CSCI - B 609 : A Theorist ’ s Toolkit , Fall 2016 Nov 8 Lecture 20 : LP Relaxation and Approximation Algorithms

When variables of constraints of an 0−1 Integer Linear Program (ILP) is replaced by a weaker constraint such that each variable belongs to the interval [0, 1] (i.e. decimal) instead of fixed values {0, 1} (integers); then this relaxation generates a new Linear Programming problem. For example, the constraint xi ∈ {0, 1} becomes 0 ≤ xi ≤ 1. This relaxation technique is useful to convert an NP-hard optimization problem (ILP) into a related problem that is solvable in polynomial time. Moreover, the solution to the relaxed linear program can be employed to acquire information about the optimal solution to the original problem. On the other hand, Approximation Algorithms are algorithms used to find approximate solutions to the optimization problems. Linear programming relaxation is an established technique for designing such approximation algorithms for the NP-hard optimization problems (ILP).