When can $n$ given numbers be combined using arithmetic operators from a given subset of $\{+, -, \times, ÷\}$ to obtain a given target number? We study three variations of this problem of Arithmetic Expression Construction: when the expression (1) is unconstrained; (2) has a specified pattern of parentheses and operators (and only the numbers need to be assigned to blanks); or (3) must match a specified ordering of the numbers (but the operators and parenthesization are free). For each of these variants, and many of the subsets of $\{+,-,\times,÷\}$, we prove the problem NP-complete, sometimes in the weak sense and sometimes in the strong sense. Most of these proofs make use of a "rational function framework" which proves equivalence of these problems for values in rational functions with values in positive integers.

Motivation: Segmental duplications (SDs) or low-copy repeats, are segments of DNA > 1 Kbp with high sequence identity that are copied to other regions of the genome. SDs are among the most important sources of evolution, a common cause of genomic structural variation and several are associated with diseases of genomic origin including schizophrenia and autism. Despite their functional importance, SDs present one of the major hurdles for de novo genome assembly due to the ambiguity they cause in building and traversing both state-of-the-art overlap-layout-consensus and de Bruijn graphs. This causes SD regions to be misassembled, collapsed into a unique representation, or completely missing from assembled reference genomes for various organisms. In turn, this missing or incorrect information limits our ability to fully understand the evolution and the architecture of the genomes. Despite the essential need to accurately characterize SDs in assemblies, there has been only one tool that was developed for this purpose, called Whole-Genome Assembly Comparison (WGAC); its primary goal is SD detection. WGAC is comprised of several steps that employ different tools and custom scripts, which makes this strategy difficult and time consuming to use. Thus there is still a need for algorithms to characterize within-assembly SDs quickly, accurately, and in a user friendly manner. Results: Here we introduce SEgmental Duplication Evaluation Framework (SEDEF) to rapidly detect SDs through sophisticated filtering strategies based on Jaccard similarity and local chaining. We show that SEDEF accurately detects SDs while maintaining substantial speed up over WGAC that translates into practical run times of minutes instead of weeks. Notably, our algorithm captures up to 25% 'pairwise error' between segments, whereas previous studies focused on only 10%, allowing us to more deeply track the evolutionary history of the genome. Availability and implementation: SEDEF is available at https://github.com/vpc-ccg/sedef.