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Implications for Linear and Quadratic Constraints in Graph Coloring Problems

This text explains how to represent linear and quadratic constraints in graph coloring problems using a logical PL framework. It tackles the assignment of colors to vertices such that adjacent vertices have different colors, minimizing the total number of colors. The approach involves using variables Xv,c and Yc, ensuring that vertices sharing an edge are assigned distinct colors. The method also presents a set packing problem where vertices must not share an edge, maximizing the number of selected vertices while meeting constraints.

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Implications for Linear and Quadratic Constraints in Graph Coloring Problems

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  1. If Pij & Pjk then Pik Not(Pij & Pjk) or Pik (see implication in PL) NotPij or NotPjk or Pik 1-xij + 1-xjk + yik >= 1 xij + xjk + (1-yik) <=2 This works with objective function Min sum of yik’s So object function forces yik to be 0 Unless both xij,xjk are 1 then constraint forces yik to be 1 So a quadratic term is represented by yik and problem remains linear, i.e. ILP

  2. Graph coloring • Xv,c and Yc • Xv,c + Xu,c <= 1 for all (u,v) edges, c in other words vertices u,v must be assigned different colors if they share an edge. • We need Yc to represent the objective function which is to minimize total number of colors. If Xu,c=1 then Yc =1 • Using PL, Puc=>Pc, 1-Xuc + Yc >=1 • Yc >= Xuc for all u nodes, c

  3. GUB constraint … • Yc <= sum u Xuc for all colors • This provides an upper bound on Yc • Although objective function of sum over c Yc will force Yc’s to zero anyways if the color is not being used.

  4. Set packing problem • Let Xi represent a vertex i of the graph • Select max number of vertices • No two vertices can share an edge • Ax <= 1 • Represents Xi + Xj <=1 for all (i,j) edges

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