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# Superstrings with multiplicities

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LIPIcs.CPM.2018.21.pdf
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## Cite As

Bastien Cazaux and Eric Rivals. Superstrings with multiplicities. In 29th Annual Symposium on Combinatorial Pattern Matching (CPM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 105, pp. 21:1-21:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.CPM.2018.21

## Abstract

A superstring of a set of words P = {s_1, ..., s_p } is a string that contains each word of P as substring. Given P, the well known Shortest Linear Superstring problem (SLS), asks for a shortest superstring of P. In a variant of SLS, called Multi-SLS, each word s_i comes with an integer m(i), its multiplicity, that sets a constraint on its number of occurrences, and the goal is to find a shortest superstring that contains at least m(i) occurrences of s_i. Multi-SLS generalizes SLS and is obviously as hard to solve, but it has been studied only in special cases (with words of length 2 or with a fixed number of words). The approximability of Multi-SLS in the general case remains open. Here, we study the approximability of Multi-SLS and that of the companion problem Multi-SCCS, which asks for a shortest cyclic cover instead of shortest superstring. First, we investigate the approximation of a greedy algorithm for maximizing the compression offered by a superstring or by a cyclic cover: the approximation ratio is 1/2 for Multi-SLS and 1 for Multi-SCCS. Then, we exhibit a linear time approximation algorithm, Concat-Greedy, and show it achieves a ratio of 4 regarding the superstring length. This demonstrates that for both measures Multi-SLS belongs to the class of APX problems.

## Subject Classification

##### ACM Subject Classification
• Mathematics of computing → Discrete mathematics
• Theory of computation → Approximation algorithms analysis
##### Keywords
• greedy algorithm
• approximation
• overlap
• cyclic cover
• APX
• subset system

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