The n-dimensional hypercube Q_n is a fundamental interconnection network in parallel computing. In practice, link failures and prescribed connection demands must be accommodated simultaneously. This paper studies the existence of a Hamiltonian cycle containing every edge of a prescribed set L of edges in the faulty hypercube Q_n-F , where F is a set of faulty edges. Under the combined condition |L∪ F|≤ 2n-5 , we introduce the notion of a bottleneck—a vertex whose local interaction with L and F obstructs Hamiltonian cycle extension—and establish a necessary and sufficient condition: such a Hamiltonian cycle exists if and only if L forms a linear forest, δ (Q_n-F)≥ 2 , and no vertex is a bottleneck. The bound 2n-5 is tight. This characterization generalizes the results of Dvořák (2005) and Chen (2019); Wang and Chen (2008).
The n-way number partitioning problem (dividing a list of integers among n bins so that the bin sums are balanced) is a classical NP-hard problem underlying fair division and machine scheduling. Because many instances admit no perfectly balanced partition, we relax the problem by permitting a bounded amount of splitting, and ask precisely how much splitting is needed to make it tractable. We study three relaxations: (i) at most s items may be split between bins; (ii) at most t splittings are allowed (one item split across k bins counts as k-1 splittings); and (iii) a decision version in which the largest bin sum need only lie within an interval of width u times the largest item. We treat the decision, min-max, and max-min objectives, and generalize each to asymmetric bins with entitlements r_1,… ,r_n , modeling fair division among agents with unequal rights and scheduling on uniform machines of different speeds. For every fixed n ≥ 3 we obtain a complete complexity classification, governed by sharp thresholds: the problem is polynomial-time solvable exactly when s ≥ n-2 , when t ≥ n-1 , or when u ≥ n-2 , respectively, and NP-complete below each bound. When n is unbounded, all three variants are strongly NP-complete. The positive results hold for both the asymmetric min-max and asymmetric max-min objectives. The engine behind most of these results is a two-way polynomial-time reduction between the split-item and interval-target variants, revealing that two superficially different relaxations are essentially the same problem.
Given two graphs G and F, if G contains a copy of F, then the rainbow number of F in G, denoted by rb(G, F), is the minimum positive integer t such that any t-edge-coloring of G contains a rainbow copy of F. Let ℳ_n(F) denote the family of all maximal outerplanar graphs on n vertices, each of which contains a copy of F. The rainbow number of F in maximal outerplanar graph on n vertices, denoted by rb(ℳ_n(F),F) , is defined as max{rb(G,F)|G∈ℳ_n(F)} . Denote by C_k^+ the graph on k+1 vertices obtained from C_k by adding a pendant edge to its arbitrary vertex. In this paper, we prove that rb(ℳ_n(C_k^+),C_k^+)=rb(ℳ_n(C_k),C_k)=2n-⌈n-2/k-2⌉ -2 for n≥ 2k-3 and 5≥ k≥ 4 .
Federated Learning (FL) enables collaborative model training while preserving data privacy, yet remains vulnerable to Byzantine attacks, free-riding behaviors, and unfair contribution assessment. We propose TrustAware-X extended from TrustAware++ a novel FL framework that addresses these challenges through four integrated theoretical innovations: (1) Lyapunov-stable trust dynamics with provable exponential convergence V(t+1)≤ (1-δ )V(t) ; (2) Nash equilibrium analysis proving honest participation is a strict equilibrium when 𝔼[R_mal]<β S ; (3) Federated Reinforcement Learning (Fed-RL) for adaptive trust parameter optimization with stability preservation (Lemma 1); and (4) Shapley value-based fair reward distribution satisfying efficiency, symmetry, and null-player axioms. Unlike prior work (TrustAware++), TrustAware-X provides simultaneous mathematical guarantees on convergence, incentive compatibility, adaptivity, and fairness (Theorem 3). Evaluated on Edge-IIoTset intrusion detection with 5 clients (3 honest, 1 Byzantine, 1 malicious gradient poisoning), TrustAware-X achieves: 34 slashes detecting all adversaries by round 7, 100 V(30)=14.6 , 0.900 F1-score, and and 39.4
Transportation problems are mathematical models used to plan the movement of goods from multiple sources to their destinations efficiently. If we incorporate intermediate transfer points into transportation problems, then these obtained problems are called transshipment problems. A transportation problem is called solid transportation problem if we add extra conveyances to transportation problems. However, real-world problems often involve multiple objectives and uncertain parameters, combined with the simultaneous occurrence of both intermediary nodes and additional conveyances. These problems are referred to multi-objective solid transshipment problems under uncertainty. In this paper, a multi-objective solid transshipment problem under uncertainty is formulated in which the demand (b_j) , supply (a_i) , conveyance capacities (e_l) , and decision variables (x_ijl) are assumed to be crisp, whereas the coefficients of the objective functions are represented by triangular intuitionistic fuzzy numbers. A solution methodology based on model transformation, defuzzification using a ranking function and independently optimizing each objective function using GAMS 49.4.0 is used to obtain the corresponding optimal solutions of the resulting equivalent crisp model. Numerical examples and comparative studies are presented to show the applicability and effectiveness of the proposed methodology.
