Genetic Design of Drugs Without Side-Effects

Consider two sets of strings, ${\cal B}$ (bad genes) and ${\cal G}$ (good genes), as well as two integers $d_b$ and $d_g$ ($d_b\leq d_g$). 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., such that for each string $s_i\in {\cal B}$ there exists a length-L substring t i of s i with $d(s, t_i)\leq d_b$ (close to bad strings), and for every substring u i of length L of every string $g_i\in {\cal G}$, $d(s, u_i)\geq d_g$ (far from good strings). We present a polynomial time approximation scheme to settle the problem; i.e., for any constant $\epsilon >0$, the algorithm finds a string s of length L such that for every $s_i\in {\cal B}$ there is a length-L substring t i of s i with $d(t_i, s)\leq (1+\epsilon) d_b$, and for every substring u i of length L of every $g_i\in {\cal G}$, $d(u_i, s)\geq (1-\epsilon) d_g$ if a solution to the original pair ($d_b\leq d_g$) exists. Since there is a polynomial number of such pairs $(d_b,d_g)$, we can exhaust all the possibilities in polynomial time to find a good approximation required by the corresponding application problems.

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