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In this paper, we examine two related problems of inferring the evolutionary history of n objects, either from present characters of the objects or from several partial estimates of their evolutionary history. The first problem is called the Phylogeny problem, and second is the Tree Compatibility problem. Both of these problems are central in algorithmic approaches to the study of evolution and in other problems of historical reconstruction. In this paper, we show that both of these problems can be solved by graph theoretic methods in linear time, which is time optimal, and which is a significant improvement over existing methods. PERFECT PHYLOGENYLet M be an n by m 0-1 matrix representing n objects in terms of m characters that describe the objects; cell (iJ) of M has a value of one if and only if object i has character j . A phylogenetic tree for M is a rooted tree T where each object is attached to exactly one leaf of T, where each of the m characters is associated with exactly one edge of the tree and where, for any leaf w of T, the characters associated with the edges along the unique path from the root to w exactly specify the character vector of the objects at leaf w. In the example in Figure 1. the first matrix M has a phylogenetic tree T, but the second matrix M' does not.The interpretation of a phylogenetic tree is that it gives an estimate of the evolutionary history (in terms of branching pattern, but not time) of the objects, based on the following biological assumptions:(1) The root of the tree represents an ancestral object that has none of the present m characters; that is, in the ancestral object, the state of each character is zero.(2) Each of the characters changes from the zero state to the one state exactly once, and never from the one state to the zero state.The key feature of a phylogenetic tree (without which there would be no interesting problem) is that each character is associated with exactly one edge

In an instance of size n of the stable marriage problem, each of n men and n women ranks the members of the opposite sex in order of preference. A stable matching is a complete matching of men and women such that no man and woman who are not partners both prefer each other to their actual partners under the matching. It is well known [2] that at least one stable matching exists for every stable marriage instance. However, the classical Gale-Shapley algorithm produces a marriage that greatly favors the men at the expense of the women, or vice versa. The problem arises of finding a stable matching that is optimal under some more equitable or egalitarian criterion of optimality. This problem was posed by Knuth [6] and has remained unsolved for some time. Here, the objective of maximizing the average (or, equivalently, the total) “satisfaction” of all people is used. This objective is achieved when a person's satisfaction is measured by the position of his/her partner in his/her preference list. By exploiting the structure of the set of all stable matchings, and using graph-theoretic methods, an O ( n 4 ) algorithm for this problem is derived.

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A phylogenetic network is a generalization of a phylogenetic tree, allowing structural properties that are not tree-like. In a seminal paper, Wang et al.(1) studied the problem of constructing a phylogenetic network, allowing recombination between sequences, with the constraint that the resulting cycles must be disjoint. We call such a phylogenetic network a "galled-tree". They gave a polynomial-time algorithm that was intended to determine whether or not a set of sequences could be generated on galled-tree. Unfortunately, the algorithm by Wang et al.(1) is incomplete and does not constitute a necessary test for the existence of a galled-tree for the data. In this paper, we completely solve the problem. Moreover, we prove that if there is a galled-tree, then the one produced by our algorithm minimizes the number of recombinations over all phylogenetic networks for the data, even allowing multiple-crossover recombinations. We also prove that when there is a galled-tree for the data, the galled-tree minimizing the number of recombinations is "essentially unique". We also note two additional results: first, any set of sequences that can be derived on a galled tree can be derived on a true tree (without recombination cycles), where at most one back mutation per site is allowed; second, the site compatibility problem (which is NP-hard in general) can be solved in polynomial time for any set of sequences that can be derived on a galled tree. Perhaps more important than the specific results about galled-trees, we introduce an approach that can be used to study recombination in general phylogenetic networks. This paper greatly extends the conference version that appears in an earlier work.(8) PowerPoint slides of the conference talk can be found at our website.(7).