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 the total benefit is maximized.

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

Cutting Planes for Knapsack Problem Example: Knapsack size is W=50. Size constraint for this example: 21x 1 +11x 2 +51x 3 +26x 4 +30x 5  50. IP optimum: (1, 0, 0, 1, 0) with value 87. LP-relax. optimum: (0, 0, 50/51, 0, 0) with value LP solution is too far from IP solution. How to cut the fractional optimal solution? Add the following constraint (cutting plane): x 3 = 0. Cuts off the old LP optimum: (0, 0, 50/51, 0, 0) with value The new LP-relax. optimum: (0, 0, 0, 50/26, 0) with value item12345 size benefit

Cutting Planes for Knapsack Problem More cutting planes? Add constraint x 4  1. Cuts off the old LP optimum: (0, 0, 0, 50/26, 0) with value The new LP-relax. optimum: (24/21, 0, 0, 1, 0) with value Generally, for item i we can add the following cutting plane: x i   W / w i . E.g., x 1  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: x 1 + x 4  2. (Note that this constraint is tighter than x 1  2 ). Cuts off the old LP optimum: (24/21, 0, 0, 1, 0) with value The new LP-relax. optimum: (1, 0, 0, 1, 1/10) with value 91.1.

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

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) ABCD A-5101 B5-17 C 1-4 D174-

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?

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 ABCDEF A B C -111 D111- E111 - F111 - A B C D E F.5

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 2 n ); 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).

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.