Maximum Independent Set Problem

In the previous lecture we discussed the Knapsack problem. In this lecture we discuss other packing and independent set problems. A basic graph optimization problem with many applications is the maximum (weighted) independent set problem (MIS) in graphs. Definition 1 Given an undirected graph G = (V, E) a subset of nodes S ⊆ V is an independent set (stable set) iff there is no edge in E between any two nodes in S. A subset of nodes S is a clique if every pair of nodes in S have an edge between them in G. The MIS problem is the following: given a graph G = (V, E) find an independent set in G of maximum cardinality. In the weighted case, each node v ∈ V has an associated non-negative weight w(v) and the goal is to find a maximum weight independent set. This problem is NP-Hard and it is natural to ask for approximation algorithms. Unfortunately, as the famous theorem below shows, the problem is extremely hard to approximate. 1 n 1−-approximation for MIS for any fixed > 0 where n is the number of nodes in the given graph. Remark: The maximum clique problem is to find the maximum cardinality clique in a given graph. It is approximation-equivalent to the MIS problem; simple complement the graph. The theorem basically says the following: there are a class of graphs in which the maximum independent set size is either less than n δ or greater than n 1−δ and it is NP-Complete to decide whether a given graph falls into the former category or the latter. The lower bound result suggests that one should focus on special cases, and several interesting positive results are known. First, we consider a simple greedy algorithm for the unweighted problem. Theorem 2 Greedy outputs an independent set S such that |S| ≥ n/(∆ + 1) where ∆ is the maximum degree of any node in the graph.