@inbook{e4b1b10056234425992556af04ac0f10,

title = "Parameterized Algorithms for Covering by Arithmetic Progressions",

abstract = "An arithmetic progression is a sequence of integers in which the difference between any two consecutive elements is the same. We investigate the parameterized complexity of two problems related to arithmetic progressions, called Cover by Arithmetic Progressions (CAP) and Exact Cover by Arithmetic Progressions (XCAP). In both problems, we are given a set X consisting of n integers along with an integer k, and our goal is to find k arithmetic progressions whose union is X. In XCAP we additionally require the arithmetic progressions to be disjoint. Both problems were shown to be NP-complete by Heath [IPL{\textquoteright}90]. We present a O(k2)poly(n) time algorithm for CAP and a 2O(k3)poly(n) time algorithm for XCAP. We also give a fixed parameter tractable algorithm for CAP parameterized below some guaranteed solution size. We complement these findings by proving that CAP is Strongly NP-complete in the field Zp, if p is a prime number part of the input.",

keywords = "Arithmetic Progressions, Number Theory, Set Cover, parameterized complexity theory",

author = "Ivan Bliznets and Jesper Nederlof and Krisztina Szil{\'a}gyi",

note = "Publisher Copyright: {\textcopyright} The Author(s), under exclusive license to Springer Nature Switzerland AG 2024.",

year = "2024",

month = jan,

day = "21",

doi = "10.1007/978-3-031-52113-3_9",

language = "English",

isbn = "978-3-031-52112-6",

series = "Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)",

publisher = "Springer",

pages = "125–138",

editor = "Fernau, {Henning } and Gaspers, {Serge } and Klasing, {Ralf }",

booktitle = "SOFSEM 2024: Theory and Practice of Computer Science",

edition = "1",

}