Approximation Algorithms Based on Lp Relaxation 1 Linear Programs and Linear Integer Programs

There are two fundamental approximation algorithm design techniques based on linear programming: (a) LP-relaxation and rounding, and (b) the primal-dual method. In this lecture note, we will discuss the former. The idea of LP-relaxation and rounding is quite simple. We first formulate an optimization problem as an integer program (IP), which is like a linear program (LP) with integer variables. Then, we solve the LP for an optimal solution, say x∗. From x∗, we construct a feasible solution xA to the IP. This construction step is often called rounding. Rounding can be done deterministically or randomly with some probability distribution. In the latter approach is taken, we obtain the randomized rounding method. Let cost(xA) and cost(x∗) denote the objective values of xA and x∗, respectively. Let OPT(IP ) and OPT(LP ) denote the optimal values of the the IP and the LP, respectively. (Note that OPT(LP ) = cost(x∗).) Suppose we are working on a minimization problem, then the performance ratio of this algorithm can be obtained by observing that