The Total-Neighbor-Distinguishing Index by Sums of a graph G ( TNDI_∑(G) ) is the smallest positive integer for which there exists a total coloring such that the sum of the color assigned to a vertex and the colors of all its incident edges is distinct, for every pair of adjacent vertices u and w. Pilsniak and Wozniak (2015) introduced this notion and conjectured that TNDI_∑ is less than or equal to the maximum degree plus 3. In 2011, Gravier et al. introduced the concept of generalized Sierpiński graphs, denoted S(n, G), which extends the notion of Sierpiński graphs given by Hinz et al. (2017) by replacing the complete graph in their definition with any graph G. In this paper, we verify that S(n, G) graphs satisfy the TNDI_∑ Conjecture, when G is a bipartite graph, a path, a cycle, a star or a complete graph. Furthermore, we determine the TNDI_∑(S(n,G)) when G is a complete bipartite, a path, a cycle, a star or a regular bipartite graph. Remarkably, these colorings also satisfy the corresponding conjecture for the Adjacent-Vertex-Distinguishing Total Coloring (AVDTC), and the TNDI_∑ of these graphs coincides with their AVDTC chromatic number. Moreover, we show that the TNDI_∑ Conjecture is satisfied for the graph classes of the regularizations +S_p^n and ++S_p^n of Sierpiński graphs S_p^n , which contributes to proving that the conjecture holds for their respective fractals as well.
Graph partitioning is a fundamental family of problems in mathematics and computer science. Many applications, including power grid islanding, geographical districting, and network partitioning, require balanced part sizes and also that each part induces a connected subgraph. In this work, we study the structure of balanced connected graph partitions produced by connected recursive bisection (CRB). CRB repeatedly splits each part in half while maintaining part connectivity and terminates when it reaches a specified number of parts. Given a final partition produced by CRB, recovering the sequence of bisections is equivalent to finding a sequence of perfect matchings in the quotient graph, in which each vertex represents a part and edges correspond to part adjacencies. We define this sequence of matchings as a perfect hierarchical matching (PHM). The union of the matchings induces a spanning tree in the quotient graph that we denote as a PHM tree. We first characterize PHM trees by an efficient edge labeling scheme and count the number of PHM trees of the complete graph up to isomorphism. We prove tight max-degree and diameter bounds for PHM trees, as well as a matroid-like edge exchange property. We give an efficient algorithm for recognizing series-parallel graphs with PHMs and generalize it to a slicewise polynomial (XP) algorithm on graphs with constant treewidth. We conclude with open problems.
Overnight rebalancing in dock-based bike-sharing systems requires routing a limited fleet of trucks before user activity begins. This article formulates static rebalancing problem under stochastic demand uncertainty as a bi-objective combinatorial optimization problem that selects truck routes and visited stations. The first objective minimizes total travel distance. The second objective minimizes scenario-weighted unmet demand under a finite set of demand scenarios derived from historical station-status data. A deterministic recourse evaluation simulates truck loads and station inventories along each route and computes unmet demand for visited and unvisited stations. The article applies two multi-objective evolutionary algorithms, NSGA-II and MOEA/D, using a permutation–partition encoding and relocate-based operators that implement a 1–0 relocate neighborhood between routes. A roulette-wheel-based relocation operator (BB2) biases move selection by the induced change in route distance. Experiments on the Barcelona Bicing network with 518 stations and on clustered subinstances show that NSGA-II attains higher hypervolume and larger non-dominated sets, whereas MOEA/D attains lower runtime; an ablation analysis shows that BB2 improves coverage and proximity indicators.
In this paper, we consider the min–max heterogeneous weighted delivery (MMHWD) problem. Concretely, given a weighted graph G=(V,E;w) with length function w:E→ℝ^+ satisfying the triangle inequality, a fixed depot r∈ V , m items and k vehicles having nonuniform speeds λ _1 , λ _2 , … , λ _k , respectively, each item j is initially located at its source vertex s_j and needs to be uninterruptedly delivered to its target vertex t_j for j=1,2, … , m , each vehicle i can move along edges of G and only continuously deliver one item at a time, the goal is to find a set 𝒞={C_1,C_2,… , C_k} of k tours for these k vehicles, each starting and ending at r, to collectively deliver all items such that the maximum completion time of vehicles is minimized, where the completion time of a vehicle is its total length divided by its speed. We obtain two main results. (1) We design a 134.4434· (1+δ ) -approximation algorithm to solve the MMHWD problem, and this algorithm runs in time O(n^6 + n^4/δ+log _2 w(E)) , where δ >0 is a small constant. (2) We provide a (φ +9/5-1/k) -approximation algorithm to resolve the MMHWD problem, and that algorithm runs in time O(n^3) , where φ is the ratio of the largest vehicle speed to the smallest one.
Given a polygonal domain 𝒫 comprising h pairwise disjoint convex polygonal obstacles together defined with n vertices and a positive real number ϵ < 1 , this paper presents an algorithm to preprocess 𝒫 to compute routing tables at the vertices of 𝒫 so that a data packet from any vertex of 𝒫 is routed to any other vertex of 𝒫 . At every vertex v of 𝒫 along the routing path, until the packet reaches its destination, the next hop is determined using the routing tables at v and the information stored in the packet header. The preprocessing algorithm computes a routing table at every vertex of 𝒫 of size O((h + 1/ϵ)(n)) in O(n+h/√(ϵ)(h +1/ϵ)n) time so that any packet can be routed along a path that has (1 + ϵ ) multiplicative stretch and 2kℓ additive stretch, while the header of any packet carries at most 2n bits of routing information. Here, k is the sum of the number of times each obstacle in 𝒫 is visited along the routing path, and ℓ depends on the local geometry of obstacles in 𝒫 and is upper bounded by (√(2ϵ)) (max _P_i ∈𝒫max _p, q ∈ P_i |pq|) .
Motivated by the geometric framework underlying the Sombor index, the elliptic Sombor index (ESO) and the Euler Sombor index (EU) have recently been introduced, each arising from a distinct geometric rationale. In this paper, we resolve several open problems and demonstrate the effectiveness of these methods in addressing questions that have remained unsolved until now. We first investigate the extremal properties of the Euler–Sombor index. Specifically, we determine the second-minimal Euler Sombor index among all trees and, based on this result, establish the minimum Euler Sombor index for graphs with a given clique number, along with a characterization of the extremal graphs attaining this bound. Furthermore, we characterize the graphs that achieve the maximum Euler Sombor index among all graphs with a fixed chromatic number. These findings resolve the open problems posed by Kızılırmak (2025a). We then turn to the study of the elliptic Sombor index. Recently, Ahmad et al. (2025) investigated the extremal properties of the elliptic Sombor index for trees under specific structural constraints and provided a complete characterization. They also discussed its potential applications in chemical graph theory and posed an open problem: to determine the chemical trees with the maximum elliptic Sombor index among all chemical trees with (n) vertices and (s) segments. In this work, we completely resolve this open problem. Our results provide new insights into the structural behavior of the elliptic Sombor index in chemical trees and contribute to its further development within the framework of chemical graph theory.
Large Language Models have been used extensively across the full spectrum of human life - even being used by mathematicians to solve difficult mathematical problems. The finding of all solutions to a system of non-linear equations is a longstanding and challenging problem with applications across science and engineering. The natural question to ask is then whether these large language models can adequately find all solutions to systems of nonlinear equations. An in-depth analysis of 15 variants of large language models is undertaken, with systems taken from the published literature (which theoretically the models could have been trained on), as well as systems that have been newly created and have not been published. The solution quality of all models is analyzed, and guidelines for practitioners and future directions are discussed in detail. In addition, for three well-know test systems, this research found new additional true solutions.
As a generalization of submodular functions, k-submodular functions have broad applications in machine learning, including multi-type sensor placement, multi-topic influence maximization, and coupled feature selection, etc. Many optimization problems in the real world often involve uncertainty, and the risk of violating constraints caused by these random factors needs to be strictly controlled. In this paper, we study the k-submodular maximization problem with the chance constraint and propose two algorithms with linear query complexity. The first algorithm achieves an approximation ratio close to 1/4 for monotone f, and close to 1/5 for non-monotone f, using a query complexity O(nk/ϵ) . Furthermore, the second algorithm achieves an approximation ratio close to 1/3 for monotone f, and close to 1/4 for non-monotone f, using a query complexity O(nk/ϵlog1/ϵ) , where ϵ∈ (0,1/5) is a small constant.
Meta-learning, also known as learning to learn, has garnered significant attention in the fields of artificial intelligence and machine learning over the past few years. The core idea of meta-learning is to leverage prior experience and data to enhance quality and efficiency on new tasks. To date, numerous meta-learning algorithms have been studied within continuous domains. Recently, Adibi et al. integrated the concept of meta-learning into submodular optimization for the first time, proposing a discrete meta-learning framework termed submodular meta-learning. This framework can solve a series of tasks ={ _i} , where each task _i is considered as the problem of maximizing a monotone submodular function f_i under the cardinality constraint (upper bound by k). The goal is to train a common initial set with size l (0
Submodular optimization is an important part of combinatorial optimization, and meanwhile, it has been widely used in the fields of economy, computer science, etc. In recent years, many generalizations of the submodular maximization problem have been proposed, such as the k-submodular maximization problem, which considers not only whether the elements of the ground set are selected, but also which set of k-set they are selected into. Many machine learning problems, including influence maximization with k kinds of topics and sensor placement with k kinds of sensors, can be modeled as k-submodular maximization models. In practical applications, elements usually have certain attributes, such as gender, age, race, and so on. Based on these attributes we can group elements in the ground set. For fairness, we want the number of elements in each group to be relatively balanced, which introduces group fairness. Group fairness can be conceptualized in various ways, but typically falls into two primary categories: equity-fairness and equality-fairness. At present, the existing algorithms on k-submodular do not consider group fairness constraints, which can result in the number of elements in some groups being too high or too low. This motivates us to study k-submodular maximization under group fairness constraints. In this paper, we consider intersection constraints of two different categories of group fairness constraints and total size constraint, respectively. For the k-submodular maximization problem under equity-fairness constraints and total size constraint, we design an offline approximation algorithm with an approximate ratio of 1/2-ϵ for the case of monotone, and an approximate ratio of 1/3-ϵ for non-monotone. For the k-submodular maximization problem under equality-fairness constraints and total size constraint, we get an approximate ratio of 1/2 for the case of monotone and an approximate ratio of 0.041 for the case of non-monotone.
Given a graph G=(V,E) and a function b: V→{0, 1, 2} . If b satisfies ∑ _z∈ N_G(w)b(z)≥ 2 for every vertex w with b(w)=0 , where N_G(w) is the neighborhood of w in G, then b is called a Roman 2-dominating function of G. The Roman 2-domination number γ _{R2}(G) of G is the minimum value ∑ _w∈ Vb(w) among all Roman 2-dominating functions b of G. A set S⊆ V is a 2-dominating set of G if |N_G(w)∩ S|≥ 2 for each w∈ V\ S . The minimum cardinality among all 2-dominating sets of G is called the 2-domination number γ _2(G) of G. For any graph G, Chellali et al. (2016) showed that γ _{R2}(G)≤γ _2(G) . In this article, firstly, we provide a characterization for the trees T with γ _{R2}(T)=γ _2(T) . Secondly, for a tree T, we provide a lower bound of γ _{R2}(T) relying on γ _2(T) and the number of leaves l(T) in T: γ _{R2}(T)≥γ _2(T)-l(T)+2 , and a characterization for the trees T with γ _{R2}(T)=γ _2(T)-l(T)+2 . Finally, we prove that it is NP-hard to determine whether γ _{R2}(G) and γ _2(G) are equal for a given bipartite graph G.
In the realm of wireless sensor networks (WSN), clustering schemes stand out as effective approaches to optimize resource utilization, particularly in minimizing overhead and enhancing energy efficiency to extend network lifespan. Clustering optimisation that takes energy considerations into account is a difficult NP optimisation issue. In light of this difficulty, meta-heuristic optimisation algorithms stand out as a possible solution; they may greatly enhance energy efficiency, which would lead to the sustainability of the network. This article introduces a novel solution, the Hybrid African Buffalo and Remora Optimization Algorithm (HABROA), designed to achieve energy stability and prolong network lifetime through the judicious selection of cluster heads (CH) during the clustering process. HABROA is specifically tailored to navigate the intricate trade-off between exploitation and exploration rates, thereby facilitating the efficient identification of Cluster Head Candidates (CHS). This strategic balance contributes to the sustained longevity and energy stability of the network. The proposed HABROA model exhibits notable performance metrics, surpassing alternative methods. With a Mean Packet Delivery of 96
The Maximally Diverse Grouping Problem (MDGP) is the problem of assigning a set of elements to mutually disjoint groups in order to maximise the overall diversity between the elements. The MDGP is often formulated as an Integer Quadratic Programme (IQP). Because the MDGP is NP-complete, most studies have focused on heuristic solution approaches, as compared to exact solution approaches, to the problem. Although heuristic solution approaches are often used in practice, these approaches do not guarantee a global optimal solution. Conversely, although exact solution approaches are not often used in practice, these approaches guarantee a global optimal solution, and therefore serve as useful benchmarks for the performance of heuristics. As such, a few studies have formulated the MDGP as an Integer Linear Programme (ILP), which can be solved using exact solution approaches. The present paper proposes two new ILP formulations, and compares their performances against existing formulations. The proposed formulations can be used to establish useful benchmarks for the performance of heuristics in a broader range of applications moving forward.
This paper studies a bi-criteria parallel machine scheduling problem with uncertain processing times to minimize makespan and total completion time. We adopt a scenario-based approach for uncertainty and assume that a scenario index can take any real value in an interval. This index value can represent, for example, the amount of congestion in a network. We assume that the job processing times are non-decreasing polynomial functions of fixed degrees with respect to the scenario index. To tackle multi-criteria optimization in the presence of uncertainty, researchers have proposed the concept of multi-scenario efficient set (Botte and Schöbel 2019, Engau and Sigler 2020). We adopt this framework and want to compute, in polynomial time, an approximation of this set with theoretical performance guarantees. Based on a dynamic programming framework, with carefully designed states, we obtain a Fully Polynomial-Time Approximation Scheme (FPTAS) for the problem.
A strong k-edge coloring of a graph G is an assignment of k colors to the edges of G such that for any two edges e and f with distance at most two receive different colors. The minimum number of k such that G has a strong k-edge coloring is called the strong chromatic index of G, denoted as χ '_s(G) . In this paper, we proved that for a subquartic graph G, χ _s'(G)≤ 12 if mad(G)<14/5 , where mad(G)=max{2|E(H)|/|V(H)|,H⊆ G} .
This study investigates the location-routing problem in the pallet pooling system, which is an extended combinatorial optimization problem in the field of logistics management. A mixed-integer linear programming model is formulated for the location-routing problem, incorporating simultaneous pickup and delivery, and the limitations on carbon emissions and transportation time. To address the demand uncertainty, this study extends the model using a robust optimization approach to enhance solution reliability. Due to the NP-hard nature and high computational complexity of the problem, a heuristic algorithm integrating genetic algorithm and ant colony optimization is developed to obtain efficient solutions. Numerical experiments are conducted on classical benchmark instances to validate the model feasibility, demonstrate that the integrated optimization model outperforms the two-stage model, and highlight the superior performance of the heuristic algorithm in terms of solution quality and computational efficiency. Furthermore, the study provides valuable insights for logistics managers in the pallet pooling system.
Unmanned Aerial Vehicles (UAVs) are widely used in environmental monitoring tasks such as forest surveillance, agricultural mapping, and disaster assessment, where their ability to collect dense and high-resolution observations is particularly advantageous. However, many existing approaches implicitly assume that UAV platforms and onboard sensors are homogeneous. This assumption can introduce significant estimation errors in real-world deployments. This work studies the deployment of heterogeneous UAVs equipped with sensors of different accuracies for fire-scene assessment, using a point-of-interest–based model. Research on heterogeneous UAV measurements remains relatively limited. We introduce a weighted frame potential (WFP) that captures the orthogonality structure of sensing matrices under heterogeneous noise. By reformulating the problem as a submodular maximization task, we develop a Random Greedy Algorithm that guarantees a 1/(1+c) approximation of the optimal solution, where c denotes the curvature of the submodular function. A theoretical upper bound on the reconstruction error induced by the selected UAV configuration is also established. Experiments on both small- and large-scale fire monitoring tasks show that the proposed method achieves higher reconstruction accuracy and better computational efficiency than existing algorithms.
Combinatorial group testing (CGT) is used to identify a subset of defective items from a set of items by grouping them together and performing a small number of tests on the groups. Cover-free families (CFFs), also called superimposed codes, are well-studied combinatorial structures used to design the groups in such a way that identifying the defective items from test results (decoding) can be done efficiently. This paper focuses on generalizations of CFFs that take into account a known structure among items to be tested. This structure is modeled by a hypergraph, where vertices are items to be tested and edges represent a predictable relationship among items. A typical application is testing for an infectious disease in a population where there are clusters of individuals with high contact rates, such as households within a neighbourhood or students taking courses within a school. As we aim at minimizing the number of tests, CFFs on hypergraphs yield an interesting combinatorial optimization problem and, like CFFs, have connections to coding theory, design theory and extremal set theory. In this paper, we discuss various types of CFFs on hypergraphs, decoding algorithms, bounds and constructions. In particular, we give several constructions that use the structure and properties of the underlying hypergraph.
