We study approximation algorithms for the Min-Max Rural Postmen Cover Problem (MMRPCP). Given an undirected graph [Formula: see text], and a required subset [Formula: see text] of edges, where each edge in [Formula: see text] has a nonnegative weight, the objective is to find at most [Formula: see text] closed walks covering all the edges in [Formula: see text] such that the maximum weight of the closed walks is minimum. We propose a bicriteria [Formula: see text]-approximation algorithm for the MMRPCP. More exactly, given any instance [Formula: see text] of the MMRPCP consisting of a positive integer [Formula: see text], a graph [Formula: see text] and a required edge set [Formula: see text], the algorithm can produce at most [Formula: see text] closed walks covering all the edges in [Formula: see text] such that the maximum weight of the closed walks is no more than [Formula: see text] times the optimal value of [Formula: see text]. Previously, the best-known approximation ratio for the MMRPCP is [Formula: see text]. Our result demonstrates that a moderate relaxation of the constraint on the number of closed walks is helpful to reduce the approximation ratio.

In 2018, by using the inverse Gray mapping, Su et al. proposed several new classes of optimal autocorrelation interleaved quaternary sequences with period [Formula: see text] from cyclotomic classes of order [Formula: see text] (Science China Information Sciences, 61:1–13, 2018). In this paper, we determine the 4-adic complexity of these quaternary sequences. Our results show that the [Formula: see text]-adic complexity of these sequences reaches or approaches its maximum value, so as to resist the attack of the rational approximation algorithm.

This corrigendum identifies an issue with a proof of a lemma found in the paper Inside the Class of REGEX Languages [M. L. Schmid, Int. J. Found. Comput. Sci. 24, 07 (2013)], and proposes a method to bridge the gap. The relevant part of the original paper aims to show that a certain variant of pattern expressions (where variables are substituted uniformly) is equivalent to an interesting variant of extended regular expressions, which treat undefined variable references as reference definitions (as opposed to the empty symbol or set, as is done usually). Using a new concept of necessarily initialised regex automata systems we prove this equivalence.

Let [Formula: see text] be the narrow-sense BCH code over [Formula: see text] of length [Formula: see text] with designed distance [Formula: see text]. This paper mainly investigates the parameters of the codes [Formula: see text] and their hulls for [Formula: see text]. A sufficient and necessary condition on the designed distance such that the hull has the largest dimension is presented. Finally, examples of self-orthogonal codes from their hulls are obtained.

Recently, the nonlinear complexity of periodic binary sequences with a specific structure was investigated in [Yuan et al., IEEE Transactions on Information Theory 70(7), 5376–5391 (2024)], and a remaining problem is whether the maximum value of the nonlinear complexity of all right circular shifts of the sequence with length [Formula: see text] in the structure is [Formula: see text], where [Formula: see text] is the spacing between the maximum length identical subsequences with different successors. Based on the internal structure of the sequence, we verify that the maximum nonlinear complexity of all right cyclic shifts of [Formula: see text]-ary sequences with length [Formula: see text] equals [Formula: see text] when [Formula: see text].

Using the framework for measuring distances of languages based on finite approximations of languages and distances on words introduced in [25] it is shown that two languages have infinite intersection if and only if their one-sided edit distance is zero. Additionally, the relationship between one-sided edit distance and [Formula: see text]-neighborhoods of languages (see e.g., [24]) is established.

The emergence of applications in machine learning, social networks, and sensor placement has naturally introduced challenges in maximizing submodular or non-submodular functions over integer lattices under knapsack constraints. While efficient algorithms exist for maximizing monotone non-negative submodular functions over both finite sets and integer lattices, these methods do not generalize straightforwardly to non-submodular objectives. To address this gap, we propose a two-phase greedy algorithm with a provable approximation ratio of [Formula: see text], given that the objective function satisfies the Diminishing Return ratio (DR-ratio) [Formula: see text] and the generalized curvature [Formula: see text]. To our knowledge, this represents the first non-trivial approximation guarantee for such problem. Experimental evaluations demonstrate the algorithm’s practical effectiveness.

In numerous real-world scenarios, the objective is to choose multiple sets of disjoint elements rather than just one. Thus a [Formula: see text]-submodular function, which is a natural extension of a submodular function, finds applications in a wide array of contexts. This research is driven by a scenario in which data items within a stream are categorized into distinct groups, each subject to a size constraint. Firstly, we present a multi-pass streaming algorithm that needs [Formula: see text] passes through the data stream. This algorithm achieves a [Formula: see text]-approximation for the monotone case and a [Formula: see text]-approximation for the non-monotone case, respectively, for any constant [Formula: see text]. The query complexity is [Formula: see text], and the memory complexity is [Formula: see text]. And then we propose a one-pass streaming algorithm which returns a [Formula: see text]-approximation ratio for the monotone case and a [Formula: see text]-approximation ratio for the non-monotone case. The query complexity is [Formula: see text], and the memory complexity is [Formula: see text].

Input-driven pushdown automata ([Formula: see text]) are pushdown automata where the next action on the pushdown store (push, pop, nothing) is solely governed by the current input symbol. Here, we introduce sweeping input-driven pushdown automata that process the input in multiple passes (also sweeps). That is, a sweeping input-driven pushdown automaton is a two-way device that may change the input head direction only at the endmarkers. First we show that, given an arbitrary [Formula: see text], an equivalent [Formula: see text] that halts on any input can effectively be constructed. From this result further properties follow. Then we address the determinization of [Formula: see text]s and its descriptional complexity. Furthermore, the computational capacity of [Formula: see text]s is studied. To this end, we compare the family [Formula: see text] with other well-known language families. In particular, we are interested in families that have strong relations to some kind of pushdown machines. Finally, we explore several decidability problems and show that emptiness and several other problems are non-semidecidable by reductions of non-semidecidable problems of deterministic, linearly space bounded, one-tape, one-head Turing machines.

We prove that for a given partial deterministic attributed tree transducer with monadic output, it is decidable whether or not an equivalent top-down tree transducer (with or without look-ahead) exists. We present a procedure that constructs such an equivalent top-down tree transducer (with or without look-ahead) if it exists. We then show that our results can be extended to arbitrary nondeterministic attributed tree transducer with look-around that have monadic output.