On explaining integer vectors by few homogeneous segments

Abstract

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.