Identifying periodic patterns in time series data is crucial for uncovering hidden structures and predicting future events. Recognizing meaningful periodic patterns in real data requires handling approximation criteria since periodic phenomena are usually inexact. This paper introduces a suitable criterion and focuses on detecting Consecutive Periodic Patterns (CPPs) with$k \Delta$-errors, where$k$bounds the number of errors and$\Delta$limits the size of the error. We develop efficient algorithms to detect the Longest Pure Consecutive Maximal Periodic Pattern with bounded errors in both raw and compressed data, the latter by means of the Arithmetic Progressions Tree (APT) data structure.

The n-way number partitioning problem is a classic problem in combinatorial optimization, with applications to diverse settings such as fair allocation and machine scheduling. All these problems are NP-hard, but various approximation algorithms are known. We consider three closely related kinds of approximations. The first two variants optimize the partition such that: in the first variant some fixed number s of items can be split between two or more bins and in the second variant we allow at most a fixed number t of splittings. The third variant is a decision problem: the largest bin sum must be within a pre-specified interval, parameterized by a fixed rational number u times the largest item size. When the number of bins n is unbounded, we show that every variant is strongly NP-complete. When the number of bins n is fixed, the running time depends on the fixed parameters s,t,u. For each variant, we give a complete picture of its running time. For n = 2, the running time is easy to identify. Our main results consider any fixed integer n ≥ 3. Using a two-way polynomial-time reduction between the first and the third variant, we show that n-way number-partitioning with s split items can be solved in polynomial time if s ≥ n-2, and it is NP-complete otherwise. Also, n-way number-partitioning with t splittings can be solved in polynomial time if t ≥ n-1, and it is NP-complete otherwise. Finally, we show that the third variant can be solved in polynomial time if u ≥ (n-2)/n, and it is NP-complete otherwise. Our positive results for the optimization problems consider both min-max and max-min versions. Using the same reduction, we provide a fully polynomial-time approximation scheme for the case where the number of split items is lower than n-2.