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Cutting Planes II

Cutting Planes II. The Knapsack Problem. Recall the knapsack problem: n items to be packed in a knapsack (can take multiple copies of the same item). The knapsack can hold up to W lb of items. Each item has weight w i lb and benefit b i . Goal: Pack the knapsack such that

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Cutting Planes II

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  1. Cutting Planes II

  2. The Knapsack Problem Recall the knapsack problem: • n items to be packed in a knapsack (can take multiple copies of the same item). • The knapsack can hold up to W lb of items. • Each item has weight wilb and benefit bi . • Goal: Pack the knapsack such that the total benefit is maximized.

  3. IP model for Knapsack problem • Define decision variables (i = 1, …, n): xi = number of copies of item i to take • Then the total benefit: the total weight: • Summarizing, the IP model is: max s.t. xi ≥ 0integer (i = 1, …, n)

  4. Cutting Planes for Knapsack Problem • Example: Knapsack size is W=50. • Size constraint for this example: 21x1+11x2+51x3+26x4+30x5 50. • IP optimum: (1, 0, 0, 1, 0) with value 87 . LP-relax. optimum: (0, 0, 50/51, 0, 0) with value 490.2 . LP solution is too far from IP solution. • How to cut the fractional optimal solution? Add the following constraint (cutting plane): x3 = 0 . Cuts off the old LP optimum: (0, 0, 50/51, 0, 0) with value 490.2 . The new LP-relax. optimum: (0, 0, 0, 50/26, 0) with value 96.2 .

  5. Cutting Planes for Knapsack Problem • More cutting planes? Add constraint x4 1 . Cuts off the old LP optimum: (0, 0, 0, 50/26, 0) with value 96.2 . The new LP-relax. optimum: (24/21, 0, 0, 1, 0) with value 92.3 . • Generally, for item i we can add the following cutting plane: xi W / wi . • E.g., x1  2 . Note, however, that this constraint doesn’t cut the LP-optimum: (24/21, 0, 0, 1, 0) . Can we add better (tighter) cutting planes? • Cutting planes can include more than one variable. E.g., the following constraint is a valid cutting plane: x1 + x4 2 . (Note that this constraint is tighter than x1  2 ). Cuts off the old LP optimum: (24/21, 0, 0, 1, 0) with value 92.3 . The new LP-relax. optimum: (1, 0, 0, 1, 1/10) with value 91.1 .

  6. Cutting Planes for Knapsack Problem • General form of cutting planes with more than one variable. Let k≥1 and S={ i | wi > W / k } . Then we have the following cutting plane: • Take k=3 in our example. Then S= { i | wi > W / 3 } = {1, 3, 4, 5} . Thus, we have the following cutting plane: x1 + x3 + x4 + x5 2 . (Note that this constraint is tighter than x1 + x4 2 ). Cuts off the old LP optimum: (1, 0, 0, 1, 1/10) with value 91.1 . The new LP-relax. optimum: (1, 3/11, 0, 1, 0) with value 90.3 .

  7. Pairing Problem • 2n students • n projects; each project needs two students • Value of pairing students i and j is value[i , j] • Goal: Pair up the students to work on the projects so that the total value is maximized. • Example: 4 students, 2 projects, value matrix: Possible solutions: (A, B), (C, D) with value 9 (A, C), (B, D) with value 17 (A, D), (B, C) with value 2 Optimal solution: (A, C), (B, D)

  8. IP Formulation for the Pairing Problem • Define the following variables. For any i1,…,2n and j1,…,2n (ij), • The goal is to maximize the total value: Maximize 0.5 * sum{i1,…,2n, j1,…,2n : ij}value[i,j]*x[i,j] • Need the following constraints. Symmetry constraints: x[i,j]= x[j,i] for each i1,…,2n, j1,…,2n (ij) Each student works with exactly one other student: sum{j1,…,2n : ij}x[i,j] = 1 for each i1,…,2n • Q: How good (tight) is this formulation?

  9. Solving the LP-relaxation • Consider the following example: There are multiple optimal IP solutions with value 21 The optimal solution to the LP-relaxation: x[A,B] = x[B,A] = x[A,C] = x[C,A] = x[B,C] = x[C,B] = .5 ; x[D,E] = x[E,D] = x[D,F] = x[F,D] = x[E,F] = x[F,E] = .5 ; all other variables have value 0 . Optimal LP value: 6*0.5*10 = 30 B D .5 .5 A .5 E .5 .5 .5 C F

  10. Cutting Planes for the Pairing Problem • Add the following cutting plane: sum{ i  S, j  S } x[i, j] ≥ 1 for S = {A, B, C} Cuts off the old LP optimum with value 30. The new LP-relaxation has multiple optimal solutions with value 21. • General form: For any set S  {1,…,2n} that has an odd number of elements sum{ i  S, j  S } x[i, j] ≥ 1 is a cutting plane. • Note that the number of this kind of cutting planes is exponential (order of 2n); so can’t add all of them in the IP formulation. Thus, this kind of cutting planes should be added only if necessary (to cut off a fractional solution of the current LP relaxation).

  11. Branch-and-cut algorithms • Integer programs are rarely solved based solely on cutting plane method. • More often cutting planes and branch-and-bound are combined to provide a powerful algorithmic approach for solving integer programs. • Cutting planes are added to the subproblems created in branch-and-bound to achieve tighter bounds and thus to accelerate the solution process. • This kind of methods are known as branch-and-cut algorithms.

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