Parameterized Complexity of Asynchronous Border Minimization

Robert Ganian, Martin Kronegger, Andreas Pfandler, Alexandru Popa

Abstract: Microarrays are research tools used in gene discovery as well as disease and cancer diagnostics. Two prominent but challenging problems related to microarrays are the Border Minimization Problem (BMP) and the Border Minimization Problem with given placement (P-BMP). In this paper we investigate the parameterized complexity of natural variants of BMP and P-BMP, termed $$\mathrm{BMP}^e$$ and $$\hbox {P-BMP}^{e}$$ respectively, under several natural parameters. We show that $$\mathrm{BMP}^e$$ and $$\hbox {P-BMP}^{e}$$ are in FPT under the following two combinations of parameters: ( $$1$$ ) the size of the alphabet ( $$c$$ ), the maximum length of a sequence (string) in the  input ( $$\ell $$ ) and the number of rows of the microarray ( $$r$$ ); and, ( $$2$$ ) the size of the alphabet and the size of the border length ( $$o$$ ). Furthermore, $$\hbox {P-BMP}^{e}$$ is in FPT when parameterized by $$c$$ and $$\ell $$ . We complement our tractability results with corresponding hardness results.