Radiation hybrid map construction problem parameterized

Abstract

In this paper, we study the Radiation hybrid map construction ( $$\mathsf{{RHMC} }$$ ) problem which is about reconstructing a genome from a set of gene clusters. The problem is known to be $$\mathsf{{NP} }$$ -complete even when all gene clusters are of size two and the corresponding problem ( $$\mathsf{{RHMC}_2 }$$ ) admits efficient constant-factor approximation algorithms. In this paper, for the first time, we consider the more general case when the gene clusters can have size either two or three ( $$\mathsf{{RHMC}_3 }$$ ). Let $${p\text{- }\mathsf {RHMC} }$$ be a parameterized version of $$\mathsf{{RHMC} }$$ where the parameter is the size of solution. We present a linear kernel for $${p\text{- }\mathsf {RHMC}_3 }$$ of size $$22k$$ that when combined with a bounded search-tree algorithm, gives an FPT algorithm running in $$O(6^kk+n)$$ time. For $${p\text{- }\mathsf {RHMC}_3 }$$ we present a bounded search tree algorithm which runs in $$O^*(2.45^k)$$ time, greatly improving the previous bound using weak kernels.