On explaining integer vectors by few homogeneous segments

We extend previous studies on “explaining” a nonnegative integer vector by sums of few homogeneous segments, that is, vectors where all nonzero entries are equal and consecutive. We study two NP-complete variants which are motivated by radiation therapy and database applications. In Vector Positive Explanation, the segments may have only positive integer entries; in Vector Explanation, the segments may have arbitrary integer entries. Considering several natural parameterizations such as the maximum vector entry γ and the maximum difference δ between consecutive vector entries, we obtain a refined picture of the computational (in-)tractability of these problems. For example, we show that Vector Explanation is fixed-parameter tractable with respect to δ, and that, unless NP ⊆ coNP/poly, there is no polynomial kernelization for Vector Positive Explanation with respect to the parameter γ. We also identify relevant special cases where Vector Positive Explanation is algorithmically harder than Vector Explanation. An extended abstract appeared in the Proceedings of the 13th Algorithms and Data Structures Symposium (WADS 2013), pp. 207-218, LNCS 8037, Springer, 2013. Email addresses: [email protected] (Robert Bredereck), [email protected] (Jiehua Chen), [email protected] (Sepp Hartung), [email protected] (Christian Komusiewicz), [email protected] (Rolf Niedermeier), [email protected] (Ondřej Suchý) Supported by the DFG, research project PAWS, NI 369/10. Supported by the Studienstiftung des Deutschen Volkes. The main work was done while Ondřej Suchý was at TU Berlin, supported by the DFG, research project AREG, NI 369/9. Preprint submitted to Journal of Computer and System Sciences January 29, 2015