Fast Exponential Algorithms for Maximum r-Regular Induced Subgraph Problems

Given a graph G=(V, E) on n vertices, the MAXIMUM r-REGULAR INDUCED SUBGRAPH (M-r-RIS) problems ask for a maximum sized subset of vertices R V such that the induced subgraph on R, G[R], is r-regular. We give an O(cn ) time algorithm for these problems for any fixed constant r, where c is a positive constant strictly less than 2, solving a well known open problem. These algorithms are then generalized to solve counting and enumeration version of these problems in the same time. An interesting consequence of the enumeration algorithm is, that it shows that the number of maximal r-regular induced subgraphs for a fixed constant r on any graph on n vertices is upper bounded by o(2n ). We then give combinatorial lower bounds on the number of maximal r-regular induced subgraphs possible on a graph on n vertices and also give matching algorithmic upper bounds. We use the techniques and results obtained in the paper to obtain an improved exact algorithm for a special case of INDUCED SUBGRAPH I...

• Type: Conference
• Year: 2006
• Where: FSTTCS
• Authors: Sushmita Gupta, Venkatesh Raman, Saket Saurabh
• Keywords: Algorithms, Fixed Constant R, FSTTCS 2006, R-regular Induced Subgraphs, Software Engineering