findMIS is an heuristic algorithm for solving Maximum Independent Set problem (MIS). An independent set of a graph is a subset of vertices in which no two vertices are adjacent. Given a set of vertices, the maximum independent set problem calls for finding the independent set of maximum cardinality.
Algorithm run in O(n^2) time, where n is the number of vertices (worst case). Experimentally: time = 8.1e-007*n^2 + 2.2e-005*n + 0.00012 seconds, (see screenshot)
The algorithm has been independently developped but is similar to: Balaji, Swaminathan, Kannan, "A Simple Algorithm to Optimize Maximum Independent Set", Advanced Modeling and Optimization, Volume 12, Number 1, 2010 Notation: The neighborhood of v is defined by N(v) ={u in V such that u is adjacent to v} The DEGREE of a vertex v in V, denoted by deg(v) and is defined by the number of neighbors of v. The SUPPORT of a vertex v in V is defined by the sum of the degree of the vertices which are adjacent to v, i.e., support(v) = s(v) = sum( deg(u) ) where u are all vertices in N(v). INPUTS: "AD" is the adjacency matrix. It must be of logical values! "priority" is used to break the ties (parity) situations that occur when more than one maximum independent set can be selected. Consider for example the trivial case of two nodes connected by one edge: there are two possible maximum independent sets. By using priority you can decide which vertex has to selected first. Try for example: x=findMIS(logical([0 1; 1 0]),[1 2]) %Higher priority to node 1 and x=findMIS(logical([0 1; 1 0]),[2 1]) %Higher priority to node 2 OUTPUTS: x is an binary array where x(i)=1 if vertex i belongs to the Maximum independent set and x(i)=0 if belongs to the Minimum vertex cover. |