Inapproximability of Flip-Cut, Shift-Cut and Other Problems from Optical Mapping

• Authors: L. Parida
• Venue: Inapproximability of Flip-Cut, Shift-Cut and Other Problems from Optical Mapping
• Year: 1997

Optical Mapping is an emerging technology for constructing ordered restriction maps of DNA molecules. The study of the complexity of the problems arising in Optical Mapping has generated considerable interest amongst computer science researchers. In this paper we examine the complexity of these problems. Optical Mapping leads to various computational problems such as the Binary Flip Cut (BFC) problem, the Weighted Flip Cut (WFC) problem the Exclusive Binary Flip Cut (EBFC) problem \cite{parida1, parida2}, the Binary Shift Cut (BSC) problem, the Binary Partition Cut (BPC) problem and others. The complexity and the hardness of the BFC problem, the WFC problem were not known. Using the technique of {\em gap-preserving} reduction of the max-cut problem, we show that BFC and WFC problems are MAX SNP-hard and achieving an approximation ratio $1-\Upsilon/7$ for these problems is NP-hard, where $\Upsilon$ denotes the upper bound on the polynomial time approximation factor of the well-known max cut problem. A slight variation of BFC, BFC$_{\max K}$, had been shown to be NP-hard; we improve the result to show that BFC$_{\max K}$ is MAX SNP-hard and achieving an approximation ratio $(1-\Upsilon/7)\frac{p_{max}}{p_{min}}$ for BFC$_{\max K}$ is NP-hard, where $p_{\min}$ and $p_{\max}$ are the minimum and maximum of the digestion rates in the given problem. The EBFC problem was shown to be NP-Complete; we improve this result to show that EBFC is MAX SNP-hard and achieving an approximation ratio $1-\Upsilon/7$ for EBFC is NP-hard. However, a dense instance of the EBFC problem does have a PTAS. The Binary Partition Cut (modeling spurious molecules) problem has been shown to be NP-Complete: we show, in this paper, that a (reasonable) unrestrained version of it has an efficient polynomial time algorithm. A variation of the Binary Shift Cut (modeling missing fragments) BSC$_{\max K}$, had been shown to be NP-hard \cite{Tom}; we show both the versions of this problem to be MAX SNP-hard and achieving an approximation ratio $1-\Upsilon/6$ for BSC and a ratio $(1-\Upsilon/6)\frac{p_{max}}{p_{min}}$ for BSC$_{\max K}$ is NP-hard. In addition, we show that $d$-wise Match ($d$M) problem is MAX SNP-hard and achieving an approximation ratio $1-\Upsilon$ is NP-hard.