Let T be a perfect binary tree and I be its edge ideal in the polynomial ring S. We determine the vertex cover number, the independence number, and establish the recursive formula to compute the number of minimal vertex covers. As a consequence, we compute the depth and projective dimension of S / I and show that the total Betti number of S / I at the highest homological degree always equals one.

Graph neural networks (GNNs) with unsupervised learning can provide high-quality approximate solutions to large-scale combinatorial optimization problems (COPs) with efficient time complexity, making them versatile for various applications. However, since this method maps the COP to the training process of a GNN, and the current mainstream backpropagation-based training algorithms are prone to falling into local minima, the optimization performance is still inferior to the current state-of-the-art (SOTA) COP methods. To address this issue, inspired by the possibility of learning through chaotic dynamics of the real brain, we introduce a chaotic training algorithm, i.e., chaotic graph backpropagation (CGBP), which introduces a local loss function in GNN that makes the training process not only chaotic but also highly efficient. Different from existing methods, we show that the global ergodicity and pseudorandomness with fractal structure of such chaotic dynamics enable CGBP to learn GNNs effectively and globally, thus solving the COP efficiently. We have applied CGBP to solve various COPs, such as the maximum independent set (MIS), maximum cut (MC), and graph coloring (GC). Results on several large-scale benchmark datasets showcase that CGBP can compete with or outperform SOTA methods. In addition, CGBP can be easily integrated into any existing learning method as an additional universal plug-in module to improve the searching ability and performance.

In this paper we consider the problem of efficiently finding a (small) set of stocks taken from an index that can replicate the index performance. Furthermore, we add the requirement that the set’s returns have weak correlation with each other. Such a selection of stocks may be useful for investors who want to simplify their analysis of the stock index, trying to capture market movement with reduced risk. To solve this problem, we use maximum independent set, a concept from graph theory. As a case study we consider IDX80 in the year 2024.

Abstract. We present egenioussBench, a visual localisation benchmark built on geospatial reference data: a city-scale airborne 3D mesh and a CityGML LoD2 model. This pairing reflects deployable mapping assets and supports true scalability beyond traditional SfM-based approaches. The query data comprise smartphone images with centimetre-accurate, map-independent ground truth obtained via PPK and GCP/CP-aided adjustment. From 2,709 images, we derive a non-co-visible subset by estimating the full co-visibility matrix from rendered depth and selecting a maximum independent set; the released data include a test split of 42 non-co-visible images with withheld ground truth and a validation split of 412 sequential images with poses, e.g. for training of pose regressors and self-validation. The benchmark features a public leaderboard evaluated with binning metrics at multiple pose-error thresholds alongside global statistics (median, RMSE, outlier ratio), ensuring fair, like-for-like comparison across mesh- and LoD2-based methods. Together, these design choices expose realistic cross-view and cross-domain challenges while providing a rigorous, scalable path for advancing large-scale visual localisation. We make the evaluation code and data availeable at https://github.com/fratopa/egenioussBench and https://www.egeniouss.eu/

Maximum Independent Set when Excluding an Induced Minor: $$K_1 + tK_2$$ and $$tC_3 \uplus C_4$$

Abstract Quantum annealing (QA) offers a promising approach for solving constrained combinatorial optimization on near-term quantum devices. It encodes solutions into the ground states of the Ising problem Hamiltonians through penalty terms and penalty parameters to enforce constraints. We propose a variational determination framework to address the issue of penalty parameter selection through three progressively generalized methods: the Frozen method variationally adjusts parameters in a tunable Hamiltonian to prepare evolved states minimizing the energy of a target Hamiltonian with frozen parameters; the Time‑Transfer method applies optimized parameters to longer annealing times; and the Full‑Transfer method extends this approach across both system sizes and annealing durations. The effectiveness of these methods stems from energy minimization steering the evolution toward the low-energy subspace. Evaluated on vertex cover problems over 40 randomly generated 12‑vertex 3‑regular graphs, the Frozen method improves the average single‑run success probability from 0.27^{±0.02} to 0.62^{±0.03} at ta = 10, and reduces the average number of runs required for 99.9% success probability by 3.8 times at ta = 2 (including optimization overhead). Crucially, both Time‑Transfer and Full‑Transfer methods closely match this single-run success probability improvements and 6.63-time and 6.4-time reductions in required runs at ta = 2.5 to reach 99.9% success probability, where Time-Transfer excludes the initial optimization cost at ta = 2 while Full-Transfer provides end-to-end acceleration. Our variational framework demonstrates that short-time quantum annealing in optimization can achieve performances comparable to longer schedules, offering a practical approach for current quantum devices.

The Maximum Independent Set (MaxIS) problem is a well-known NP-hard problem. This paper presents a fixed order configuration deterministic algorithm that improves complexities by achieving a time complexity of O ( m + nlog ⁡ n ) , which simplifies to O ( nlog ⁡ n ) for sparse graphs and O ( n 2 ) for dense graphs. Experimental results in DIMACS and other benchmark datasets confirm that our algorithm not only delivers faster execution, but also maintains high solution quality, making it a superior choice for large-scale MaxIS problems.

Traditional visual SLAM methods are built on the strong assumption that the system operates in static environments, with limited consideration of moving objects. This assumption often leads to significant performance degradation when dynamic elements are present. To mitigate the impact of moving objects and enhance both localization accuracy and mapping quality, we propose a visual SLAM framework that explicitly removes dynamic object interference from the visual odometry and mapping modules. First, we refine the data association process in visual odometry by introducing motion consistency constraints, which reduce incorrect feature matches and thereby improve pose estimation accuracy. At the same time, depth information from RGB-D sensors is used to validate potentially dynamic feature points. Second, within the mapping module, we formulate keyframe selection as a vertex cover problem to ensure the local representativeness of keyframes. This approach not only reduces mapping artifacts but also enables the comprehensive detection and removal of dynamic objects. Finally, experiments conducted on the TUM RGB-D dataset demonstrate that our system achieves higher accuracy, robustness, and stability compared to baseline methods.

Maximum Independent Sets Using Hybrid Approach of Grey Wolf Optimizer and Genetic Algorithm

ABSTRACT Recently, Hadfield et al. proposed the quantum alternating operator ansatz algorithm (QAOA+), an extension of the quantum approximate optimization algorithm (QAOA), to solve constrained combinatorial optimization problems (CCOPs). Compared with QAOA, QAOA+ enables the search for optimal solutions within a feasible solution space by encoding problem constraints into the mixer Hamiltonian, thereby reducing the search space and eliminating the possibility of yielding infeasible solutions. However, QAOA+ may incur high overall gate costs when the mixer is applied to all qubits in each layer, and each mixer is costly to implement. To address this challenge, an adaptive mixer allocation strategy is tailored for QAOA+. The resulting algorithm, which integrates this strategy into the original QAOA+ framework, is referred to as AMA‐QAOA+. Unlike QAOA+, AMA‐QAOA+ adaptively applies the mixer to a subset of qubits in each layer of the mixer unitary operator based on an evaluation function. The performance of AMA‐QAOA+ is evaluated on the maximum independent set problem. Numerical simulation results show that, under the same number of optimization runs, AMA‐QAOA+ achieves better solution quality than QAOA+, with the optimal approximation ratio improved by on Erdős–Rényi random graphs and on 3‐regular graphs. Moreover, AMA‐QAOA+ significantly reduces the CNOT gate consumption, requiring only and of the CNOT gates used by QAOA+ on Erdős–Rényi and 3‐regular random graphs, respectively. These results demonstrate that AMA‐QAOA+ enhances solution quality and computational efficiency, enabling the design of more compact and resource‐efficient quantum circuits.

Let denote the minimum cardinality among all maximal independent sets of containing . Then is called the independent dom-saturation number of . In this paper, we study the effect of removal of an edge on the independent dom-saturation number of a graph. We also study the concept of -subdivision number of a graph and initiate a study of edge independent dom-saturation number of a graph

A vertex cover \( S \subseteq V(G) \) is called a \textit{$2$-path geodetic vertex cover} of \( G \) if for every \( v \in V(G) \setminus S \), there exist vertices \( u, w \in S \) such that $d_G(u,w) = 2$ and \( v \in I_G(u, w) \). The \textit{$2$-path geodetic vertex covering number} of \( G \), denoted \( \beta_{2pg}(G) \), is the minimum cardinality of a $2$-path geodetic vertex covering of \( G \). In this paper, we show that given two positive integer $a$ and $b$ such that $2\leq a\leq b,$ there exist a connected graph $G$ such that $\beta(G)=a$ and $\beta_{2pg}=b.$ As a consequence, the difference between the $2$-path geodetic vertex covering number and the classical vertex covering number of a graph can be made arbitrarily large. We characterize graphs with small and large values of the $2$-path closure absorbing vertex covering number. Furthermore, we provide necessary and sufficient conditions for the $2$-path geodetic vertex covers in certain graph operations. The exact values of $2$-path geodetic vertex cover numbers of these graphs are also determined.

Mobile agents on chordal graphs: Maximum independent set and beyond

Let n = p 1 n 1 p 2 n 2 … p t n t be a positive composite integer with distinct primes p 1 , …, p t . Consider the graph G′ n , introduced by Rather and Ganie [ RAIRO-Oper. Res. 59 (2025) 1605–1616], whose vertices are the proper divisors of n and where two vertices are adjacent if and only if they are coprime. A complete description of maximal independent sets in G′ n is obtained via a correspondence with maximal intersecting families of subsets of [ t ]. This correspondence is applied to determine the independence number for various classes of n , including the square-free case, where the exact value and an explicit formula are established. For general n with at least one exponent n i > 1, the independence number is expressed in terms of combinatorial properties of intersecting families, yielding a unified approach to both the structural characterization and enumeration of maximal independent sets.

Eternal connected vertex cover problem in graphs: Complexity and algorithms

Computing eternal vertex cover number of maximal outerplanar graphs in linear time

A lower bound for the energy of graphs in terms of the vertex cover number

3-Path Vertex Cover Problem based on the Variable Neighborhood Search algorithm and the Artificial Bee Colony algorithm

Max-SAT with cardinality constraint ( CC-Max-Sat ) is one of the classical NP-complete problems, that generalizes Maximum Coverage , Partial Vertex Cover , Max-2-SAT with bisection constraints, and has been extensively studied across all algorithmic paradigms. In this problem, we are given a CNF formula \(\Phi\) , and a positive integer \( k \) , and the goal is to find an assignment \(\beta\) with at most \( k \) variables set to true (also called a \( k \) -weight assignment) such that the number of clauses satisfied by \(\beta\) is maximized. The problem is known to admit an approximation algorithm with factor \(1-\frac{1}{e}\) , which is probably optimal. Furthermore, assuming Gap-Exponential Time Hypothesis (Gap-ETH), for any \(\epsilon > 0\) and any function \( h \) , no \(h(k)(n+m)^{o(k)}\) time algorithm can approximate Maximum Coverage (a monotone version of CC-Max-Sat ) with \( n \) elements and \( m \) sets to within a factor \((1-\frac{1}{e}+\epsilon)\) , even with a promise that there exist \( k \) sets that fully cover the whole universe. In fact, the problem is hard to approximate within 0.929, assuming Unique Games Conjecture, even when the input formula is 2-CNF. These intractable results lead us to explore families of formula, where we can circumvent these barriers. Toward this, we consider \(K_{d,d}\) -free formulas (that is, the clause-variable incidence bipartite graph of the formula excludes \(K_{d,d}\) as an induced subgraph). We show that for every \(\epsilon > 0\) , there exists an algorithm for CC-Max-Sat on \(K_{d,d}\) -free formulas with approximation ratio \((1-\epsilon)\) and running in time \(2^{{\mathcal{O}}((\frac{dk}{\epsilon})^{d})}(n+m)^{{\mathcal{O}}(1)}\) (these algorithms are called FPT-AS). For Maximum Coverage on \(K_{d,d}\) -free set families, we obtain FPT-AS with running time \((\frac{dk}{\epsilon})^{{\mathcal{O}}(dk)}n^{{\mathcal{O}}(1)}\) . Our second result considers “optimizing \( k \) ,” with fixed covering constraint for the Maximum Coverage problem. To explain our result, we first recast the Maximum Coverage problem as the Max Red Blue Dominating Set with Covering Constraint problem. Here, the input is a bipartite graph \(G=(A,B,E)\) , a positive integer \( t \) , and the objective is to find a minimum sized subset \(S\subseteq A\) , such that \(|N(S)|\) (the size of the set of neighbors of \( S \) ) is at least \( t \) . We design an additive approximation algorithm for Max Red Blue Dominating Set with Covering Constraint , on \(K_{d,d}\) -free bipartite graphs, running in FPT time. In particular, if \( k \) denotes the minimum size of \(S\subseteq A\) , such that \(|N(S)|\geq t\) , then our algorithm runs in time \((kd)^{{\mathcal{O}}(kd)}n^{{\mathcal{O}}{(1)}}\) and returns a set \(S^{\prime}\) such that \(|N(S^{\prime})|\geq t\) and \(|S^{\prime}|\leq k+1\) . This is in sharp contrast to the fact that, even a special case of our problem, namely, the Partial Vertex Cover problem (or Max \( k \) -VC ) is W[1]-hard, parameterized by \( k \) . Thus, we get the best possible parameterized approximation algorithm for the Maximum Coverage problem on \(K_{d,d}\) -free bipartite graphs.

Context The threatened subspecies of thick-billed grasswren, Amytornis modestus raglessi, occupies dense chenopod shrublands on the lower slopes and peripheral drainages of the North Flinders Ranges, South Australia. A decline in grasswren numbers was observed around 2012, after two preceding years of exceptionally high rainfall. Profound reduction in observed numbers was also evident in 2019, the second successive year of exceptionally hot and dry conditions, in areas where they had previously been numerous. Aims To identify environmental factors influencing A. m. raglessi habitat suitability using habitat suitability modelling, and to better understand possible drivers of grasswren decline and distributional changes between pre- and post-2012 periods. Methods Random forest modelling was used to predict grasswren habitat suitability in response to mapped environmental variables including remotely sensed vegetation, soil and landscape properties. Habitat suitability maps were produced for two separate periods, 1994–2011 and 2012–2023, and compared. An ornithological field survey was undertaken to validate the modelling, and vegetation time-series used to examine areas showing contrasting habitat suitability changes. Key results Mapped soil properties and the minimum green vegetation cover value were the most important habitat suitability predictors. The overall predicted area of habitat (suitability >50%) declined by 25% between the 1994–2011 and 2012–2023 periods. Changes included an expansion of high-suitability habitat in the west, and habitat contraction in south-eastern areas of the distribution. Time-series vegetation data showed that lower bare ground cover and higher non-green vegetation cover occurred in an area with marked reduction in predicted habitat suitability. Conclusions Habitat suitability modelling successfully identified key environmental drivers and demonstrated habitat shifts between periods. Soil properties and minimum green vegetation cover confirmed that water stress responses are fundamental to grasswren distribution. Modelling identified areas of habitat contraction that highlight conservation priorities, while also demonstrating management effectiveness in improving habitat quality. Implications These methods provide spatially explicit guidance for prioritizing conservation efforts for this subspecies and thick-billed grasswrens broadly. Demonstrated habitat improvement following reduced grazing at Witchelina illustrates the practical value of this modeling approach. Such methods are increasingly essential for land managers to understand biodiversity responses and species distributions under climate change.