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By Le Bellac M.

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MSMDd can be solved in O((d + 1)6k (6k + 1)d n2 ), O((d + 2 k−1 2 1)2Cg gk (2Cg gk + 1)d n2 ) and O((d + 1)2b(b−1) (2b(b − 1)k−1 + 1)d n2 ) time in planar graphs, graphs of genus g, and graphs of degree at most b, respectively. 24 O. Amini, I. Sau, and S. Saurabh vertices in Xt (apices ) Bp(t) Bt ^ Bt At As 1 ^ Bp(t) As 2 Bs Bs 1 ^ 2 Bs 1 ^ Bs 2 Fig. 2. Tree-decomposition of a minor free graph. e. the apices) are depicted by . Note that Bs1 and Bs2 could have non-empty intersection (in Bt ). 2 M -Minor Free Graphs In this section we give the results for the class of M -minor free graphs.

Theorem 1 (Chen et al. [7]). A binary matrix M admits a perfect phylogeny if and only if the corresponding character graph G does not contain an induced M-graph. With Theorem 1 the Minimum-Flip Problem is equivalent to the following graph-theoretical problem: Find a minimum set of edge modiﬁcations, that is, edge deletions and edge insertions, which transform the character graph of the input matrix into an M-free bipartite graph. Using this characterization, Chen et al. [6, 7] introduce a simple ﬁxed-parameter algorithm with running time O(6k mn) where k is the minimum number of ﬂips.

In: FOCS, pp. 273– 282 (2006) 13. : Deciding ﬁrst-order properties of locally tree-decomposable structures. J. ACM 48(6), 1184–1206 (2001) 14. : Local Tree-width, Excluded Minors and Approximation Algorithms. Combinatorica 23(4), 613–632 (2003) 15. : Ruling out PTAS for graph min-bisection, densest subgraph and bipartite clique. In: FOCS, pp. 136–145 (2004) 16. : Approximation algorithms for ﬁnding low-degree subgraphs. Networks 44(3), 203–215 (2004) 17. : Approximation Algorithms for Minimum-Cost Low-Degree Subgraphs, PhD Thesis (2003) 18.

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### A short introduction to quantum information and quantum computation: solutions of exercises by Le Bellac M.

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