\limits_{{j = 1}}^{n}{x}_{ij} & = & 1, & i = 1,\ldots n \\ \mathop{\sum }\limits_{{i = 1}}^{n}{x}_{ij} & = & 1, & j = 1,\ldots n \\ {x}_{ij} & \geq & 0, & i, j = 1,\ldots n \end{ma... | We show that \( \dim \left( P\right) = {n}^{2} - {2n} + 1 \) .\n\nLet \( {Ax} = \mathbf{1} \) be the system comprising the \( {2n} \) equations in the definition of \( P \), where \( \mathbf{1} \) denotes the vector of all ones. Thus \( A \) is a \( {2n} \times {n}^{2} \) matrix. We first show that \( \operatorname{ran... | Yes |
The 0,1 knapsack polytope is the convex hull of the 0,1 knapsack set defined in Sect. 2.1. That is, the polyhedron\n\n\[ P \mathrel{\text{:=}} \operatorname{conv}\left( \left\{ {x \in \{ 0,1{\} }^{n} : {ax} \leq b}\right\} \right) \]\n\nwhere \( a \in {\mathbb{R}}_{ + }^{n} \) and \( b > 0 \) . | Let \( J \subseteq \{ 1,\ldots, n\} \) be the set of indices \( j \) such that \( {a}_{j} > b \) . Note that \( P \subseteq \left\{ {x \in {\mathbb{R}}^{n} : {x}_{j} = 0, j \in J}\right\} \) . This shows that \( \dim \left( P\right) \leq n - \left| J\right| \) . On the other hand, for every \( j \notin J \), the \( j \... | Yes |
The permutahedron \( {\Pi }_{n} \subset {\mathbb{R}}^{n} \) is the convex hull of the set \( {S}_{n} \) of the \( n \) ! vectors that can be obtained by permuting the entries of the vector \( \left( {1,2,\ldots, n - 1, n}\right) \). We will show that \( \dim \left( {\Pi }_{n}\right) = n - 1 \). | Note that, for every vector in \( {S}_{n} \), the sum of the components is \( 1 + \) \( 2 + \cdots + n = \left( \begin{matrix} n + 1 \\ 2 \end{matrix}\right) \), thus \( {\Pi }_{n} \) is contained in the hyperplane \( \left\{ {x \in {\mathbb{R}}^{n}}\right. \) : \( \left. {\mathop{\sum }\limits_{{i = 1}}^{n}{x}_{i} = \... | Yes |
The dimension of the Hamiltonian-path polytope of the complete graph on \( n \) nodes is \( \left( \begin{array}{l} n \\ 2 \end{array}\right) - 1 \) . | Let \( P \subset {\mathbb{R}}^{\left( \begin{matrix} n \\ 2 \end{matrix}\right) } \) be the Hamiltonian-path polytope of the complete graph \( G \) on \( n \) nodes. Note that, since all Hamiltonian paths on a graph with \( n \) nodes have \( n - 1 \) edges, the Hamiltonian-path polytope is contained in the hyperplane ... | Yes |
Theorem 3.22. Let \( P : = \{ x \in {\mathbb{R}}^{n} : {Ax} \leq b\} \) be a nonempty polyhedron. An inequality \( {cx} \leq \delta \) is valid for \( P \) if and only if there exists \( u \geq 0 \) such that \( {uA} = c \) and \( {ub} \leq \delta \) . | Proof. Let \( c \in {\mathbb{R}}^{n} \) and \( \delta \in \mathbb{R} \) . Assume \( {uA} = c,{ub} \leq \delta, u \geq 0 \) is feasible. Then, for all \( x \in P \), we have \( {cx} = {uAx} \leq {ub} \leq \delta \) . This shows that \( {cx} \leq \delta \) is valid for \( P \) .\n\nConversely, assume that the inequality ... | Yes |
Theorem 3.24 (Characterization of the Faces). Let \( P \mathrel{\text{:=}} \left\{ {x \in {\mathbb{R}}^{n} : {a}^{i}x \leq }\right. \) \( \left. {{b}_{i}, i \in M}\right\} \) be a nonempty polyhedron. For any \( I \subseteq M \), the set\n\n\[ \n{F}_{I} \mathrel{\text{:=}} \left\{ {x \in {\mathbb{R}}^{n} : {a}^{i}x = {... | Proof. For the first part of the statement, let \( c \mathrel{\text{:=}} \mathop{\sum }\limits_{{i \in I}}{a}^{i},\delta \mathrel{\text{:=}} \mathop{\sum }\limits_{{i \in I}}{b}_{i} \) . Then \( {cx} \leq \delta \) is a valid inequality. Furthermore, given \( x \in P, x \) satisfies \( {cx} = \delta \) if and only if i... | Yes |
Proposition 3.25. Given a polyhedron \( P \), the following hold.\n\n(i) The number of faces of \( P \) is finite.\n\n(ii) For every nonempty face \( F \) of \( P,\operatorname{lin}\left( F\right) = \operatorname{lin}\left( P\right) \) .\n\n(iii) Given faces \( F \) and \( {F}^{\prime } \) of \( P, F \cap {F}^{\prime }... | Proof. Let \( P \mathrel{\text{:=}} \left\{ {x \in {\mathbb{R}}^{n} : {a}^{i}x \leq {b}_{i}, i \in M}\right\} \).\n\n(i) By Theorem 3.24, the number of faces of \( P \) does not exceed \( {2}^{\left| M\right| } \).\n\n(ii) It follows from Proposition 3.15 and Theorem 3.24 that \( \operatorname{lin}\left( F\right) = \{ ... | Yes |
Lemma 3.26. Let \( P \mathrel{\text{:=}} \left\{ {x \in {\mathbb{R}}^{n} : {Ax} \leq b}\right\} \) be a nonempty polyhedron. Given \( j \in {I}^{ < } \) such that the inequality \( {a}^{j}x \leq {b}_{j} \) is irredundant, let \( F \mathrel{\text{:=}} \{ x \in \) \( \left. {P : {a}^{j}x = {b}_{j}}\right\} \) . The follo... | Proof. (i) By Theorem 3.24, \( F \) is a face of \( P \) . By Remark 3.16, there exists \( \bar{x} \in P \) satisfying \( {a}^{i}\bar{x} < {b}_{i} \) for all \( i \in {I}^{ < } \) . Furthermore, since \( {a}^{j}x \leq {b}_{j} \) is not redundant, there exists a point \( \widetilde{x} \) satisfying\n\n\[ \n{a}^{i}\widet... | Yes |
Theorem 3.27 (Characterization of the Facets). Let \( P \mathrel{\text{:=}} \left\{ {x \in {\mathbb{R}}^{n}}\right. \) : \( {Ax} \leq b\} \) be a nonempty polyhedron, and let \( f \) be the number of its facets.\n\n(i) For each facet \( F \) of \( P \), there exists \( j \in {I}^{ < } \) such that the inequality \( {a}... | Proof. (i) Given a facet \( F \) of \( P \), by Theorem 3.24 there exists \( I \subseteq {I}^{ < } \) such that \( F = \left\{ {x \in P : {a}^{i}x = {b}_{i}, i \in I}\right\} \) . Since \( F \neq P \), it follows that \( I \neq \varnothing \) . Choose \( j \in I \) and let \( {F}^{\prime } \mathrel{\text{:=}} \left\{ {... | Yes |
The stable set problem has been introduced in Chap. 2. Given a graph \( G = \left( {V, E}\right) \) the stable set polytope \( \operatorname{STAB}\left( G\right) \) is the convex hull of the characteristic vectors of all the stable sets of \( G \) . \( \operatorname{STAB}\left( G\right) \) is a full-dimensional polytop... | Since \( K \) is a maximal clique of \( G \), every node in \( V \smallsetminus K \) is contained in a stable set of size 2 containing one node of \( K \) . Consider the \( \left| V\right| \) stable sets consisting of the \( \left| {V \smallsetminus K}\right| \) stable sets just defined, together with \( \left| K\right... | Yes |
Consider the permutahedron \( {\Pi }_{n} \) and the set \( {S}_{n} \) of permutation vectors defined in Example 3.20. Note that, for any \( K \subset \{ 1,\ldots, n\} \) , letting \( k = \left| K\right| \), the inequality\n\n\[ \mathop{\sum }\limits_{{i \in K}}{x}_{i} \geq \left( \begin{matrix} k + 1 \\ 2 \end{matrix}\... | Indeed, given any permutation vector \( \bar{x} \in {S}_{n},\mathop{\sum }\limits_{{i \in K}}{\bar{x}}_{i} \geq \) \( 1 + 2 + \cdots + k = \left( \begin{matrix} k + 1 \\ 2 \end{matrix}\right) \), where equality holds if and only if \( \left\{ {{\bar{x}}_{i} : i \in K}\right\} = \) \( \{ 1,\ldots, k\} \) . We show that ... | Yes |
Theorem 3.30 (Uniqueness of the Minimal Representation).\n\nLet \( P \subseteq {\mathbb{R}}^{n} \) be a nonempty polyhedron. Let \( k : = \dim \left( P\right) \) and \( f \) be the number of facets of \( P \) . Let \( {A}^{ = }x = {b}^{ = },{A}^{ < }x \leq {b}^{ < } \) and \( {C}^{ = }x = {d}^{ = },{C}^{ < }x \leq {d}^... | Proof. (i) Both matrices \( {A}^{ = },{C}^{ = } \) have full row-rank, otherwise \( {A}^{ = }x = {b}^{ = } \) or \( {C}^{ = }x = {d}^{ = } \) would contain some redundant equation. By Theorem 3.17, \( \left\{ {x \in {\mathbb{R}}^{n} : {A}^{ = }x = {b}^{ = }}\right\} = \left\{ {x \in {\mathbb{R}}^{n} : {C}^{ = }x = {d}^... | Yes |
Theorem 3.33. Let \( P \mathrel{\text{:=}} \left\{ {x \in {\mathbb{R}}^{n} : {Ax} \leq b}\right\} \) be a nonempty polyhedron.\n\n(i) A nonempty face \( F \) of \( P \) is minimal if and only if \( F = \left\{ {x \in {\mathbb{R}}^{n}}\right. \) : \( \left. {{A}^{\prime }x = {b}^{\prime }}\right\} \) for some system \( ... | Proof. We prove (i). Let \( F \) be a nonempty face of \( P \) . We first show the \ | No |
Theorem 3.34. Let \( P \mathrel{\text{:=}} \left\{ {x \in {\mathbb{R}}^{n} : {Ax} \leq b}\right\} \) be a pointed polyhedron, and let \( \bar{x} \in P \) . The following statements are equivalent.\n\n(i) \( \bar{x} \) is a vertex.\n\n(ii) \( \bar{x} \) satisfies at equality \( n \) linearly independent inequalities of ... | Proof. By Theorem 3.33, (i) and (ii) are equivalent. We show next that (i) implies (iii). Indeed, suppose \( \bar{x} \) is a vertex, and let \( {cx} \leq \delta \) be a valid inequality for \( P \) such that \( \{ \bar{x}\} = P \cap \{ x : {cx} = \delta \} \) . Given \( {x}^{\prime },{x}^{\prime \prime } \in P \) and \... | Yes |
Theorem 3.35. Let \( C \mathrel{\text{:=}} \left\{ {x \in {\mathbb{R}}^{n} : {Ax} \leq 0}\right\} \) be a pointed cone, and let \( \bar{r} \) be a ray of \( C \) . The following are equivalent.\n\n(i) \( \bar{r} \) is an extreme ray of \( C \) .\n\n(ii) \( \bar{r} \) satisfies at equality \( n - 1 \) linearly independe... | Proof. Consider \( \bar{r} \in C \smallsetminus \{ 0\} \) . Let \( \alpha \in {\mathbb{R}}^{n} \) be the sum of the rows of \( A \) and let \( P \mathrel{\text{:=}} \{ x \in C : {\alpha x} \geq - 1\} \) . Since \( C \) is pointed, \( \operatorname{rank}\left( A\right) = n \) (Proposition 3.15). Therefore, \( {\alpha x}... | Yes |
We show that the cut polytope is a neighborly polytope, that is, any pair of vertices of \( {P}_{n}^{\text{cut }} \) is contained in an edge of \( {P}_{n}^{\text{cut }} \) (in other words, the skeleton of \( {P}_{n}^{\text{cut }} \) is a clique on \( {2}^{n - 1} \) nodes). | The vertices of \( {P}_{n}^{\text{cut }} \) are precisely the characteristic vectors of the cuts of \( G \) (this follows from Exercise 3.25). Let \( S, T \subseteq V \) such that \( \delta \left( S\right) \neq \delta \left( T\right) \) . We show that the characteristic vectors \( {x}^{S} \) of \( \delta \left( S\right... | Yes |
Theorem 3.37 (Decomposition Theorem for Polyhedra). Let \( P \subseteq {\mathbb{R}}^{n} \) be a nonempty polyhedron, and let \( t \mathrel{\text{:=}} \dim \left( {\operatorname{lin}\left( P\right) }\right) \) . Let \( {F}_{1},\ldots ,{F}_{p} \) be the family of minimal faces of \( P \), and \( {R}_{1},\ldots ,{R}_{q} \... | Proof. Assume first that \( P \) is pointed, i.e., \( \dim \left( {\operatorname{lin}\left( P\right) }\right) = 0 \) . Then minimal faces are the vertices of \( P \) and one-dimensional faces of \( \operatorname{rec}\left( P\right) \) are the extreme rays of \( \operatorname{rec}\left( P\right) \) . In this case the ab... | Yes |
Theorem 3.38. Let \( P \mathrel{\text{:=}} \left\{ {x \in {\mathbb{R}}^{n} : {Ax} \leq b}\right\} \) be a pointed polyhedron where \( A \) and \( b \) have rational entries. Let \( {v}^{1},\ldots ,{v}^{p} \in {\mathbb{Q}}^{n} \) be the vertices of \( P \), and \( {r}^{1},\ldots ,{r}^{q} \in {\mathbb{Q}}^{n} \) its extr... | Proof. By Theorem 3.34, a point \( \bar{x} \in P \) is a vertex if and only if there exists a system \( \bar{A}x \leq \bar{b} \) comprising \( n \) linearly independent inequalities from \( {Ax} \leq b \) such that \( \bar{x} \) is the unique solution of \( \bar{A}x = \bar{b} \) . By Proposition 1.2, the encoding size ... | Yes |
Theorem 3.39. Let \( P = \operatorname{conv}\left( {{v}^{1},\ldots ,{v}^{p}}\right) + \operatorname{cone}\left( {{r}^{1},\ldots ,{r}^{q}}\right) \) be a polyhedron, where \( {v}^{1},\ldots ,{v}^{p} \) and \( {r}^{1},\ldots ,{r}^{q} \) are given rational vectors in \( {\mathbb{R}}^{n} \) . If the encoding size of each v... | Proof. By Theorem 3.27, it suffices to show that, given any facet \( F \) of \( P \) , there exists a valid inequality \( {\alpha x} \leq \beta \) for \( P \) defining \( F \) whose encoding size is polynomially bounded by \( n \) and \( L \) . Let \( d = \dim \left( P\right) \) . Let \( {q}^{1},\ldots ,{q}^{d} \) be \... | Yes |
Theorem 3.40 (Carathéodory). If a vector \( v \in {\mathbb{R}}^{n} \) is a conic combination of vectors in some set \( X \subseteq {\mathbb{R}}^{n} \), then it is a conic combination of at most \( \dim \left( X\right) \) linearly independent vectors in \( X \) . | Proof. We can assume that \( X \) is finite, say \( X = \left\{ {{v}^{1},\ldots ,{v}^{k}}\right\} \) . Since \( v \in \operatorname{cone}\left( X\right) \), it follows that the polyhedron \( P \mathrel{\text{:=}} \left\{ {\lambda \in {\mathbb{R}}_{ + }^{k} : \mathop{\sum }\limits_{{i = 1}}^{k}{\lambda }_{i}{v}^{i} = v}... | Yes |
Corollary 3.41. If a point \( v \in {\mathbb{R}}^{n} \) is a convex combination of points in some set \( X \subseteq {\mathbb{R}}^{n} \), then it is a convex combination of at most \( \dim \left( X\right) + 1 \) affinely independent points in \( X \) . | Proof. If \( v \in \operatorname{conv}\left( X\right) \), then \( \left( \begin{array}{l} v \\ 1 \end{array}\right) \in {\mathbb{R}}^{n + 1} \) is a conic combination of points in \( X \times \{ 1\} \) . By Theorem 3.40, there exist \( {v}^{1},\ldots ,{v}^{k} \in X \) and \( \lambda \in {\mathbb{R}}_{ + }^{k} \) such t... | Yes |
Theorem 3.44 (Helly). Let \( {C}_{1},{C}_{2},\ldots ,{C}_{h} \) be convex sets in \( {\mathbb{R}}^{d} \) such that \( {C}_{1} \cap {C}_{2} \cap \cdots \cap {C}_{h} = \varnothing \), where \( h \geq d + 1 \) . Then there exist \( d + 1 \) sets among \( {C}_{1},{C}_{2},\ldots ,{C}_{h} \) whose intersection is empty. | The proofs are left as an exercise, see Exercises 3.30, 3.31. | No |
Theorem 3.46. Consider a polyhedron \( P \mathrel{\text{:=}} \left\{ {\left( {x, z}\right) \in {\mathbb{R}}^{n} \times {\mathbb{R}}^{p} : {Ax} + {Bz} \leq }\right. \) \( b\} \), and let \( {r}^{1},\ldots ,{r}^{q} \) be the extreme rays of \( {C}_{P} \) . Then \( {\operatorname{proj}}_{x}\left( P\right) = \left\{ {x \in... | Proof. It suffices to show that, for any \( \bar{x} \in {\mathbb{R}}^{n},\bar{x} \notin {\operatorname{proj}}_{x}\left( P\right) \) if and only if \( {r}^{t}A\bar{x} > {r}^{t}b \) for some \( t = 1,\ldots, q \) . By definition, \( \bar{x} \notin {\operatorname{proj}}_{x}\left( P\right) \) if and only if the system \( {... | Yes |
Theorem 3.49. Given \( {a}^{1},\ldots ,{a}^{m} \in {\mathbb{R}}^{n} \) and \( 0 \leq k \leq m \), let \( P : = \{ x \in {\mathbb{R}}^{n} : \) \( \left. {{a}^{i}x \leq 1, i = 1\ldots, k;{a}^{i}x \leq 0, i = k + 1\ldots, m}\right\} \) and \( Q \mathrel{\text{:=}} \operatorname{conv}\left( {0,{a}^{1},\ldots ,{a}^{k}}\righ... | Proof. We first show that \( Q \subseteq {P}^{ * } \) and \( P \subseteq {Q}^{ * } \) . It is sufficient to prove that, given \( x \in P \) and \( y \in Q,{yx} \leq 1 \) . If \( y \in Q \) then there exists \( \nu \in {\mathbb{R}}^{m} \) such that \( y = \mathop{\sum }\limits_{{i = 1}}^{m}{\nu }_{i}{a}^{i} \), where \(... | Yes |
Corollary 3.50. Let \( P \subseteq {\mathbb{R}}^{n} \) be a polyhedron containing the origin. The following hold.\n\n(i) \( {P}^{* * } = P \) .\n\n(ii) \( {P}^{ * } \) is bounded if and only if \( P \) contains the origin in its interior.\n\n(iii) \( \operatorname{aff}\left( {P}^{ * }\right) \) is the orthogonal comple... | Proof. Let \( A \in {\mathbb{R}}^{m \times n} \) and \( b \in {\mathbb{R}}^{m} \) such that \( P = \left\{ {x \in {\mathbb{R}}^{n} : {Ax} \leq b}\right\} \) . Since \( 0 \in P \), it follows that \( b \geq 0 \) . Thus we can assume that \( P = \left\{ {x \in {\mathbb{R}}^{n} : {a}^{i}x \leq 1}\right. \) , \( i = 1\ldot... | Yes |
Lemma 3.51. Let \( \left( {{A}_{i},{R}_{i}}\right) \) be an MW-pair and let \( {A}_{i + 1} \) be obtained from \( {A}_{i} \) by adding row \( {a}^{i + 1} \) . Let \( {R}_{i + 1} \) be the matrix constructed by the double description method. Then \( \left( {{A}_{i + 1},{R}_{i + 1}}\right) \) is an MW-pair. | Proof. Let \( C \mathrel{\text{:=}} \left\{ {x : {A}_{i + 1}x \leq 0}\right\} \) and \( Q \mathrel{\text{:=}} \operatorname{cone}\left( {R}_{i + 1}\right) \) . By construction, every column \( r \) of \( {R}_{i + 1} \) satisfies \( {A}_{i + 1}r \leq 0 \), therefore \( Q \subseteq C \). To prove \( Q \supseteq C \), con... | Yes |
Theorem 4.1. Let \( P \) be a rational polyhedron. The following conditions are equivalent.\n\n(i) \( P \) is an integral polyhedron.\n\n(ii) Every minimal face of \( P \) contains an integral point.\n\n(iii) \( \max \{ {cx} : x \in P\} \) is attained by an integral vector \( x \) for each \( c \in {\mathbb{R}}^{n} \) ... | Proof. (i) \( \Rightarrow \) (ii) Since \( P \) is integral, every \( x \in P \) is a convex combination of integral points in \( P \) . Given a minimal face \( F \) of \( P \), let \( x \in F \) . Then \( x = \mathop{\sum }\limits_{{j = 1}}^{k}{\lambda }_{j}{x}^{j} \), where \( {x}^{1},\ldots ,{x}^{k} \in P \cap {\mat... | Yes |
Corollary 4.2. A rational polyhedron \( P \) is integral if and only if every rational supporting hyperplane for \( P \) contains an integral point. | Proof. \ | No |
Theorem 4.3. Let \( P \subseteq {\mathbb{R}}^{n} \times {\mathbb{R}}^{p} \) be a rational polyhedron, and let \( S \mathrel{\text{:=}} P \cap \left( {{\mathbb{Z}}^{n} \times {\mathbb{R}}^{p}}\right) \) . The following are equivalent.\n\n(i) \( P = \operatorname{conv}\left( S\right) \) .\n\n(ii) Every minimal face of \(... | The proof of this theorem is similar to the proof of Theorem 4.1 (we leave it to the reader to check this). | No |
Theorem 4.4 (Hoffman and Kruskal). Let \( A \) be an \( m \times n \) integral matrix. The polyhedron \( \{ x : {Ax} \leq b, x \geq 0\} \) is integral for every \( b \in {\mathbb{Z}}^{m} \) if and only if \( A \) is totally unimodular. | Proof. For all \( b \in {\mathbb{Z}}^{m} \), let \( P\left( b\right) \mathrel{\text{:=}} \{ x : {Ax} \leq b, x \geq 0\} \) . Since \( P\left( b\right) \) is a rational pointed polyhedron for all \( b \in {\mathbb{Z}}^{m} \), by Theorem 4.1 \( P\left( b\right) \) is an integral polyhedron if and only if all its vertices... | No |
A matrix \( A \) is totally unimodular if and only if every column submatrix of \( A \) admits an equitable bicoloring. | For the \ | No |
Corollary 4.8. \( {A0}, \pm 1 \) matrix \( A \) with at most two nonzero elements in each column is totally unimodular if and only if \( A \) admits an equitable row-bicoloring. | This is because, when \( A \) has at most two nonzero entries per column, an equitable row-bicoloring of \( A \) trivially induces one for any row submatrix. | No |
Theorem 4.9. Incidence matrices of digraphs are totally unimodular. | Proof. Since the incidence matrix of a digraph has two nonzero entries in each column, the statement follows from Corollary 4.8 by coloring all rows the same color. | Yes |
Lemma 4.10. Let \( D \) be a digraph. The extreme rays of the circulation cone are the characteristic vectors of the simple circuits of \( D \) . | Proof. Let \( \bar{x} \) be an extreme ray of \( \left\{ {x \in {\mathbb{R}}^{A} : {A}_{D}x = 0, x \geq 0}\right\} \) . Possibly by multiplying \( \bar{x} \) by a positive scalar, we may assume that \( \bar{x} \leq \mathbf{1} \) and that \( \bar{x} \) is a vertex of the polytope \( Q \mathrel{\text{:=}} \left\{ {x \in ... | Yes |
Theorem 4.11. \( {P}_{\text{subtour }} \subset {\operatorname{proj}}_{x}\left( {P}_{MTZ}\right) \) . | Proof. By Theorem 3.46 we have that\n\n\( {\operatorname{proj}}_{x}\left( {P}_{MTZ}\right) = \left\{ {x \in {\mathbb{R}}^{A} : x}\right. \) satisfies \( \left( {4.2}\right) \) and \( v\left( {\left( {n - 1}\right) \mathbf{1} - {nx}}\right) \geq 0 \) for all \( v \in Q\}\ \)\n\nwhere \( Q \mathrel{\text{:=}} \{ v \geq 0... | Yes |
Lemma 4.14. Given a digraph \( D \), two distinct nodes \( s, t \in V \), nonnegative capacities \( {c}_{e}, e \in A \), a feasible \( s, t \) -flow \( x \) and an \( s, t \) -cut \( \Gamma = {\delta }^{ + }\left( S\right) \), we have that \( \operatorname{val}\left( x\right) \leq c\left( \Gamma \right) \) . Furthermor... | Proof. Since \( \Gamma \mathrel{\text{:=}} {\delta }^{ + }\left( S\right) \) is an \( s, t \) -cut, \( s \in S \) and \( t \notin S \) . Therefore adding all the equations (4.8) for the nodes in \( S \) one obtains\n\n\[ \operatorname{val}\left( x\right) = \mathop{\sum }\limits_{{e \in {\delta }^{ + }\left( S\right) }}... | Yes |
Theorem 4.15 (Max-Flow Min-Cut Theorem). Given a digraph \( D \), two distinct nodes \( s, t \in V \), and nonnegative capacities \( {c}_{e}, e \in A \) , \[ \max \{ \operatorname{val}\left( x\right) : x\text{is a feasible}s, t\text{-flow}\} = \min \{ c\left( \Gamma \right) : \Gamma \text{is an}s, t\text{-cut}\} \text{... | Proof. Let \( {x}^{ * } \) be a feasible \( s, t \) -flow of maximum value. By Lemma 4.14 it suffices to show that there exists an \( s, t \) -cut \( \Gamma \) such that \( \operatorname{val}\left( {x}^{ * }\right) \geq c\left( \Gamma \right) \) . Let \( \left( {{y}^{ * },{z}^{ * }}\right) \) be an optimal solution for... | Yes |
Theorem 4.16. Given a digraph \( D \), two distinct nodes \( s, t \in V \), and nonnegative capacities \( {c}_{e}, e \in A \), a feasible flow \( x \) is maximum if and only if the residual digraph \( {D}_{x} = \left( {V,{A}_{x}}\right) \) contains no \( s, t \) -path. | Proof. Assume first \( {D}_{x} = \left( {V,{A}_{x}}\right) \) contains an \( s, t \) -path \( P \) . Let\n\n\[ \varepsilon \mathrel{\text{:=}} \min \left\{ {\mathop{\min }\limits_{{{uv} \in {\overrightarrow{A}}_{x} \cap P}}{c}_{uv} - {x}_{uv},\mathop{\min }\limits_{{{vu} \in {\overleftarrow{A}}_{x} \cap P}}{x}_{uv}}\ri... | Yes |
Theorem 4.18. Let \( {A}_{G} \) be the incidence matrix of a graph \( G \) . Then \( {A}_{G} \) is totally unimodular if and only if \( G \) is bipartite. | Proof. Since \( {A}_{G} \) has two nonzero entries in each column, by Corollary 4.8 \( {A}_{G} \) is totally unimodular if and only if it has an equitable row-bicoloring. Note that an equitable row-bicoloring of \( {A}_{G} \) corresponds to a bipartition of the nodes of \( G \) such that each edge has an endnode in eac... | Yes |
Corollary 4.19. If \( G \) is a bipartite graph, then the matching polytope of \( G \) is the set \( \left\{ {x \in {\mathbb{R}}^{E} : {A}_{G}x \leq \mathbf{1}, x \geq 0}\right\} \), and the perfect matching polytope of \( G \) is the set \( \left\{ {x \in {\mathbb{R}}^{E} : {A}_{G}x = \mathbf{1}, x \geq 0}\right\} \). | This shows that in the bipartite case finding the maximum cardinality matching amounts to solving the linear relaxation of (4.14). We will see in Sect. 4.4.4 that a set of inequalities describing the matching polytope can be given even for general graphs, but it is considerably more involved. | No |
Theorem 4.20. A matching \( M \) has maximum cardinality if and only if there is no \( M \) -augmenting path. | Proof. We already proved the \ | No |
Theorem 4.26. Let \( {Ax} \leq b \) be a totally dual integral system and \( b \) be an integral vector. Then \( P \mathrel{\text{:=}} \{ x : {Ax} \leq b\} \) is an integral polyhedron. | Proof. For every \( c \) for which the value \( {z}_{c} \mathrel{\text{:=}} \max \{ {cx} : {Ax} \leq b\} \) is finite, we have that \( {z}_{c} \) is integer since \( {z}_{c} = {by} \) for some integral optimal solution \( y \) to the dual \( \min \{ {yb} : {yA} = c, y \geq 0\} \) . By Theorem 4.1(iv), \( P \) is an int... | Yes |
Given a graph \( G = \left( {V, E}\right) \), the cut function of \( G \) is the function \( f : {2}^{V} \rightarrow \mathbb{R} \) defined by \( f\left( S\right) = \left| {\delta \left( S\right) }\right| \) for all \( S \subseteq V \). An easy counting argument shows that, for any \( S, T \subseteq V \) the following h... | \[ \left| {\delta \left( S\right) }\right| + \left| {\delta \left( T\right) }\right| = \left| {\delta \left( {S \cap T}\right) }\right| + \left| {\delta \left( {S \cup T}\right) }\right| + 2\left| \left( {T \smallsetminus S : S \smallsetminus T}\right) \right| \] where \( \left( {T \smallsetminus S : S \smallsetminus T... | Yes |
Given rational matrices \( A, G \) and a rational vector \( b \), let \( P \mathrel{\text{:=}} \{ \left( {x, y}\right) : {Ax} + {Gy} \leq b\} \) and let \( S \mathrel{\text{:=}} \{ \left( {x, y}\right) \in P \) : \( x \) integral \( \} \) . 1. There exist rational matrices \( {A}^{\prime },{G}^{\prime } \) and a ration... | Proof. The theorem is obvious if \( S \) is empty, so we assume that \( S \) is nonempty. By Theorem 3.13, there exist \( {v}^{1},\ldots ,{v}^{t} \) and \( {r}^{1},\ldots ,{r}^{q} \) such that \( P = \operatorname{conv}\left( {{v}^{1},\ldots ,{v}^{t}}\right) + \operatorname{cone}\left( {{r}^{1},\ldots ,{r}^{q}}\right) ... | Yes |
Corollary 4.31. Let \( P \subseteq {\mathbb{R}}^{n + p} \) be a rational polyhedron and \( S \mathrel{\text{:=}} P \cap \) \( \left( {{\mathbb{Z}}^{n} \times {\mathbb{R}}^{p}}\right) \) . There exist finitely many rational polytopes \( {P}_{1},\ldots ,{P}_{k} \subseteq {\mathbb{R}}^{n + p} \) and vectors \( {r}^{1},\ld... | Proof. Following the proof of Theorem 4.30, by (4.26) \( S = {T}_{I} + {R}_{I} \), where \( {R}_{I} = \) intcone \( \left\{ {{r}^{1},\ldots ,{r}^{q}}\right\} \) and \( {T}_{I} \) is the union of finitely many rational polytopes. | Yes |
Proposition 4.33. The number of vertices of \( {P}^{\text{mix }} \) is the number of distinct values in the sequence \( {f}_{0},{f}_{1},\ldots ,{f}_{n} \) . Furthermore, the vertices of \( {P}^{\text{mix }} \) are among the \( n + 1 \) points \( {v}^{0},\ldots ,{v}^{n} \) defined by\n\n\[ \n{v}_{i}^{t} = \left\{ {\begi... | Proof. Let \( \bar{x} \) be a vertex of \( {P}^{\operatorname{mix}} \) . We first show \( {\bar{x}}_{0} < 1 \) . Suppose not. Then \( {P}^{\text{mix }} \) contains both points \( \bar{x} + {r}^{0} \) and \( \bar{x} - {r}^{0} \) . Since \( \bar{x} = \frac{1}{2}\left( {\left( {\bar{x} + {r}^{0}}\right) + \left( {\bar{x} ... | Yes |
Lemma 4.35. Given \( A \in {\mathbb{Q}}^{m \times n}, G \in {\mathbb{Q}}^{m \times p} \), and \( b \in {\mathbb{Q}}^{m} \), let \( S \mathrel{\text{:=}} \{ \left( {x, y}\right) \in \) \( {\mathbb{Z}}^{n} \times {\mathbb{R}}^{p} : {Ax} + {Gy} \leq b, x \geq 0, y \geq 0\} \) . If the encoding size of every coefficient of... | Proof. Let \( P \mathrel{\text{:=}} \{ \left( {x, y}\right) : {Ax} + {Gy} \leq b, x \geq 0, y \geq 0\} \) . Let \( {v}^{1},\ldots ,{v}^{t} \) be the vertices of \( P \) and \( {r}^{1},\ldots ,{r}^{q} \) be its extreme rays. By Theorem 3.38, \( {v}^{1},\ldots ,{v}^{t} \) and \( {r}^{1},\ldots ,{r}^{q} \) can be written ... | Yes |
Theorem 4.36. The MILP feasibility problem is in NP. | Proof. We need to show that, if an instance of the MILP feasibility problem has a \ | No |
Corollary 4.37. Given \( A \in {\mathbb{Q}}^{m \times n} \) and \( b \in {\mathbb{Q}}^{m} \), let \( L \) be the maximum encoding size of the coefficients of \( \left( {A, b}\right) \) . If the system \( {Ax} \leq b \) has an integral solution, then it has an integral solution \( \bar{x} \) whose encoding size is polyn... | Proof. The only difficulty arises from the fact that \( P \mathrel{\text{:=}} \left\{ {x \in {\mathbb{R}}^{n} : {Ax} \leq b}\right\} \) might not be pointed. However, \( {Ax} \leq b \) has an integral solution if and only if the system \( A{x}^{ + } - A{x}^{ - } \leq b,{x}^{ + },{x}^{ - } \geq 0 \) has an integral solu... | Yes |
Lemma 4.38. Given \( A \in {\mathbb{Q}}^{m \times n}, G \in {\mathbb{Q}}^{m \times p} \), and \( b \in {\mathbb{Q}}^{m} \), let \( S \mathrel{\text{:=}} \{ \left( {x, y}\right) \in \) \( {\mathbb{Z}}^{n} \times {\mathbb{R}}^{p} : {Ax} + {Gy} \leq b, x \geq 0, y \geq 0\} \) . If the encoding size of every coefficient of... | Proof. By Lemma 4.35 the encoding size of the vertices of \( \operatorname{conv}\left( S\right) \) is poly-nomially bounded by \( n + p \) and \( L \) . By Theorem 4.30, the recession cone of \( \operatorname{conv}\left( S\right) \) is \( \{ \left( {x, y}\right) : {Ax} + {Gy} \leq 0, x \geq 0, y \geq 0\} \), therefore ... | Yes |
Theorem 4.39 (Balas [24,26]). Given \( k \) polyhedra \( {P}_{i} \mathrel{\text{:=}} \left\{ {x \in {\mathbb{R}}^{n} : {A}_{i}x \leq }\right. \) \( \left. {b}^{i}\right\}, i = 1,\ldots k \), let \( {C}_{i} \mathrel{\text{:=}} \left\{ {x : {A}_{i}x \leq 0}\right\} \), and let \( {R}^{i} \subset {\mathbb{R}}^{n} \) be a ... | Proof. Assume first that \( P = \varnothing \) . This implies \( {P}_{i} = \varnothing \) for all \( i = 1,\ldots, k \) . Since the system describing \( Y \), includes \( {\delta }_{i} \geq 0, i = 1,\ldots, k \), and \( \mathop{\sum }\limits_{{i = 1}}^{k}{\delta }_{i} = 1 \) , then at least one of the variables \( {\de... | Yes |
Theorem 4.42 (Balas [24,26]). Let \( {P}_{i} \mathrel{\text{:=}} \left\{ {x \in {\mathbb{R}}^{n} : {A}_{i}x \leq {b}^{i}}\right\} \) be \( k \) polyhedra such that \( { \cup }_{i = 1}^{k}{P}_{i} \neq \varnothing \), and let \( Y \) be the polyhedron defined in Theorem 4.39. Let \( {C}_{i} \mathrel{\text{:=}} \left\{ {x... | Proof. For every \( i \in \{ 1,\ldots, k\} \) such that \( {P}_{i} \neq \varnothing \), let \( {V}^{i} \subset {\mathbb{R}}^{n} \) be a finite set such that \( {P}_{i} = \operatorname{conv}\left( {V}^{i}\right) + {C}_{i} \) . Let \( P \mathrel{\text{:=}} \operatorname{conv}\left( {\mathop{\bigcup }\limits_{{i : {P}_{i}... | Yes |
Corollary 4.44. If \( {P}_{1},\ldots ,{P}_{k} \) are nonempty polyhedra with identical recession cones, then \( \operatorname{conv}\left( {{ \cup }_{i = 1}^{k}{P}_{i}}\right) \) is a polyhedron. | Proof. We leave it as an exercise for the reader to check how the last part of the proof of Lemma 4.41 simplifies to show \( Q + C \subseteq \operatorname{conv}\left( {{ \cup }_{i = 1}^{k}{P}_{i}}\right) \) . | No |
A set \( S \subseteq {\mathbb{R}}^{n} \) is mixed integer linear representable if and only if there exist rational polytopes \( {P}_{1},\ldots ,{P}_{k} \subseteq {\mathbb{R}}^{n} \) and vectors \( {r}^{1},\ldots ,{r}^{t} \in \) \( {\mathbb{Z}}^{n} \) such that\n\n\[ S = \mathop{\bigcup }\limits_{{i = 1}}^{k}{P}_{i} + \... | Proof. We first prove the \ | No |
Corollary 4.48. Let \( S \) be a set of the form (4.34). For \( i = 1,\ldots, k \), let \( {A}_{i}x \leq {b}^{i} \) be a rational linear system describing \( {P}_{i} \) . Let \( Q \) be the polyhedron defined by the inequalities in (4.35). Then \( \operatorname{conv}\left( S\right) = {\operatorname{proj}}_{x}\left( Q\r... | Proof. If (4.34) holds, then clearly\n\n\[ S \subseteq \operatorname{conv}\left( {{P}_{1} \cup \cdots \cup {P}_{k}}\right) + \operatorname{cone}\left\{ {{r}^{1},\ldots ,{r}^{t}}\right\} \subseteq \operatorname{conv}\left( S\right) . \]\n\nBecause the set in the middle is a polyhedron and \( \operatorname{conv}\left( S\... | Yes |
Lemma 4.49. Let \( {Ax} \leq b \) and \( {Cx} \leq d \) be systems describing the same polytope \( P \) and let \( S,{S}^{\prime } \) be the corresponding slack matrices. Then the nonnegative ranks of \( S \) and \( {S}^{\prime } \) coincide. | In particular, adding redundant inequalities to a linear system \( {Ax} \leq b \) does not modify the nonnegative rank of the slack matrix. To prove Lemma 4.49 , we will need the following result on polytopes. | No |
Lemma 4.50. Let \( P \mathrel{\text{:=}} \left\{ {x \in {\mathbb{R}}^{n} : {Ax} \leq b}\right\} \) be a polytope of dimension at least one. An inequality \( {cx} \leq \delta \) is valid for \( P \) if and only if there exists \( u \geq 0 \) such that \( {uA} = c \) and \( {ub} = \delta \) . | Proof. Let \( {cx} \leq \delta \) be valid for \( P \) . By Theorem 3.22 there exists \( u \geq 0 \) such that \( {uA} = c \) and \( {ub} \leq \delta \) . Since \( P \) is a polytope and \( \dim \left( P\right) \geq 1 \), there exists \( r \in {\mathbb{R}}^{n} \) such that both \( {\delta }_{1} \mathrel{\text{:=}} \min... | Yes |
Lemma 4.52. Let \( S \) be an \( m \times n \) nonnegative matrix with at least one positive entry. The nonnegative rank of \( S \) is the smallest number \( t \) such that \( S \) is the sum of \( t \) nonnegative rank-1 matrices. | Proof. Let \( {\phi }^{1},\ldots ,{\phi }^{t} \in {\mathbb{R}}^{m} \) be nonnegative column vectors, and \( {\omega }^{1},\ldots ,{\omega }^{t} \in \) \( {\mathbb{R}}^{n} \) be nonnegative row vectors. Let \( F \in {\mathbb{R}}^{m \times t} \) be the matrix with columns \( {\phi }^{1},\ldots ,{\phi }^{t}, W \in {\mathb... | Yes |
Lemma 4.53. Let \( P \subset {\mathbb{R}}^{n} \) be a polytope. Every extended formulation of \( P \) has a number of constraints at least equal to the rectangle covering number of the slack matrix of any system of linear inequalities describing \( P \) . | Proof. Let \( S \in {\mathbb{R}}^{m \times n} \) be the slack matrix of a system describing \( P \), and let \( t \) be its nonnegative rank. The lemma holds when \( S = 0 \), so we assume \( S \neq 0 \) . By Lemma 4.52, \( S = \mathop{\sum }\limits_{{h = 1}}^{t}{T}^{h} \) where \( {T}^{1},\ldots ,{T}^{t} \) are \( m \... | Yes |
Lemma 4.55. For all \( n,{P}_{n}^{\text{corr }} \) and \( {P}_{n + 1}^{\text{cut }} \) are linearly isomorphic. | Proof. Let \( f : {\mathbb{R}}^{n \times n} \rightarrow {\mathbb{R}}^{{E}_{n + 1}} \) be the linear function that maps each \( x \in {\mathbb{R}}^{n \times n} \) to the element \( y \in {\mathbb{R}}^{{E}_{n + 1}} \) defined by\n\n\[ \n{y}_{ij} = \left\{ \begin{array}{ll} {x}_{ii} & \text{ if }1 \leq i \leq n, j = n + 1... | Yes |
For all column vectors \( a \in \{ 0,1{\} }^{n} \), the inequality\n\n\[ \left\langle {2\operatorname{Diag}\left( a\right) - a{a}^{T}, Y}\right\rangle \leq 1 \]\n\n(4.38)\n\nis valid for \( {P}_{n}^{\text{corr }} \) . Furthermore, the slack of each vertex \( Y = b{b}^{T} \) with \( b \in \{ 0,1{\} }^{n} \), is \( {\lef... | Proof. For all column vectors \( a, b \in \{ 0,1{\} }^{n} \), we have\n\n\[ 1 - \langle 2\text{Diag}(a) - a{a}^{T}, b{b}^{T}\rangle = 1 - 2{a}^{T}b + \text{tr}(a{a}^{T}b{b}^{T}) = 1 - 2{a}^{T}b + \text{tr}({b}^{T}a{a}^{T}b) = (1 - {a}^{T}b{)}^{2} \]\n\nwhere the first equality follows from the fact that \( b \in \{ 0,1... | Yes |
Theorem 4.57. Every extended formulation for \( {P}_{n}^{\text{cut }} \) has at least \( {1.5}^{n} \) constraints. | Proof. Let \( {Ay} \leq b \) be a system of inequalities such that \( {P}_{n}^{\text{corr }} = \left\{ {y \in {\mathbb{R}}^{n \times n}}\right. \) : \( {Ay} \leq b\} \) and such that all the inequalities (4.38) are included in \( {Ay} \leq b \) . Define \( S \) to be the slack matrix of \( {Ay} \leq b \) and let \( M \... | Yes |
Theorem 4.58. Let \( S \) be an \( m \times n \) nonnegative matrix and let \( t \) be its nonnegative rank. Then, for every \( m \times n \) matrix \( W \) , \[ t \geq \frac{\langle W, S\rangle }{\alpha \cdot \parallel S{\parallel }_{\infty }} \] where \( \alpha \mathrel{\text{:=}} \max \{ \langle W, R\rangle : R \) i... | Proof. Let \( {\alpha }^{ * } = \max \{ \langle W, R\rangle : R \) is a rank-1 matrix in \( {\left\lbrack 0,1\right\rbrack }^{m \times n}\} \) be the linear relaxation of \( \alpha \) . Since any rank-1 matrix \( R \in {\left\lbrack 0,1\right\rbrack }^{m \times n} \) can be written as \( R = x{y}^{T} \), where \( x \in... | Yes |
Proposition 5.2. Assume that the polyhedron \( P \) is nonempty.\n\n(i) Let \( \left( {\pi ,{\pi }_{0}}\right) \) be a split. If \( {\Pi }_{1},{\Pi }_{2} \neq \varnothing \) and \( {V}_{1},{V}_{2} \subseteq {\mathbb{R}}^{n} \) are finite sets such that \( {\Pi }_{1} = \operatorname{conv}\left( {V}_{1}\right) + \operato... | Proof. By Lemma 4.45, \( {P}^{\left( \pi ,{\pi }_{0}\right) } \) is a polyhedron, i.e., \( {P}^{\left( \pi ,{\pi }_{0}\right) } \mathrel{\text{:=}} \operatorname{conv}\left( {{\Pi }_{1} \cup }\right. \) \( \left. {\Pi }_{2}\right) = \overline{\operatorname{conv}}\left( {{\Pi }_{1} \cup {\Pi }_{2}}\right) \) . Therefore... | Yes |
Lemma 5.3. Given \( \bar{x} \in P \) such that \( {\pi }_{0} < \pi \bar{x} < {\pi }_{0} + 1,\bar{x} \) belongs to \( {P}^{\left( \pi ,{\pi }_{0}\right) } \) if and only if there exists \( \widetilde{x} \in {\Pi }_{2} \) such that\n\n\[ b - A\widetilde{x} \leq \frac{b - A\bar{x}}{\pi \bar{x} - {\pi }_{0}}. \] | Proof. For the \ | No |
Lemma 5.4. Let \( u \in {\mathbb{R}}^{m} \) such that \( u{A}_{I} \) is integral, \( u{A}_{C} = 0 \), and \( {ub} \notin \mathbb{Z} \) . Define \( \pi \mathrel{\text{:=}} {uA} \) and \( {\pi }_{0} \mathrel{\text{:=}} \lfloor {ub}\rfloor \) . The inequality (5.6) is valid for \( {P}^{\left( \pi ,{\pi }_{0}\right) } \) ,... | Proof. By definition, \( {\pi }_{I} \) is integral and \( {\pi }_{C} = 0 \), thus \( \left( {\pi ,{\pi }_{0}}\right) \) is a split. It suffices to show that (5.6) is valid for \( {P}^{\left( \pi ,{\pi }_{0}\right) } \) . We show that (5.6) is valid for \( {\Pi }_{1} \), the argument for \( {\Pi }_{2} \) being symmetric... | Yes |
Corollary 5.7. \( {P}^{\text{split }} = \mathop{\bigcap }\limits_{{B \in \mathcal{B}}}{P}_{B}^{\text{split }} \) . | Note that, given \( B \in \mathcal{B} \), the polyhedron \( {P}_{B} \mathrel{\text{:=}} \left\{ {x : {a}^{i}x \leq {b}_{i}, i \in B}\right\} \) has a unique minimal face, namely \( {F}_{B} \mathrel{\text{:=}} \left\{ {x : {a}^{i}x = {b}_{i}, i \in B}\right\} \) . In particular \( {P}_{B} \) is a translate of its recess... | No |
Lemma 5.9. Let \( \Delta \) be the largest among 1 and the absolute values of all sub-determinants of \( {A}_{C} \) . Let \( u \in {\mathbb{R}}^{m} \) be such that \( u{A}_{I} \in {\mathbb{Z}}^{I}, u{A}_{C} = 0,{ub} \notin \mathbb{Z} \) , and the inequality (5.6) is undominated. Then \( \left| {u}_{i}\right| \leq {m\De... | Proof. Let \( u \in {\mathbb{R}}^{m} \) be as in the statement, and let \( f \mathrel{\text{:=}} {ub} - \lfloor {ub}\rfloor \) . Consider the set of indices \( {M}^{ - } \mathrel{\text{:=}} \left\{ {i \in \left\lbrack m\right\rbrack : {u}_{i} < 0}\right\} \) and \( {M}^{ + } \mathrel{\text{:=}} \left\{ {i \in \left\lbr... | Yes |
Theorem 5.10 (Cook et al. [90]). Let \( P \subseteq {\mathbb{R}}^{n} \) be a rational polyhedron and let \( S \mathrel{\text{:=}} P \cap \left( {{\mathbb{Z}}^{I} \times {\mathbb{R}}^{C}}\right) \) . Then \( {P}^{\text{split }} \) is a rational polyhedron. | Proof. Let \( \left( {A, b}\right) \) be an integral matrix such that \( P = \left\{ {x \in {\mathbb{R}}^{n} : {Ax} \leq b}\right\} \) . By Theorem 5.5 and Lemma 5.9, \( {P}^{\text{split }} \) is the intersection of all polyhedra \( {P}^{\left( \pi ,{\pi }_{0}\right) } \) where \( \left( {\pi ,{\pi }_{0}\right) \) is a... | Yes |
Let \( S \mathrel{\text{:=}} \left\{ {\left( {x, y}\right) \in {\mathbb{Z}}_{ + }^{2} \times {\mathbb{R}}_{ + } : {x}_{1} \geq y,{x}_{2} \geq y,{x}_{1} + {x}_{2} + }\right. \) \( {2y} \leq 2\} \) . Starting from \( P \mathrel{\text{:=}} \left\{ {\left( {{x}_{1},{x}_{2}, y}\right) \in {\mathbb{R}}_{ + }^{3} : {x}_{1} \g... | To see this, note that \( P \) is a simplex with vertices \( O = \left( {0,0,0}\right) \) , \( A = \left( {2,0,0}\right), B = \left( {0,2,0}\right) \) and \( C = \left( {\frac{1}{2},\frac{1}{2},\frac{1}{2}}\right) \) (see Fig. 5.3). \( S \) is contained in the plane \( y = 0 \) . So \( \operatorname{conv}\left( S\right... | Yes |
Theorem 5.12. \( {P}^{\mathrm{{MIR}}} = {P}^{\text{split }} \) . | Proof. By construction, the mixed integer rounding inequality (5.16) is a split inequality relative to the split \( \left( {\pi ,\lfloor \beta \rfloor }\right) \), thus \( {P}^{\mathrm{{MIR}}} \supseteq {P}^{\text{split }} \) . To prove the converse, by Theorem 5.5 it suffices to show that any inequality of the form (5... | Yes |
Lemma 5.13. \( {P}^{Ch} \) is the set of all points in \( P \) satisfying the Chvátal inequalities \( {uAx} \leq \lfloor {ub}\rfloor \) for all \( u \) such that \( {uA} \in {\mathbb{Z}}^{n} \) and \( 0 \leq u \leq \mathbf{1} \) . | Proof. Given \( u \geq 0 \) such that \( {uA} \) is integral, let \( {u}^{1} \mathrel{\text{:=}} u - \lfloor u\rfloor \) and \( {u}^{2} = \lfloor u\rfloor \) . Note that \( {u}^{2} \geq 0 \), thus \( \left( {{u}^{2}A}\right) x \leq {u}^{2}b \) is valid for \( P \) . Since \( {u}^{2} \) is an integral vector and \( A, b... | Yes |
Theorem 5.14 (Chvátal [73]). \( {P}^{Ch} \) is a rational polyhedron. | Proof. Since \( \left\{ {{uA} \in {\mathbb{R}}^{n} : 0 \leq u < \mathbf{1}}\right\} \) is a bounded set, the set \( \{ {uA} \in \) \( \left. {{\mathbb{Z}}^{n} : 0 \leq u < \mathbf{1}}\right\} \) is finite. It follows from Lemma 5.13 that \( {P}^{Ch} \) is a polyhedron. | Yes |
Lemma 5.15. Let \( P \subseteq {\mathbb{R}}^{n} \) be a nonempty rational polyhedron such that \( \operatorname{aff}\left( P\right) \cap {\mathbb{Z}}^{n} \neq \varnothing \) . If \( {P}_{I} = \varnothing \), then \( \dim \left( {\operatorname{rec}\left( P\right) }\right) < \dim \left( P\right) \) . | Proof. Let \( d \mathrel{\text{:=}} \dim \left( P\right) = \dim \left( {\operatorname{aff}\left( P\right) }\right) \) and assume \( {P}_{I} = \varnothing \) . Suppose, by contradiction, that \( \dim \left( {\operatorname{rec}\left( P\right) }\right) = d \) . Then, since \( P \) is a rational polyhedron, there exist \( ... | Yes |
Lemma 5.16. Let \( P \subseteq {\mathbb{R}}^{n} \) be a nonempty rational polyhedron such that \( \operatorname{aff}\left( P\right) \cap {\mathbb{Z}}^{n} \neq \varnothing \) . Then \( {P}_{I} = \{ x : {Ax} \leq b\} \cap \operatorname{aff}\left( P\right) \) for some integral \( A \) and \( b \) such that, for every row ... | Proof. Assume first \( {P}_{I} \neq \varnothing \) . Then by Meyer’s theorem (Theorem 4.30) there exist an integral matrix \( A \) and an integral vector \( b \) such that \( {P}_{I} = \{ x : {Ax} \leq \) \( b\} \cap \operatorname{aff}\left( P\right) \) and no row of \( A \) is orthogonal to \( \operatorname{aff}\left(... | Yes |
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