Parameterizing by the Number of Numbers

13 years 4 months ago

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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 abovementioned 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 ...

Michael R. Fellows, Serge Gaspers, Frances A. Rosa

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