Minimizing the Minimizers via Alphabet Reordering

TL;DR

The paper studies the problem of minimizing the number of minimizers $|\mathcal{M}_{w,k}(S)|$ by choosing an optimal total order on the alphabet $\Sigma$. It proves NP-completeness of Minimizing the Minimizers (Decision) for all $w\ge 2$, $k\ge 1$ via two main reductions from Feedback Arc Set: one handling $w\ge 3$ (with three gadget cases) and another handling $w=2$ using Eulerian graphs. The reductions encode graph structure into string gadgets so that the optimal alphabet order corresponds to a minimum feedback arc set, thereby aligning $|\mathcal{M}_{w,k}(S)|$ with the graph's feedback arc set size. The paper also notes that if $|\Sigma|$ is constant, the problem becomes polynomial-time solvable, and discusses the relationship to orderings on substrings $\Sigma^k$. Overall, it establishes a complete hardness landscape for alphabet reordering in minimizer sampling, explaining why exact algorithms are unlikely and justifying reliance on heuristics in practice.

Abstract

Minimizers sampling is one of the most widely-used mechanisms for sampling strings [Roberts et al., Bioinformatics 2004]. Let $S=S[1]\ldots S[n]$ be a string over a totally ordered alphabet $Σ$. Further let $w\geq 2$ and $k\geq 1$ be two integers. The minimizer of $S[i\mathinner{.\,.} i+w+k-2]$ is the smallest position in $[i,i+w-1]$ where the lexicographically smallest length-$k$ substring of $S[i\mathinner{.\,.} i+w+k-2]$ starts. The set of minimizers over all $i\in[1,n-w-k+2]$ is the set $\mathcal{M}_{w,k}(S)$ of the minimizers of $S$. We consider the following basic problem: Given $S$, $w$, and $k$, can we efficiently compute a total order on $Σ$ that minimizes $|\mathcal{M}_{w,k}(S)|$? We show that this is unlikely by proving that the problem is NP-hard for any $w\geq 2$ and $k\geq 1$. Our result provides theoretical justification as to why there exist no exact algorithms for minimizing the minimizers samples, while there exists a plethora of heuristics for the same purpose.

Figures (5)

• Figure 1: The $\min$ and $\max$ values of the size of the minimizers sample, among some of the possible orderings of $[1,|\Sigma|^k]$, on two real datasets using a range of $(w,k)$ parameter values.
• Figure 2: Illustration of the structure of string $S$, with the different gadgets for different arcs in $G$. The highlighted blocks are the ones for which the minimizers are counted in $M_{\texttt{a}<\texttt{b}}$ and $M_{\texttt{b}<\texttt{a}}$.
• Figure 3: Illustration of $3$ copies of $T_{\texttt{a}\texttt{b}}$ in $S$ for $w = 7$ and $k = 4$, along with its respective minimizers when $\texttt{a} < \texttt{b}$ (top) and when $\texttt{b} < \texttt{a}$ (bottom). It can be seen that $M_{\texttt{a}<\texttt{b}} = 1$ and $M_{\texttt{b}<\texttt{a}} = 3$.
• Figure 4: $T_{\texttt{a}\texttt{b}}$ for $w = 3$ and $k = 3$, with its respective minimizers. The last $\texttt{b}$ is a minimizer even when $\texttt{a} < \texttt{b}$, because $w = 3$. In this situation, $M_{\texttt{a}<\texttt{b}} = 4$ and $M_{\texttt{b}<\texttt{a}} = 5$.
• Figure 7: $T_{\texttt{a}\texttt{b}}$ for $w = 4$ and $k = 4$, showing its minimizers for $\texttt{a} < \texttt{b}$ (top) and $\texttt{b} < \texttt{a}$ (bottom). In this situation, $M_{\texttt{a}<\texttt{b}} = 4$ and $M_{\texttt{b}<\texttt{a}} = 5$.

Theorems & Definitions (14)

• Example 1
• Example 2
• Theorem 3
• Theorem 4
• Corollary 5
• Lemma 6
• Lemma 7
• Lemma 8
• Lemma 9
• Lemma 10