Fig. 2

The parsimonious clone integration (pci) problem. a Given clones \(\Pi _1\) and \(\Pi _2\) and corresponding proportions \(U_1\) and \(U_2\), we seek clones \(\Pi \subseteq \Pi _1 \times \Pi _2\) and corresponding proportions U consistent with \(U_1\) and \(U_2\). b There always exists a consistent proportion matrix \(U'\) for the trivial solution \(\Pi ' = \Pi _1 \times \Pi _2\), which can be identified by solving a maximum flow problem. c We seek the solution \(\Pi\) with minimum number \(|\Pi |\) of clones. Here, \(|\Pi |=4\), which is smaller than ground truth (see panel (a)). The corresponding matrix U follows from solving the illustrated maximum flow problem. However, incorporating tree constraints, as in the pcti problem, will lead to ground truth (Fig. 1)