求解多棵有界度树上的最大一致性SubTree和最大相容树问题
Solving the Maximum Agreement SubTree and the Maximum Compatible Tree problems on many bounded degree trees
摘要：
给定一组具有相同叶集的叶标记树，著名的“最大一致性子树”问题（MAST）包括找到一个子树同胚地包含在所有输入树中，并具有最大数量的叶子。它的变体称为“最大兼容树”（MCT），不太严格，因为它允许输入树被细化。这两个问题在计算生物学中特别有趣，在计算生物学中，遇到的树通常度数很小。在本文中，我们研究的参数化复杂性的MAST和MCT的最大程度，表示为D，输入树。已知MAST对有界D是多项式。作为对应，我们证明了该问题对于参数D是W[1]-困难的.此外，依赖于参数化复杂度的最新进展，我们得到了一个严格的下限：虽然MAST可以在O（N^{O（D）}）时间内解决，其中N表示输入长度，但我们证明了O（N^{O（D）}）界是不可实现的，除非SE中包含SNP。我们还证明了MCT对于D是W[1]-困难的，MCT不能在O（N^{o（2^{D/2}）}）时间内求解，SNP包含在SE中.
英文摘要：
Given a set of leaf-labeled trees with identical leaf sets, the well-known "Maximum Agreement SubTree" problem (MAST) consists of finding a subtree homeomorphically included in all input trees and with the largest number of leaves. Its variant called "Maximum Compatible Tree" (MCT) is less stringent, as it allows the input trees to be refined. Both problems are of particular interest in computational biology, where trees encountered have often small degrees. In this paper, we study the parameterized complexity of MAST and MCT with respect to the maximum degree, denoted by D, of the input trees. It is known that MAST is polynomial for bounded D. As a counterpart, we show that the problem is W[1]-hard with respect to parameter D. Moreover, relying on recent advances in parameterized complexity we obtain a tight lower bound: while MAST can be solved in O(N^{O(D)}) time where N denotes the input length, we show that an O(N^{o(D)}) bound is not achievable, unless SNP is contained in SE. We also show that MCT is W[1]-hard with respect to D, and that MCT cannot be solved in O(N^{o(2^{D/2})}) time, SNP is contained in SE.
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