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6.36. Let G = (V,E) be a graph with no isolated nodes and consider the polytope P defined by the system of inequalities xu xv ≤1, for every edge uvЄ E (6.23) บ x, > 0, for every node v Є V. Let be a 0-1 vector in P. Then the set of nodes vЄ V such that x,, = 1 is a stable set of G, that is, no two nodes in the set are adjacent in G. The convex hull of the incidence vectors of the stable sets of G is called the stable set polytope of G, and denoted by S(G). Clearly, S(G) = P1. For each of the following classes of valid inequalities for S(G), give a cutting- plane proof starting from the system (6.23). (i) Let C CV be the node set of an odd circuit in G. Then the inequality x(C) ≤ (|C| − 1)/2 is valid for S(G). - SEPARATION AND OPTIMIZATION 237 (ii) The graph illustrated in Figure 6.13 is called a 5-wheel. If W is the node set of a 5-wheel in G and r is the center node, then 2x, +x(W \ {r}) ≤ 2 is valid for S(G). (iii) A set of nodes K CV is called a clique of G if each pair of nodes in K are adjacent in G. For any clique K, the inequality (K) ≤ 1 is valid for S(G).

6.36. Let G = (V,E) be a graph with no isolated nodes and consider the polytope P defined by the system of inequalities xu xv ≤1, for every edge uvЄ E (6.23) บ x, > 0, for every node v Є V. Let be a 0-1 vector in P. Then the set of nodes vЄ V such that x,, = 1 is a stable set of G, that is, no two nodes in the set are adjacent in G. The convex hull of the incidence vectors of the stable sets of G is called the stable set polytope of G, and denoted by S(G). Clearly, S(G) = P1. For each of the following classes of valid inequalities for S(G), give a cutting- plane proof starting from the system (6.23). (i) Let C CV be the node set of an odd circuit in G. Then the inequality x(C) ≤ (|C| − 1)/2 is valid for S(G). - SEPARATION AND OPTIMIZATION 237 (ii) The graph illustrated in Figure 6.13 is called a 5-wheel. If W is the node set of a 5-wheel in G and r is the center node, then 2x, +x(W \ {r}) ≤ 2 is valid for S(G). (iii) A set of nodes K CV is called a clique of G if each pair of nodes in K are adjacent in G. For any clique K, the inequality (K) ≤ 1 is valid for S(G).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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6.36. Let G = (V,E) be a graph with no isolated nodes and consider the polytope P defined by the system of inequalities xu xv ≤1, for every edge uvЄ E (6.23) บ x, > 0, for every node v Є V. Let be a 0-1 vector in P. Then the set of nodes vЄ V such that x,, = 1 is a stable set of G, that is, no two nodes in the set are adjacent in G. The convex hull of the incidence vectors of the stable sets of G is called the stable set polytope of G, and denoted by S(G). Clearly, S(G) = P1. For each of the following classes of valid inequalities for S(G), give a cutting- plane proof starting from the system (6.23). (i) Let C CV be the node set of an odd circuit in G. Then the inequality x(C) ≤ (|C| − 1)/2 is valid for S(G). - SEPARATION AND OPTIMIZATION 237 (ii) The graph illustrated in Figure 6.13 is called a 5-wheel. If W is the node set of a 5-wheel in G and r is the center node, then 2x, +x(W \ {r}) ≤ 2 is valid for S(G). (iii) A set of nodes K CV is called a clique of G if each pair of nodes in K are adjacent in G. For any clique K, the inequality (K) ≤ 1 is valid for S(G).
Transcribed Image Text:6.36. Let G = (V,E) be a graph with no isolated nodes and consider the polytope P defined by the system of inequalities xu xv ≤1, for every edge uvЄ E (6.23) บ x, > 0, for every node v Є V. Let be a 0-1 vector in P. Then the set of nodes vЄ V such that x,, = 1 is a stable set of G, that is, no two nodes in the set are adjacent in G. The convex hull of the incidence vectors of the stable sets of G is called the stable set polytope of G, and denoted by S(G). Clearly, S(G) = P1. For each of the following classes of valid inequalities for S(G), give a cutting- plane proof starting from the system (6.23). (i) Let C CV be the node set of an odd circuit in G. Then the inequality x(C) ≤ (|C| − 1)/2 is valid for S(G). - SEPARATION AND OPTIMIZATION 237 (ii) The graph illustrated in Figure 6.13 is called a 5-wheel. If W is the node set of a 5-wheel in G and r is the center node, then 2x, +x(W \ {r}) ≤ 2 is valid for S(G). (iii) A set of nodes K CV is called a clique of G if each pair of nodes in K are adjacent in G. For any clique K, the inequality (K) ≤ 1 is valid for S(G).
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