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23
Perspectives of Monge Properties in Optimization
, 1995
"... An m × n matrix C is called Monge matrix if c ij + c rs c is + c rj for all 1 i ! r m, 1 j ! s n. In this paper we present a survey on Monge matrices and related Monge properties and their role in combinatorial optimization. Specifically, we deal with the following three main topics: (i) f ..."
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Cited by 70 (2 self)
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An m &times; n matrix C is called Monge matrix if c ij + c rs c is + c rj for all 1 i ! r m, 1 j ! s n. In this paper we present a survey on Monge matrices and related Monge properties and their role in combinatorial optimization. Specifically, we deal with the following three main topics: (i) fundamental combinatorial properties of Monge structures, (ii) applications of Monge properties to optimization problems and (iii) recognition of Monge properties.
Approximation Algorithms for Asymmetric TSP by Decomposing Directed Regular Multigraphs
, 2006
"... A directed multigraph is said to be dregular if the indegree and outdegree of every vertexis exactly d. By Hall's theorem one can represent such a multigraph as a combination of atmost n2 cycle covers each taken with an appropriate multiplicity. We prove that if the dregular multigraph does ..."
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Cited by 68 (2 self)
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A directed multigraph is said to be dregular if the indegree and outdegree of every vertexis exactly d. By Hall's theorem one can represent such a multigraph as a combination of atmost n2 cycle covers each taken with an appropriate multiplicity. We prove that if the dregular multigraph does not contain more than b d/2c copies of any 2cycle then we can find asimilar decomposition into n2 pairs of cycle covers where each 2cycle occurs in at most onecomponent 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 whoseweight 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 therounding 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 previous5/8 approximation with a simpler algorithm. Utilizing a reduction from maximum TSP to the shortest superstring problem we obtain a 2.5approximation algorithm for the latter problemwhich is again much simpler than the previous one. For minimum asymmetric TSP the same technique gives 2cycle covers, not sharing a 2cycle, with weight at most twice the weight of the optimum. Assuming triangle inequality, we then show how to obtain from this pair of cycle covers a tour whose weight is at most0.842 log2 n larger than optimal. This improves upon a previous approximation algorithm with approximation guarantee of 0.999 log2 n. Other applications of the rounding procedure are approximation algorithms for maximum 3cycle cover (factor 2/3, previously 3/5) and maximum
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 27 (1 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...
A 2 2/3Approximation Algorithms for the Shortest Superstring Problem
 DIMACS WORKSHOP ON SEQUENCING AND MAPPING
, 1995
"... Given a collection of strings S = fs1; : : : ; sng over an alphabet, a superstring of S is a string containing each si as a substring; that is, for each i, 1 i n, contains a block of jsij consecutive characters that match si exactly. The shortest superstring problem is the problem of nding a superst ..."
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Given a collection of strings S = fs1; : : : ; sng over an alphabet, a superstring of S is a string containing each si as a substring; that is, for each i, 1 i n, contains a block of jsij consecutive characters that match si exactly. The shortest superstring problem is the problem of nding a superstring of minimum length. The shortest superstring problem has applications in both data compression and computational biology. In data compression, the problem is a part of a general model of string compression proposed by Gallant, Maier and Storer (JCSS '80). Much of the recent interest in the problem is due to its application to DNA sequence assembly. The problem has been shown to be NPhard; in fact, it was shown by Blum et al.(JACM '94) to be MAX SNPhard. The rst O(1)approximation was also due to Blum et al., who gave an algorithm that always returns a superstring no more than 3 times the length of an optimal solution. Several researchers have published results that improve on the approximation ratio; of these, the best previous result is our algorithm ShortString, which achieves a 2 3
Parallel and Sequential Approximations of Shortest Superstrings
 In Proceedings of Fourth Scandinavian Workshop on Algorithm Theory
, 1994
"... Abstract. Superstrings have many applications in data compression and genetics. However the decision version of the shortest superstring problem is N P�complete. In this paper we examine the complexity of approximating a shortest superstring. There are two basic measures of the approximations� the c ..."
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Cited by 14 (1 self)
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Abstract. Superstrings have many applications in data compression and genetics. However the decision version of the shortest superstring problem is N P�complete. In this paper we examine the complexity of approximating a shortest superstring. There are two basic measures of the approximations� the compression ratio and the approximation ratio. The well known and practical approximation algorithm is the sequential algorithm GREEDY. It approximates the shortest superstring with the compression ratio of 1 2 and with the approximation ratio of 4. Our main results are� �1 � An N C algorithm which achieves the compression ratio of 1 4� �. �2 � The proof that the algorithm GREEDY is not parallelizable � the com� putation of its output is P�complete. �3 � An improved sequential algorithm � the approximation ratio is reduced to 2.83. Previously it was reduced by Teng and Yao from 3 to 2.89. �4 � The design of an RN C algorithm with constant approximation ratio and an N C algorithm with logarithmic approximation ratio. 1
On the Greedy Superstring Conjecture
 In Proceedings of the 23rd FSTTCS
, 2003
"... We investigate the greedy algorithm for the shortest common superstring problem. We show that the length of the greedy superstring is upperbounded by the sum of the lengths of an optimal superstring and an optimal cycle cover, provided the greedy algorithm happens to merge the strings in a part ..."
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We investigate the greedy algorithm for the shortest common superstring problem. We show that the length of the greedy superstring is upperbounded by the sum of the lengths of an optimal superstring and an optimal cycle cover, provided the greedy algorithm happens to merge the strings in a particular way. Thus, when restricting inputs correspondingly, we verify the well known greedy conjecture, namely that the approximation ratio of the greedy algorithm is within a factor of two of the optimum, and actually extend the conjecture considerably.
Sequential and Parallel Approximation of Shortest Superstrings
, 1997
"... Superstrings have many applications in data compression and genetics. However, the decision version of the shortest superstring problem is N Pcomplete. In this paper we examine the complexity of approximating shortest superstrings. There are two basic measures of the approximations: the length fact ..."
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Cited by 4 (1 self)
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Superstrings have many applications in data compression and genetics. However, the decision version of the shortest superstring problem is N Pcomplete. In this paper we examine the complexity of approximating shortest superstrings. There are two basic measures of the approximations: the length factor and the compression factor. The well known and practical approximation algorithm is the sequential algorithm GREEDY. It approximates the shortest superstring with the compression 1 factor of and with the length factor of 4. Our main results are: Ž. 2 1 A sequential length approximation algorithm which achieves a length factor of 2.83. This result improves the best previously known bound of 2.89 due to Teng and Yao. Very recently, this bound was improved by Kosaraju, Park, and Stein to 2.79, and by Armen and Stein to 2.75. Ž. 2 A proof that the algorithm GREEDY is not paralleliz
Improved Lower Bounds for the Shortest Superstring and Related Problems
 CORR
, 2012
"... We study the approximation hardness of the Shortest Superstring, the Maximal Compression and the Maximum Asymmetric Traveling Salesperson (MAXATSP) problem. We introduce a new reduction method that produces strongly restricted instances of the Shortest Superstring problem, in which the maximal orbi ..."
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We study the approximation hardness of the Shortest Superstring, the Maximal Compression and the Maximum Asymmetric Traveling Salesperson (MAXATSP) problem. We introduce a new reduction method that produces strongly restricted instances of the Shortest Superstring problem, in which the maximal orbit size is eight (with no character appearing more than eight times) and all given strings having length at most six. Based on this reduction method, we are able to improve the best up to now known approximation lower bound for the Shortest Superstring problem and the Maximal Compression problem by an order of magnitude. The results imply also an improved approximation lower bound for the MAXATSP problem.
ASSEMBLY ALGORITHMS FOR NEXTGENERATION SEQUENCE DATA
, 2009
"... *Signatures are on file in the Graduate School. ii Nextgeneration sequencing is revolutionizing genomics, promising higher coverage at a lower cost per base when compared to Sanger sequencing. Shorter reads and higher error rates from these new instruments necessitate the development of new algorit ..."
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*Signatures are on file in the Graduate School. ii Nextgeneration sequencing is revolutionizing genomics, promising higher coverage at a lower cost per base when compared to Sanger sequencing. Shorter reads and higher error rates from these new instruments necessitate the development of new algorithms and software. This dissertation describes approaches to tackle some problems related to genome assembly with these short fragments. We describe YASRA (Yet Another Short Read Assembler), that performs comparative assembly of short reads using a reference genome, which can differ substantially from the genome being sequenced. We explain the algorithm and present the results of assembling one ancientmitochondrial and one plastid dataset. Comparing the performance of YASRA with the AMOScmpshortReads and Newbler mapping assemblers (version 2.0.00.17) as template genomes are varied, we find that YASRA generates fewer contigs with higher coverage and fewer errors. We also analyze situations where the use of comparative assembly outperforms de novo assembly, and
A 2 2/3 Superstring Approximation Algorithm
, 1998
"... Given a collection of strings S = {s_1, ..., s_n} over an alphabet \Sigma, a superstring \alpha of S is a string containing each s_i as a substring; that is, for each i, 1<=i<=n, \alpha contains a block of s_i consecutive characters that match s_i exactly. The shortest superstring problem is ..."
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Given a collection of strings S = {s_1, ..., s_n} over an alphabet \Sigma, a superstring \alpha of S is a string containing each s_i as a substring; that is, for each i, 1<=i<=n, \alpha contains a block of s_i consecutive characters that match s_i exactly. The shortest superstring problem is the problem of finding a superstring \alpha of minimum length.
The shortest superstring problem has applications in both data compression and computational biology. It was shown by Blum et al. [3] to be MAX SNPhard. The first O(1)approximation algorithm also appeared in [3], which returns a superstring no more than 3 times the length of an optimal solution. Prior to the algorithm described in this paper, there were several published results that improved on the approximation ratio; of these, the best was our algorithm ShortString, a 2 3/4 approximation [1].
We present our new algorithm, GShortString, which achieves an approximation ratio of 2 2/3. Our approach builds on the work in [1], in which we identified classes of strings that have a nested periodic structure, and which must be present in the worst case for our algorithms. We introduced machinery to describe these strings and proved strong structural properties about them. In this paper we extend this study to strings that exhibit a more relaxed form of the same structure, and we use this understanding to obtain our improved result.