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Approximation Algorithms for Asymmetric TSP by Decomposing Directed Regular Multigraphs
 Proc. 44th Annual Symposium on Foundations of Computer Science (FOCS
, 2003
"... A directed multigraph is said to be dregular if the indegree and outdegree of every vertex is exactly d. By Hall’s theorem one can represent such a multigraph as a combination of at most n2 cycle covers each taken with an appropriate multiplicity. We prove that if the dregular multigraph does not ..."
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Cited by 53 (1 self)
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A directed multigraph is said to be dregular if the indegree and outdegree of every vertex is exactly d. By Hall’s theorem one can represent such a multigraph as a combination of at most n2 cycle covers each taken with an appropriate multiplicity. We prove that if the dregular multigraph does not contain more than ⌊d/2 ⌋ copies of any 2cycle then we can find a similar decomposition into n2 pairs of cycle covers where each 2cycle occurs in at most one component of each pair. Our proof is constructive and gives a polynomial algorithm to find such a decomposition. Since our applications only need one such a pair of cycle covers whose weight is at least the average weight of all pairs, we also give an alternative, simpler algorithm to extract a single such pair. This combinatorial theorem then comes handy in rounding a fractional solution of an LP relaxation of the maximum Traveling Salesman Problem (TSP) problem. The first stage of the rounding procedure obtains 2cycle covers that do not share a 2cycle with weight at least twice the weight of the optimal solution. Then we show how to extract a tour from the 2 cycle covers, whose weight is at least 2/3 of the weight of the longest tour. This improves upon the previous
Rotation of Periodic Strings and Short Superstrings
, 1996
"... This paper presents two simple approximation algorithms for the shortest superstring problem, with approximation ratios 2 2 3 ( 2:67) and 2 25 42 ( 2:596), improving the best previously published 2 3 4 approximation. The framework of our improved algorithms is similar to that of previous a ..."
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Cited by 26 (0 self)
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This paper presents two simple approximation algorithms for the shortest superstring problem, with approximation ratios 2 2 3 ( 2:67) and 2 25 42 ( 2:596), improving the best previously published 2 3 4 approximation. The framework of our improved algorithms is similar to that of previous algorithms in the sense that they construct a superstring by computing some optimal cycle covers on the distance graph of the given strings, and then break and merge the cycles to finally obtain a Hamiltonian path, but we make use of new bounds on the overlap between two strings. We prove that for each periodic semiinfinite string ff = a1a2 \Delta \Delta \Delta of period q, there exists an integer k, such that for any (finite) string s of period p which is inequivalent to ff, the overlap between s and the rotation ff[k] = ak ak+1 \Delta \Delta \Delta is at most p+ 1 2 q. Moreover, if p q, then the overlap between s and ff[k] is not larger than 2 3 (p+q). In the previous shortes...
Greedy Algorithms For The Shortest Common Superstring That Are Asymptotically Optimal
, 1997
"... There has recently been a resurgence of interest in the shortest common superstring problem due to its important applications in molecular biology (e.g., recombination of DNA) and data compression. The problem is NPhard, but it has been known for some time that greedy algorithms work well for this ..."
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Cited by 5 (4 self)
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There has recently been a resurgence of interest in the shortest common superstring problem due to its important applications in molecular biology (e.g., recombination of DNA) and data compression. The problem is NPhard, but it has been known for some time that greedy algorithms work well for this problem. More precisely, it was proved in a recent sequence of papers that in the worst case a greedy algorithm produces a superstring that is at most fi times (2 fi 4) worse than optimal. We analyze the problem in a probabilistic framework, and consider the optimal total overlap O opt n and the overlap O gr n produced by various greedy algorithms. These turn out to be asymptotically equivalent. We show that with high probability lim n!1 O opt n n log n = lim n!1 O gr n n log n = 1 H where n is the number of original strings, and H is the entropy of the underlying alphabet. Our results hold under a condition that the lengths of all strings are not too short.
Algorithms for Three Versions of the Shortest Common Superstring Problem
"... Abstract. The input to the Shortest Common Superstring (SCS) problem is a set S of k words of total length n. In the classical version the output is an explicit word SCS(S) in which each s ∈ S occurs at least once. In our paper we consider two versions with multiple occurrences, in which the input i ..."
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Cited by 1 (0 self)
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Abstract. The input to the Shortest Common Superstring (SCS) problem is a set S of k words of total length n. In the classical version the output is an explicit word SCS(S) in which each s ∈ S occurs at least once. In our paper we consider two versions with multiple occurrences, in which the input includes additional numbers (multiplicities), given in binary. Our output is the word SCS(S) given implicitly in a compact form, since its real size could be exponential. We also consider a case when all input words are of length two, where our main algorithmic tool is a compact representation of Eulerian cycles in multigraphs. Due to exponential multiplicities of edges such cycles can be exponential and the compact representation is needed. Other tools used in our paper are a polynomial case of integer linear programming and a minplus product of matrices. 1
be a set of strings over some alphabet \Sigma. A
"... ily a good approximation for the maximum overlap in the superstring, and vice versa. The first constantapproximation algorithm for the length of the shortest superstring was given by Blum et al. [4], who discovered a 3approximation algorithm and proved that the "Greedy" algorithm by Tarhio and Uk ..."
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ily a good approximation for the maximum overlap in the superstring, and vice versa. The first constantapproximation algorithm for the length of the shortest superstring was given by Blum et al. [4], who discovered a 3approximation algorithm and proved that the "Greedy" algorithm by Tarhio and Ukkonen [9] achieves 4approximation. Their algorithms and analysis rely on the close relation between the shortest superstring problem, that was shown by Turner [11] to be reducible to the travelling salesman problem, and the cycle cover problem. The same relation was exploited in subsequent papers [10] ( 2:89), [5] ( 2:83), [7] ( 2:79) and [1, 2] ( 2:75). Armen and Stein [3] have also recently obtained a 2 2 3 approximation algorithm, independently of our work. Here we continue this li
ScoringandUnfolding Trimmed Tree Assembler: Algorithms for Assembling Genome Sequences Accurately and Efficiently
, 2011
"... My family iii lowed me to clarify several important aspects of the thesis for the general reader. I thank Professor Michael Schatz for making sure that the theoretical framework presented in the dissertation was correctly presented and related to the relevant prior art extensively. I also express my ..."
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My family iii lowed me to clarify several important aspects of the thesis for the general reader. I thank Professor Michael Schatz for making sure that the theoretical framework presented in the dissertation was correctly presented and related to the relevant prior art extensively. I also express my thanks to Professor Alan Siegel for improving the presentation and stile of the thesis. He has been not just a scientific mentor but also a tutoring figure in virtue of his high commitment to the value of education. Finally, I would like to thank Professor Raul Rabadan for suggesting many areas of application of the tools developed in this thesis. One of the contributions of the dissertation (TotalReCaller) is the result of the joint work with Fabian Menges. I am particularly grateful to him for such collaboration as well as for all the valuable and energetic discussions that we had on many of the topics presented in this thesis. I also would like to thank all the members of the NYU Bioinformatics Group for creating a unique and
Approximating the Shortest Superstring Problem Using de Bruijn Graphs
"... For strings s and t by overlap(s, t) we denote the longest suffix of s that is also a prefix of t. By prefix(s, t) we denote the first s  − overlap(s, t)  symbols of s. Similarly, suffix(s, t) is the last t  − overlap(s, t)  symbols of t. Clearly, for any strings s and t, 1 ..."
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For strings s and t by overlap(s, t) we denote the longest suffix of s that is also a prefix of t. By prefix(s, t) we denote the first s  − overlap(s, t)  symbols of s. Similarly, suffix(s, t) is the last t  − overlap(s, t)  symbols of t. Clearly, for any strings s and t, 1
Author manuscript, published in "CPM, NewYork: United States (2010)" Cover array string reconstruction
, 2013
"... Abstract. A proper factor u of a string y is a cover of y if every letter of y is within some occurrence of u in y. The concept generalises the notion of periods of a string. An integer array C is the minimalcover (resp. maximalcover) array of y if C[i] is the minimal (resp. maximal) length of cov ..."
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Abstract. A proper factor u of a string y is a cover of y if every letter of y is within some occurrence of u in y. The concept generalises the notion of periods of a string. An integer array C is the minimalcover (resp. maximalcover) array of y if C[i] is the minimal (resp. maximal) length of covers of y[0.. i], or zero if no cover exists. In this paper, we present a constructive algorithm checking the validity of an array as a minimalcover or maximalcover array of some string. When the array is valid, the algorithm produces a string over an unbounded alphabet whose cover array is the input array. All algorithms run in linear time due to an interesting combinatorial property of cover arrays: the sum of important values in a cover array is bounded by twice the length of the string.