Lattices in Computer Science Lecture 5 Integer Programming Lecturer : Oded Regev Scribe : Ishay Haviv

In this lecture we present another application of the LLL algorithm, namely, integer programming in fixed dimension. The Integer Programming problem (IP) is that of deciding whether there exists an integer solution to a given set of m rational inequalities on n variables. Equivalently, given a matrix A ∈ Q m×n and b ∈ Q m , decide if there is a z ∈ Z n such that Az ≤ b. Yet another equivalent formulation is: given a matrix A ∈ Q m×n decide whether the set Z n ∩ {x ∈ R n | Ax ≤ b} is non-empty. The integer programming problem is quite powerful, and many combinatorial problems can be formulated as instances of IP. In fact, it is 'too powerful' since it is NP-complete, as the following claim shows. REMARK 1 Without the requirement on an integer solution (i.e., if we allow z ∈ R n), the problem is known as Linear Programming, and has a polynomial time solution (such as the ellipsoid method). REMARK 2 There are many equivalent formulations of integer programming. We could, for instance, allow equalities in addition to inequalities. Moreover, we could ask to find an integer solution and not just decide if one exists. CLAIM 1 The integer programming problem is NP-complete. PROOF: IP is in NP because the integer solution can be used as a witness and can be verified in polynomial time. 1 We now prove that IP is NP-hard by reduction from SAT. A SAT instance is described by a set of Boolean variables and clauses. We reduce it to an Integer Programming instance with the same number of variables. In addition, for each variable v i we have the constraints 0 ≤ v i ≤ 1. For each clause we have a constraint that correspond to it; for example, for the clause v 1 ∨ v 3 ∨ v 7 in the SAT instance, we have the constraint v 1 + (1 − v 3) + v 7 ≥ 1. Clearly, this reduction can be done in polynomial time. Moreover, it is easy to verify that if the given SAT instance has a satisfying assignment then the corresponding IP instance has an integer solution and vice versa. 2 Although it is NP-complete, one might hope to obtain efficient algorithms for the case where the dimension (i.e., the number of variables) is fixed. For n = 1, …