Question

0-1 Integer Programming is similar to linear programming except the variables can only be 0 and 1. Consider the 0-1 Integer Programming Decision Problem (0-1 IPD) as defined below:

Instance: A set X of 0-1 integer variables (x_i = 0 or x_i = 1), a set of inequalities over these variables, a function f(x) to maximize and integer K.

Question: Does there exist an assignment of values to X such that all inequalities are true and f(X) >= K?

An example of an instance of 0-1 IP-D: : x₁ + x₂ + x₃ < 3, x₁ - x₃ >= 1, f(x) = 4x₁ + x₃, K = 2. The answer is yes, given certificate {x₁ = 1, x₂ = 0, x₃ =0}

Prove that 0-1 IPD Problem is NP-Complete.

Answer 1

ANSWER:

   GIVEN THAT

-> 0-1 Integer Programming is similar to linear programming except the variables can only be 0 and 1. Consider the 0-1 Integer Programming Decision Problem (0-1 IPD)

Where,

x₁ + x₂ + x₃ < 3, x₁ - x₃ >= 1, f(x) = 4x₁ + x₃, K = 2.

Here, The answer is yes,

->The given certificate is {x₁ = 1, x₂ = 0, x₃ =0}

Here ,we have to prove that 0-1 IPD Problem is NP-Complete.

-> If Π is NP-complete and Π ≺ Π0 and Π0 ∈ NP, then Π0 is NP-complete.(Corollary)

• It is known that, SAT ≺ 0-1 Integer Programming....................................................................(1)
• By Cook-Levin Theorem, SAT is NP-complete.........................................................................(2)
• Also, 0-1 Integer Programming is in NP....................................................................................(3)

->Therefore, using above corollary,(1),(2) and (3), we get that 0-1 Integer Programming is NP-complete.