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 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 RN C algorithm with constant approximation ratio and an N C algorithm with logarithmic approximation ratio.
Boolean cardinality constraints (CCs) state that at most (at least, or exactly) k out of n propositional literals can be true. We propose a new, arc-consistent, easy to implement and efficient encoding of CCs based on a new class of selection networks. Several comparator networks have been recently proposed for encoding CCs and experiments have proved their efficiency (Abío et al. 2013, Asín et al. Constraints 12(2): 195-221, 2011, Codish and Zazon-Ivry 2010, Eén and Sörensson Boolean Modeling and Computation 2: 1 -26, 2006). In our construction we use the idea of the multiway merge sorting networks by Lee and Batcher (1995) that generalizes the technique of odd-even sorting ones by merging simultaneously more than two subsequences. The new selection network merges 4 subsequences in that way. Based on this construction, we can encode more efficiently comparators in the combine phase of the network: instead of encoding each comparator separately by 3 clauses and 2 additional variables, we propose an encoding scheme that requires 5 clauses and 2 variables on average for each pair of comparators. We also extend the model of comparator networks so that the basic components are not only comparators (2-sorters) but more general m-sorters, for m ∈ {2, 3, 4}, that can also be encoded efficiently. We show that with small overhead (regarding implementation complexity) we can achieve a significant improvement in SAT-solver runtime for many test cases. We prove that the new encoding is competitive to the other state-of-the-art encodings.
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