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Maximum agreement and compatible supertrees
 Proceedings of the 15th Combinatorial Pattern Matching Symposium (CPM’O4), volume 3109 of LNCS
, 2004
"... Given a set of leaflabelled trees with identical leaf sets, the MAST problem, respectively MCT problem, consists of finding a largest subset of leaves such that all input trees restricted to these leaves are isomorphic, respectively compatible. In this paper, we propose extensions of these problem ..."
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Given a set of leaflabelled trees with identical leaf sets, the MAST problem, respectively MCT problem, consists of finding a largest subset of leaves such that all input trees restricted to these leaves are isomorphic, respectively compatible. In this paper, we propose extensions of these problems to the context of supertree inference, where input trees have nonidentical leaf sets. This situation is of particular interest in phylogenetics. The resulting problems are called SMAST and SMCT. A sufficient condition is given that identifies cases where these problems can be solved by resorting to MAST and MCT as subproblems. This condition is met, for instance, when only two input trees are considered. Then we give algorithms for SMAST and SMCT that benefit from the link with the subtree problems. These algorithms run in time linear to the time needed to solve MAST, respectively MCT, on an instance of the same or smaller size. It is shown that arbitrary instances of SMAST and SMCT can be turned in polynomial time into instances composed of trees with a bounded number of leaves. SMAST is shown to be W[2]hard when the considered parameter is the number of input leaves that have to be removed to obtain the agreement of the input trees. A simlar result holds for SMCT. Moreover, the corresponding optimization problems, that is the complements of SMAST and SMCT, can not be approximated in polynomial time within a constant factor, unless P = NP. These results also hold when the input trees have a bounded number of leaves. The presented results apply to both collections of rooted and unrooted trees. Preprint submitted to Elsevier Science 17 November 2006 1
LOCAL OPTIMIZATION FOR GLOBAL ALIGNMENT OF PROTEIN INTERACTION NETWORKS
"... We propose a novel algorithm, PISwap, for computing global pairwise alignments of protein interaction networks, based on a local optimization heuristic that has previously demonstrated its effectiveness for a variety of other NPhard problems, such as the Traveling Salesman Problem. Our algorithm be ..."
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Cited by 19 (2 self)
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We propose a novel algorithm, PISwap, for computing global pairwise alignments of protein interaction networks, based on a local optimization heuristic that has previously demonstrated its effectiveness for a variety of other NPhard problems, such as the Traveling Salesman Problem. Our algorithm begins with a sequencebased network alignment and then iteratively adjusts the alignment by incorporating network structure information. It has a worstcase pseudopolynomial runningtime bound and is very efficient in practice. It is shown to produce improved alignments in several wellstudied cases. In addition, the flexible nature of this algorithm makes it suitable for different applications of network alignments. Finally, this algorithm can yield interesting insights into the evolutionary history of the compared species.
An even faster and more unifying algorithm for comparing trees via unbalanced bipartite matchings
 Journal of Algorithms
"... A widely used method for determining the similarity of two labeled trees is to compute a maximum agreement subtree of the two trees. Previous work on this similarity measure is only concerned with the comparison of labeled trees of two special kinds, namely, uniformly labeled trees (i.e., trees with ..."
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A widely used method for determining the similarity of two labeled trees is to compute a maximum agreement subtree of the two trees. Previous work on this similarity measure is only concerned with the comparison of labeled trees of two special kinds, namely, uniformly labeled trees (i.e., trees with all their nodes labeled by the same symbol) and evolutionary trees (i.e., leaflabeled trees with distinct symbols for distinct leaves). This paper presents an algorithm for comparing trees that are labeled in an arbitrary manner. In addition to this generality, this algorithm is faster than the previous algorithms. Another contribution of this paper is on maximum weight bipartite matchings. We show how to speed up the best known matching algorithms when the input graphs are nodeunbalanced or weightunbalanced. Based on these enhancements, we obtain an efficient algorithm for a new matching problem called the hierarchical bipartite matching problem, which is at the core of our maximum agreement subtree algorithm. 1
On the approximation of computing evolutionary trees
 in Proceedings of the 11th International Computing and Combinatorics Conference (COCOON’05
, 2005
"... Abstract. Given a set of leaflabelled trees with identical leaf sets, the wellknown MAST problem consists of finding a subtree homeomorphically included in all input trees and with the largest number of leaves. MAST and its variant called MCT are of particular interest in computational biology. Th ..."
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Abstract. Given a set of leaflabelled trees with identical leaf sets, the wellknown MAST problem consists of finding a subtree homeomorphically included in all input trees and with the largest number of leaves. MAST and its variant called MCT are of particular interest in computational biology. This paper presents positive and negative results on the approximation of MAST, MCT and their complement versions, denoted CMAST and CMCT. For CMAST and CMCT on rooted trees we give 3approximation algorithms achieving significantly lower running times than those previously known. In particular, the algorithm for CMAST runs in linear time. The approximation threshold for CMAST, resp. CMCT, is shown to be the same whenever collections of rooted trees or of unrooted trees are considered. Moreover, hardness of approximation results are stated for CMAST, CMCT and MCT on small number of trees, and for MCT on unbounded number of trees.
Approximate parameterized matching
 In Proc. 12th European Symposium on Algorithms (ESA
, 2004
"... Abstract Two equal length strings s and s0, over alphabets \Sigma s and \Sigma s0, parameterize match if thereexists a bijection ss: \Sigma s! \Sigma s0, such that ss(s) = s0, where ss(s) is the renaming of each characterof s via ss. Parameterized matching is the problem of finding all parameterize ..."
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Abstract Two equal length strings s and s0, over alphabets \Sigma s and \Sigma s0, parameterize match if thereexists a bijection ss: \Sigma s! \Sigma s0, such that ss(s) = s0, where ss(s) is the renaming of each characterof s via ss. Parameterized matching is the problem of finding all parameterized matches of apattern string p in a text t and approximate parameterized matching is the problem of finding,at each location, a bijection ss that maximizes the number of characters that are mapped from p to the appropriate plength substring of t.Parameterized matching was introduced as a model for software duplication detection in software maintenance systems and also has applications in image processing and computationalbiology. For example, approximate parameterized matching models image searching with variable color maps in the presence of errors.We consider the problem for which an error threshold, k, is given and the goal is to find alllocations in t for which there exists a bijection ss which maps p into the appropriate plengthsubstring of t with at most k mismatched mappedelements.We show that (1) the approximate parameterized matching, when  p=t, is equivalent tothe maximum matching problem on graphs, implying that (2) maximum matching is reducible to the approximate parameterized matching with threshold k, up till an O(log t) factor (thiscan be achieved by reducing approximate parameterized matching to the problem by using a binary search on the k's). Given the best known maximum matching algorithms an O(m1.5),where m = p  = t, is implied for approximate parameterized matching. We show that (3) forthe k threshold problem we can do this in O(m + k1.5).Our main result (4) is an O(nk1.5 + mk log m) time algorithm where m = p  and n = t. 1 Introduction In the traditional pattern matching model [11, 19], one seeks exact occurrences of a given pattern pin a text t, i.e. text locations where every text symbol is equal to its corresponding pattern symbol.For two equal length strings
A scaling algorithm for maximum weight matching in bipartite graphs
 IN: PROCEEDINGS 23RD ACMSIAM SYMPOSIUM ON DISCRETE ALGORITHMS (SODA
"... Given a weighted bipartite graph, the maximum weight matching (MWM) problem is to find a set of vertexdisjoint edges with maximum weight. We present a new scaling algorithm that runs in O(m √ n log N) time, when the weights are integers within the range of [0, N]. The result improves the previous b ..."
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Given a weighted bipartite graph, the maximum weight matching (MWM) problem is to find a set of vertexdisjoint edges with maximum weight. We present a new scaling algorithm that runs in O(m √ n log N) time, when the weights are integers within the range of [0, N]. The result improves the previous bounds of O(Nm √ n) by Gabow and O(m √ n log (nN)) by Gabow and Tarjan over 20 years ago. Our improvement draws ideas from a not widely known result, the primal method by Balinski and Gomory.
Improved Parameterized Complexity of the Maximum Agreement Subtree and . . .
 IEEE/ACM TRANSACTIONS ON COMPUTATIONAL BIOLOGY AND BIOINFORMATICS
, 2006
"... Given a set of evolutionary trees on a same set of taxa, the maximum agreement subtree problem (MAST), respectively maximum compatible tree problem (MCT), consists of finding a largest subset of taxa such that all input trees restricted to these taxa are isomorphic, respectively compatible. These ..."
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Cited by 9 (4 self)
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Given a set of evolutionary trees on a same set of taxa, the maximum agreement subtree problem (MAST), respectively maximum compatible tree problem (MCT), consists of finding a largest subset of taxa such that all input trees restricted to these taxa are isomorphic, respectively compatible. These problems
Algorithmic Applications of BaurStrassen’s Theorem: Shortest Cycles, Diameter and Matchings
"... Abstract—Consider a directed or undirected graph with integral edge weights in [−W, W]. This paper introduces a general framework for solving problems on such graphs using matrix multiplication. The framework is based on the BaurStrassen Theorem and Strojohann’s determinant algorithm. For directed ..."
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Abstract—Consider a directed or undirected graph with integral edge weights in [−W, W]. This paper introduces a general framework for solving problems on such graphs using matrix multiplication. The framework is based on the BaurStrassen Theorem and Strojohann’s determinant algorithm. For directed and undirected graphs without negative cycles we obtain simple Õ(Wnω) running time algorithms for finding a shortest cycle, computing the diameter or radius, and detecting a negative weight cycle. For each of these problems we unify and extend the class of graphs for which Õ(Wnω) time algorithms are known. In particular no such algorithms were known for any of these problems in undirected graphs with (potentially) negative weights. We also present an Õ(Wnω) time algorithm for minimum weight perfect matching. This resolves an open problem posed by Sankowski in 2006, who presented such an algorithm for bipartite graphs. Our algorithm uses a novel combinatorial interpretation of the linear program dual for minimum perfect matching. We believe this framework will find applications for finding larger spectra of related problems. As an example we give a simple Õ(Wnω) time algorithm to find all the vertices that lie on cycles of length at most t, for given t. This improves an Õ(Wn ω t) time algorithm of Yuster. Keywordsshortest cycles; diameter; radius; minimum weight perfect matchings; matrix multiplication I.
Scaling algorithms for approximate and exact maximum weight matching
, 2011
"... The maximum cardinality and maximum weight matching problems can be solved in time Õ(m √ n), a bound that has resisted improvement despite decades of research. (Here m and n are the number of edges and vertices.) In this article we demonstrate that this “m √ n barrier ” is extremely fragile, in the ..."
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Cited by 9 (0 self)
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The maximum cardinality and maximum weight matching problems can be solved in time Õ(m √ n), a bound that has resisted improvement despite decades of research. (Here m and n are the number of edges and vertices.) In this article we demonstrate that this “m √ n barrier ” is extremely fragile, in the following sense. For any ɛ> 0, we give an algorithm that computes a (1 − ɛ)approximate maximum weight matching in O(mɛ −1 log ɛ −1) time, that is, optimal linear time for any fixed ɛ. Our algorithm is dramatically simpler than the best exact maximum weight matching algorithms on general graphs and should be appealing in all applications that can tolerate a negligible relative error. Our second contribution is a new exact maximum weight matching algorithm for integerweighted bipartite graphs that runs in time O(m √ n log N). This improves on the O(Nm √ n)time and O(m √ n log(nN))time algorithms known since the mid 1980s, for 1 ≪ log N ≪ log n. Here N is the maximum integer edge weight. 1
Incremental assignment problem
 Information Sciences
, 2007
"... In this paper we introduce the incremental assignment problem. In this problem, a new pair of vertices and their incident edges are added to a weighted bipartite graph whose maximum weighted matching is already known, and the maximum weighted matching of the extended graph is sought. We propose an O ..."
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In this paper we introduce the incremental assignment problem. In this problem, a new pair of vertices and their incident edges are added to a weighted bipartite graph whose maximum weighted matching is already known, and the maximum weighted matching of the extended graph is sought. We propose an O(V  2) algorithm for the problem.