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 NC algorithm which achieves the compression ratio of 1/4+ε. (2) The proof that the algorithm GREEDY is not parallelizable, the computation 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 RNC algorithm with constant approximation ratio and an NC algorithm with logarithmic approximation ratio.
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