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資料結構與演算法

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國立臺灣大學 2025 · 考試 · 第 16 題

考科演算法

(5 points) Consider the following 0-1 integer linear program PP:

maximize2x1+0⋅x2+2x3+5x4subject to2x1+x2+3x3+4x4≤84x1+x2+2x3+3x4≤8x1+8x2+3x3+7x4≤8x1,x2,x3,x4∈{0,1}\begin{aligned} \text{maximize} \quad & 2x_1 + 0 \cdot x_2 + 2x_3 + 5x_4 \\ \text{subject to} \quad & 2x_1 + x_2 + 3x_3 + 4x_4 \le 8 \\ & 4x_1 + x_2 + 2x_3 + 3x_4 \le 8 \\ & x_1 + 8x_2 + 3x_3 + 7x_4 \le 8 \\ & x_1, x_2, x_3, x_4 \in \{0, 1\} \end{aligned}

Let α\alpha be the number of feasible solutions to PP. Let β\beta be the number of optimal solutions to PP. Let γ\gamma be the integrality gap. Let P′P' be the relaxed linear program of PP, let ν4\nu_4 be the four-dimensional volume of the convex polytope that consists of all feasible solutions to P′P'. Let ν3\nu_3 be the three-dimensional volume of the convex polyhedron that consists of all optimal solutions to P′P'. Select all correct statement(s) below.

  • (A)
    α=8\alpha = 8
  • (B)
    β≠1\beta \neq 1
  • (C)
    γ=1\gamma = 1
  • (D)
    ν4<1\nu_4 < 1
  • (E)
    ν3=0\nu_3 = 0

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