Exact Algorithms for Maximum Independent Set

Years and Authors of Summarized Original Work

1977; Tarjan, Trojanowski

1985; Robson

1999; Beigel

2009; Fomin, Grandoni, Kratsch

Problem Definition

Let G = (V, E) be an n-node undirected, simple graph without loops. A set $I \subseteq V$ is called an independent set of G if the nodes of I are pairwise not adjacent. The maximum independent set (MIS) problem asks to determine the maximum cardinality α(G) of an independent set of G. MIS is one of the best studied NP-hard problems.

We will need the following notation. The (open) neighborhood of a vertex v is N(v) = { u ∈ V : uv ∈ E}, and its closed neighborhood is $N[v] = N(v) \cup \{ v\}$. The degree deg(v) of v is | N(v) |. For $W \subseteq V$, $G[W] = (W,E \cap \binom{W}{2})$ is the graph induced by W. We let $G - W = G[V - W]$.

Key Results

A very simple algorithm solves MIS (exactly) in $O^{{\ast}}(2^{n})$time: it is sufficient to enumerate all the subsets of nodes, check in polynomial time whether each subset is an...