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For an arbitrary finite family of graphs, the distance labeling problem asks to assign labels to all nodes of every graph in the family in a way that allows one to recover the distance between any two nodes of any graph from their labels. The main goal is to minimize the number of unique labels used. We study this problem for the families (mathcal {C}_n) consisting of cycles of all lengths between 3 and n. We observe that the exact solution for directed cycles is straightforward and focus on the undirected case. We design a labeling scheme requiring (frac{nsqrt{n}}{sqrt{6}}+O(n)) labels, which is almost twice less than is required by the earlier known scheme. Using the computer search, we find an optimal labeling for each (nle 17), showing that our scheme gives the results that are very close to the optimum.

In this paper, we continue the research on the power of contextual grammars with selection languages from subfamilies of the family of regular languages. In the past, two independent hierarchies have been obtained for external and internal contextual grammars, one based on selection languages defined by structural properties (finite, monoidal, nilpotent, combinational, definite, ordered, non-counting, power-separating, suffix-closed, commutative, circular, or union-free languages), the other one based on selection languages defined by resources (number of non-terminal symbols, production rules, or states needed for generating or accepting them). In a previous paper, the language families of these hierarchies for external contextual grammars were compared and the hierarchies merged. In the present paper, we compare the language families of these hierarchies for internal contextual grammars and merge these hierarchies.

This paper introduces sweeping permutation automata, which move over an input string in alternating left-to-right and right-to-left sweeps and have a bijective transition function. It is proved that these automata recognize the same family of languages as the classical one-way permutation automata (Thierrin, “Permutation automata”, Mathematical Systems Theory, 1968). The proof is constructive: an n-state two-way permutation automaton is transformed to a one-way permutation automaton with at most (F(n)=max _{k+ell =n, m leqslant ell } k cdot { ell atopwithdelims ()m} cdot {k - 1 atopwithdelims ()ell - m} cdot (ell - m)!) states (here the maximum is over all partitions of n into k states with transitions to the right and (ell) states with transitions to the left, where m states have no transitions on the left end-marker). This number of states is proved to be necessary in the worst case, and its growth rate is estimated as (F(n) = n^{frac{n}{2} - frac{1 + ln 2}{2}frac{n}{ln n}(1 + o(1))}).

A complete weighted graph (G= (V, E, w)) is called (Delta _{beta })-metric, for some (beta ge 1/2), if G satisfies the (beta)-triangle inequality, i.e., (w(u,v) le beta cdot (w(u,x) + w(x,v))) for all vertices (u,v,xin V). Given a (Delta _{beta })-metric graph (G=(V, E, w)), the (Delta _{beta })-Weighted Densest k-Subgraph ((Delta _{beta })-WDkS) problem is to find an induced subgraph G[C] with exactly k vertices such that the total edge weight of G[C] is maximized. For (beta = 1), this problem, (Delta)-WDkS, is known NP-hard and admits a (frac{1}{2})-approximation algorithm. In this paper, we show the NP-hardness and the inapproximability of the (Delta _{beta })-WDkS problem for any (beta> 1/2). We prove that (Delta _{beta })-WDkS can be approximated to within a factor ((frac{1}{2beta }+frac{2beta -1}{2beta cdot (2k-3)})) for any (beta> frac{1}{2}) and show that the analysis of the approximation ratio is asymptotically tight. Additionally, we describe a method to adapt any (alpha)-approximation algorithm for (Delta)-WDkS to obtain a (delta _{alpha ,beta })-approximation algorithm for (Delta _{beta })-WDkS with (delta _{alpha ,beta }> alpha) for every (beta <1). This allows for improved approximations for (Delta _{beta })-WDkS instances, offering potential enhancements to existing algorithms for this problem.

In the present paper, we introduce a message recovery attack applicable to NTRU cryptosystem. Our methodology uses a reduction from the NTRU-lattice to a Voronoi First Kind (VFK) lattice, enabling the use of a polynomial Closest Vector Problem (CVP) exact algorithm, which is vital for successful message recovery. This approach assumes knowledge of the Most Significant Bit of the coefficients of a polynomial that is a multiple of the nonce. Finally, extensive experimental results for the NTRU-HPS variants submitted to NIST are presented. The findings highlight the need to properly protect NTRU schemes against potential leakage.

In this work, we present hypernode automata as a specification formalism for hyperproperties of systems whose executions may be misaligned among themselves, such as concurrent systems. These automata consist of nodes labeled with hypernode logic formulas and transitions marked with synchronizing actions. Hypernode logic formulas establish relations between sequences of variable values among different system executions. This logic enables both synchronous and asynchronous analysis of traces. In its asynchronous view on execution traces, hypernode formulas establish relations on the order of value changes for each variable without correlating their timing. In both views, the analysis of different execution traces is synchronized through the transitions of hypernode automata. By combining logic’s declarative nature with automata’s procedural power, hypernode automata seamlessly integrate asynchronicity requirements at the node level with synchronicity between node transitions. We show that the model-checking problem for hypernode automata is decidable for specifications where each node specifies either a synchronous or an asynchronous requirement for the system’s executions, but not both.

We present a method for finding envy-free prices in a combinatorial auction where the consumers’ number n coincides with that of distinct items for sale, each consumer can buy one single item and each item has only one unit available. This is a particular case of the unit-demand envy-free pricing problem, and was recently revisited by Arbib et al. (Discr Appl Math 261:22–27, 2019, https://doi.org/10.1016/j.dam.2018.03.034). These authors proved that using a Fibonacci heap for solving the maximum weight perfect matching and the Bellman-Ford algorithm for getting the envy-free prices, the overall time complexity for solving the problem is (O(n^3)). We propose a method based on dynamic programming design strategy that seeks the optimal envy-free prices by increasing the consumers’ utilities, which has the same cubic complexity time as the aforementioned approach, but whose theoretical and empirical results indicate that our method performs faster than the shortest paths strategy, obtaining an average time reduction in determining optimal envy-free prices of approximately 48%.

Many important hyperliveness properties, such as refinement and generalized non-interference, fall into the class of (forall exists) hyperproperties, and require, for each execution trace of a system, the existence of another execution trace relating to the first one in a certain way. The alternation of quantifiers in the specification renders these hyperproperties extremely difficult to verify, or even just to test. Indeed, contrary to trace properties, where it suffices to find a single counterexample trace, refuting a (forall exists) hyperproperty requires not only to find a trace, but also a proof that no second trace exists that satisfies the specified relation with the first trace. As a consequence, automated testing of (forall exists) hyperproperties falls out of the scope of existing automated testing tools. In this paper, we present a fully automated approach to detect violations of (forall exists) hyperproperties in synchronous and asynchronous infinite-state systems. Our approach extends bug-finding techniques based on symbolic execution with support for trace quantification. We provide a prototype implementation of our approach, and demonstrate its effectiveness on a set of challenging examples.

We investigate the relationship of languages characterized by variants of string assembling systems and by Watson-Crick finite automata with a small number of states. Besides the general variant, we consider so-called free, and pure string assembling systems and compare their language generating power to Watson-Crick finite automata having one state (also called stateless) and two or three states in their state sets. We also study restricted variants of models that describe unary languages.