Dynamic submodular maximization tackles the challenge of maintaining approximate solutions for evolving datasets, where elements are inserted or deleted over time. Unlike static or streaming models, this setting requires efficiently updating the solution after each data modification without reprocessing the entire dataset, ensuring responsiveness to real-time changes. A key challenge lies in balancing strong approximation guarantees with low amortized update time, as traditional greedy approaches become computationally prohibitive under frequent updates. This framework prioritizes time efficiency over space complexity, reflecting modern computational demands in applications like renewable energy systems, real-time summarization, recommendation systems, and adaptive clustering. [Formula: see text]Our work investigates the dynamic maximization of a monotone non-submodular function [Formula: see text] over a ground set [Formula: see text] of size [Formula: see text] under cardinality and matroid constraints. For the cardinality constraint [Formula: see text] (a positive integer), we present a randomized algorithm that with high probability maintains an [Formula: see text]-approximation solution with [Formula: see text] amortized update time, where [Formula: see text] denotes the generic submodularity ratio introduced by Gong et al. [21]. For the matroid constraint [Formula: see text] of rank [Formula: see text], we propose an algorithm achieving a [Formula: see text]-approximation for any [Formula: see text] with [Formula: see text] amortized update time.
Let [Formula: see text] be a simple connected graph and [Formula: see text] a subset of [Formula: see text] with [Formula: see text]. An [Formula: see text]-path in [Formula: see text] is a path that connects all vertices of [Formula: see text]. Two [Formula: see text]-paths [Formula: see text] and [Formula: see text] are said to be internally disjoint if [Formula: see text] and [Formula: see text]. Denote by [Formula: see text] the maximum number of internally disjoint [Formula: see text]-paths in [Formula: see text]. For an integer [Formula: see text], the [Formula: see text]-path-connectivity [Formula: see text] of [Formula: see text] is defined as [Formula: see text] and [Formula: see text]. Let [Formula: see text] denote the Cayley graph generated by the [Formula: see text]-vertex wheel graph. In this paper, we investigate the 3-path-connectivity of [Formula: see text] and prove that [Formula: see text] for all [Formula: see text].
Finding a routing path that satisfies two independent additive constraints (e.g., delay and cost) is a critical requirement in quality of service (QoS) routing. While this two-constraint path problem is NP-Hard, it is prevalent in practical network applications. Existing solutions typically face a trade-off: heuristics lack feasibility guarantees, while exact algorithms suffer from exponential computational complexity. In this paper, we propose a dual-guided exact algorithm that effectively bridges the gap between the computational efficiency of Lagrangian relaxation and the optimality guarantees of combinatorial search. Our method first solves the Lagrangian dual problem to derive the optimal multiplier, which subsequently serves as the optimal aggregate coefficient to guide a heuristic A*-prune search. This hybrid mechanism allows the algorithm to efficiently prune the search space while guaranteeing the identification of a cost-efficient feasible solution. Numerical experiments on random network topologies demonstrate that the proposed algorithm significantly outperforms the standard A*-prune algorithm while maintaining exactness. Specifically, in networks with up to 500 nodes, our method reduces the execution time by orders of magnitude compared to traditional exact methods.
The Sierpi & nacute;ski network S(n,K-k) has become a prominent model for multiprocessor interconnection systems. Among hierarchical interconnection structures, the WK-recursive networks K(l,n,K-k) are notable for their advantageous properties, making them highly suitable for applications in parallel computing and VLSI design. These networks exhibit desirable features such as small diameter, high connectivity, and strong scalability. In this paper, we focus on solving the NP-complete Maximum Subgraph Problem (MSP) within the WK-recursive framework K(2,n,K-k), where n >= 2 and k >= 3. We propose a computational method based on lexicographic ordering, implemented in SageMath, to determine the maximum number of edges that can be induced by any subset of r vertices, for 1 <= r <= 2k(n). Furthermore, we explore the Minimum Linear Arrangement (MinLA) problem by investigating embeddings of WK-recursive networks into host structures specifically path graphs, caterpillars, 1-hierarchical caterpillars, windmill, and necklace graph that aim to minimize total wirelength.
Graph spectrum has applications in computer science, chemistry, physics and mathematics itself. Graph energy is a significant parameter to characterize the structural attributes of graphs. Let G be an n-vertex connected graph. The generalized energy of G is expressed as E-alpha(G) = alpha(2)& sum; (u is an element of V) ((G))d(G)(2)(u) + (1 - alpha)(2)& sum; (u is an element of V (G))d(G)(u), where alpha is an element of [0, 1]. Graph operation serves as a powerful method to generate all manner of larger graphs from simpler factor graphs. Let F(G) (F is an element of {S,R,Q,T}) represent four typical graph operations applied to G. In this paper, we investigate the generalized energy of a class of composite graphs known as F-sum graphs, constructed via the lexicographic product of a transformed graph F(G(1)) and a graph G(2). We derive the closed-form formulas for the generalized energy of these four classes of F-sum graphs in terms of the generalized energy and structural parameters of factor graphs. Furthermore, we apply these results to derive the generalized energy and quasi-Laplacian energy of the F-sum graphs generated by popular graphs, such as path, cycle, star, and complete graph. The deduction suggests an efficient approach to compute the energy of complex graphs without direct eigenvalue computations.
