Instructor : Chandra Chekuri Scribe : GuoJun
نویسنده
چکیده
Many discrete optimization problems are naturally modeled as an integer (linear) programming (ILP) problem. An ILP problem is of the form max cx Ax ≤ b x is an integer vector. (1) It is easy to show that ILP is NP-hard via a reduction from say SAT. The decision version of ILP is the following: Given rational matrix A and rational vector b, does Ax ≤ b have an integral solution x? Theorem 1 Decision version of ILP is in NP, and hence it is NP-Complete. The above theorem requires technical work to show that if there is an integer vector in Ax ≤ b then there is one whose size is polynomial in size(A, b). A special case of interest is when the number of variables, n, is a fixed constant but the number of constraints, m, is part of the input. The following theorem is known. Theorem 2 (Lenstra's Algorithm) For each fixed n, there is a polynomial time algorithm for ILP in n variables.
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