A directed multigraph is said to be d-regular 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 n 2 cycle covers each taken with an appropriate multiplicity. We prove that if the d-regular multigraph does not contain more than ⌊d/2 ⌋ copies of any 2-cycle then we can find a similar decomposition into n 2 pairs of cycle covers where each 2-cycle 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 two cycle covers that do not share a 2-cycle 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 5/8 approximation with a simpler algorithm. Utilizing a reduction from maximum TSP to the shortest superstring problem we obtain a 2.5-approximation algorithm for the latter problem which is again much simpler than the previous one. For minimum asymmetric TSP the same technique gives two cycle covers, not sharing a 2-cycle, with weight at most twice the weight of the optimum. Assuming triangle

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 semi-infinite 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...

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 NP-hard, 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.

ily a good approximation for the maximum overlap in the superstring, and vice versa. The first constant-approximation algorithm for the length of the shortest superstring was given by Blum et al. [4], who discovered a 3-approximation algorithm and proved that the "Greedy" algorithm by Tarhio and Ukkonen [9] achieves 4-approximation. 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

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 min-plus product of matrices. 1

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