A PTAS for Distinguishing (Sub)string Selection

Abstract

Consider two sets of strings, $\mathcal{B}$ (bad genes) and $\mathcal{G}$ (good genes), as well as two integers d _b and d _g (d _b ≤ d _g). A frequently occurring problem in computational biology (and other fields) is to find a (distinguishing) substring s of length L that distinguishes the bad strings from good strings, i.e., for each string s _i ∈ $\mathcal{B}$ there exists a length-L substring t _i of s _i with d(s, t _i) ≤ d _b (close to bad strings) and for every substring u _i of length L of every string g _i ∈ $\mathcal{G}$ , d(s, u _i) ≥ d _g (far from good strings). We present a polynomial time approximation scheme to settle the problem, i.e., for any constant ∈ τ 0, the algorithm finds a string s of length L such that for every s _i ∈ $\mathcal{B}$ , there is a length-L substring t _i of s0_i with d(t _i, s) ≤ (1 + ∈)d _b and for every substring u _i of length L of every g _i ∈ $\mathcal{G}$ , d(u _i, s) ≥ (1 - ∈)d _g, if a solution to the original pair (d _b ≤ d _g) exists.

Fully supported by a grant from the Natural Science Foundation of China and Research Grants Council of the HKSAR Joint Research Scheme [Project No: NCityU 102/01].

Fully supported by a grant from the Research Grants Council of the Hong Knog SAR, China [Project No: CityU 1130/99E].