Covering and packing in linear space

• Published in: Information processing letters Vol. 111; no. 21-22; pp. 1033 - 1036
• Main Authors: BJÖRKLUND, Andreas, HUSFELDT, Thore, KASKI, Petteri, KOIVISTO, Mikko
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier 15.11.2011; Elsevier Sequoia S.A
• Subjects: Algorithmics. Computability. Computer arithmetics; Algorithms; Applied sciences; Combinatorics; Combinatorics. Ordered structures; Computation; Computer and Information Science; Computer Science; Computer science; control theory; systems; Covering; Data- och informationsvetenskap (Datateknik); Datavetenskap (datalogi); Designs and configurations; Exact sciences and technology; Functional analysis; Graph theory; Graphs; Mathematical analysis; Mathematics; Miscellaneous; Natural Sciences; Naturvetenskap; Optimization algorithms; Packing problem; Sciences and techniques of general use; Studies; Theoretical computing; Transforms; Universe; Polynomial; Computer theory; Chromatic polynomial; Covering problem; Vertex(graph); Universe; Space time; Computing; Chromatic number; Polynomial time; Algorithms; Packing; Information processing; Zeta transform; Fast algorithm; Algorithm analysis; Linear space
• ISSN: 0020-0190; 1872-6119
• DOI: 10.1016/j.ipl.2011.08.002

The fastest known algorithms for the k-cover and the k-packing problem rely on inclusion-exclusion and fast zeta transform, taking time and space n^2, up to a factor polynomial in the size of the universe n. Here, we introduce a new, fast zeta transform algorithm that improves the space requirement to only linear in the size of the given set family, while not increasing the time requirement. Thus, for instance, the chromatic or domatic number of an n-vertex graph can be found in time within a polynomial factor of n^2 and space O(n^1.442) or O(n^1.716), respectively. For computing the chromatic polynomial, we further reduce the space requirement to O(n^1.292). [PUBLICATION ABSTRACT]