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Parameterizing by the number of numbers

Fellows, Michael R., Gaspers, Serge and Rosamond, Frances A. (2012). Parameterizing by the number of numbers. Theory of Computing Systems,50(4):675-693.

Document type: Journal Article

IRMA ID 82794376xPUB13
Title Parameterizing by the number of numbers
Author Fellows, Michael R.
Gaspers, Serge
Rosamond, Frances A.
Journal Name Theory of Computing Systems
Publication Date 2012
Volume Number 50
Issue Number 4
ISSN 1432-4350   (check CDU catalogue open catalogue search in new window)
Scopus ID 2-s2.0-84856571260
Start Page 675
End Page 693
Total Pages 19
Place of Publication United States
Publisher Springer New York LLC
HERDC Category C1 - Journal Article (DIISR)
Abstract The usefulness of parameterized algorithmics has often depended on what Niedermeier has called “the art of problem parameterization”. In this paper we introduce and explore a novel but general form of parameterization: the number of numbers. Several classic numerical problems, such as SUBSET SUM, PARTITION, 3-PARTITION, NUMERICAL 3-DIMENSIONAL MATCHING, and NUMERICAL MATCHING WITH TARGET SUMS, have multisets of integers as input. We initiate the study of parameterizing these problems by the number of distinct integers in the input. We
rely on an FPT result for INTEGER LINEAR PROGRAMMING FEASIBILITY to show that all the above-mentioned problems are fixed-parameter tractable when parameterized in this way. In various applied settings, problem inputs often consist in part of multisets of integers or multisets of weighted objects (such as edges in a graph, or jobs to be scheduled). Such number-of-numbers parameterized problems often reduce to subproblems about transition systems of various kinds, parameterized by the size of the system description. We consider several core problems of this kind relevant
to number-of-numbers parameterization. Our main hardness result considers the problem: given a non-deterministic Mealy machine M (a finite state automaton outputting a letter on each transition), an input word x, and a census requirement c for the output word specifying how many times each letter of the output alphabet should be written, decide whether there exists a computation of M reading x that outputs a word y that meets the requirement c. We show that this problem is hard for W[1]. If the question is whether there exists an input word x such that a computation of M on x outputs a word that meets c, the problem becomes fixed-parameter tractable.
Keywords Parameterized complexity
Problem parameterization
Variety of a multiset
Numerical problems
DOI http://dx.doi.org/DOI 10.1007/s00224-011-9367-y   (check subscription with CDU E-Gateway service for CDU Staff and Students  check subscription with CDU E-Gateway in new window)
 
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