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/polyNP⊆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.