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Fig. 2 | Algorithms for Molecular Biology

Fig. 2

From: On a greedy approach for genome scaffolding

Fig. 2

Unbounded ratio of Algorithm 1 in the general case. Let \((G^*,M^*,\omega )\) be a \(2 \times k\) grid where the perfect matching (bold edges) corresponds to the edges between the two rows. Let \((x_1,\dots ,x_k)\) and \((y_1,\dots ,y_k)\) be the vertices of the first and second row, respectively. We are looking for a solution of max scaffolding with \(\sigma _c =0\) and \(\sigma _p=1\). If the algorithm chooses first the edge \(x_1x_2\), then the only feasible solution is \(S = \{x_{\ell }x_{\ell } \mid \ell \mod 2 = 1 \} \cup \{ y_{\ell }y_{\ell +1} \mid \ell \mod 2 = 0 \}\) (dashed edges). Suppose that an optimal solution is \(S_{opt}=E(G^*)\setminus (M^*\cup S)\) (solid edges). If all edges of \(S_{opt}\) and \(x_1x_2\) are valued by one and all edges of \(S \setminus \{x_1x_2\}\) are valued by zero, then we have \((k-1)\cdot \omega (S) = \omega (S_{opt})\) which leads to an unbounded ratio

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