Vertex Cover Research Articles

Parameterized complexity for iterated type partitions and modular-width

This paper deals with the complexity of some natural graph problems parameterized by some measures that are restrictions of clique-width, such as modular-width and neighborhood diversity. We introduce a novel parameter, called iterated type partition number, that can be computed in linear time and nicely places between modular-width and neighborhood diversity. We prove that the Equitable Coloring problem is W[1]-hard when parameterized by the iterated type partition number. This result extends to modular-width, answering an open question on the complexity of Equitable Coloring when parameterized by modular-width. On the contrary, we show that the Equitable Coloring problem is FPT when parameterized by neighborhood diversity.Furthermore, we present a scheme for devising FPT algorithms parameterized by iterated type partition number, which enables us to find optimal solutions for several graph problems. As an example, in this paper, we present algorithms parameterized by the iterated type partition number of the input graph for some generalized versions of the Maximum Clique, Minimum Graph Coloring, (Total) Minimum Dominating Set, Minimum Vertex Cover and Maximum Independent Set problems. Each algorithm outputs not only the optimal value but also the optimal solution. We stress that while the considered problems are already known to be FPT with respect to modular-width, the novel algorithms are both simpler and more efficient. We finally show that the proposed scheme can be used to devise polynomial kernels, with respect to iterated type partition number, for the decisional version of most of the problems mentioned above.

Tracking effects of extreme drought on coniferous forests from space using dynamic habitat indices

Terrestrial ecosystems such as coniferous forests in Central Europe are experiencing changes in health status following extreme droughts compounding with severe heat waves. The increasing temporal resolution and spatial coverage of earth observation data offer new opportunities to assess these dynamics. Dense time-series of optical satellite data allow for computing Dynamic Habitat Indices (DHIs), which have been predominantly used in biodiversity studies. However, DHIs cover three aspects of vegetation changes that could be affected by drought: annual productivity, minimum cover, and seasonality. Here, we evaluate the health status of coniferous forests in the federal state of Hesse in Germany over the period 2017–2020 including the severe drought year of 2018 using DHIs based on the Normalized Difference Vegetation Index (NDVI) for drought assessment. To identify the most important variables affecting coniferous forest die-off, a series of environmental variables together with the three DHIs components were used in a logistic regression (LR) model. Each DHI component changed significantly across non-damaged and damaged sites in all years (p-value 0.05). When comparing 2017 to 2019, DHI-based annual productivity decreased and seasonality increased. Most importantly, none of the DHI components had reached pre-drought conditions, which likely indicates a change in ecosystem functioning. We also identified spatially explicit areas highly affected by drought. The LR model revealed that in addition to common environmental parameters related to temperature, precipitation, and elevation, DHI components were the most important factors explaining the health status. Our analysis demonstrates the potential of DHIs to capture the effect of drought events on Central European coniferous forest ecosystems. Since the spaceborne data are available at the global level, this approach can be applied to track the dynamics of ecosystem conditions in other regions, at larger spatial scales, and for other Land Use/Land Cover types.

Grouped domination parameterized by vertex cover, twin cover, and beyond

A dominating set S of graph G is called an r-grouped dominating set if S can be partitioned into S1,S2,…,Sk such that the size of each unit Si is r and the subgraph of G induced by Si is connected. The concept of r-grouped dominating sets generalizes several well-studied variants of dominating sets with requirements for connected component sizes, such as the ordinary dominating sets (r=1), paired dominating sets (r=2), and connected dominating sets (r is arbitrary and k=1). In this paper, we investigate the computational complexity of r-Grouped Dominating Set, which is the problem of deciding whether a given graph has an r-grouped dominating set with at most k units. For general r, r-Grouped Dominating Set is hard to solve in various senses because the hardness of the connected dominating set is inherited. We thus focus on the case in which r is a constant or a parameter, but we see that r-Grouped Dominating Set for every fixed r>0 is still hard to solve. From the observations about the hardness, we consider the parameterized complexity concerning well-studied graph structural parameters. We first see that r-Grouped Dominating Set is fixed-parameter tractable for r and treewidth, which is derived from the fact that the condition of r-grouped domination for a constant r can be represented as monadic second-order logic (MSO2). This fixed-parameter tractability is good news, but the running time is not practical. We then design an O⁎(min⁡{(2τ(r+1))τ,(2τ)2τ})-time algorithm for general r≥2, where τ is the twin cover number, which is a parameter between vertex cover number and clique-width. For paired dominating set and trio dominating set, i.e., r∈{2,3}, we can speed up the algorithm, whose running time becomes O⁎((r+1)τ). We further argue the relationship between FPT results and graph parameters, which draws the parameterized complexity landscape of r-Grouped Dominating Set.

Earthworms as soil health indicators in no-tillage agroecosystems

No-tillage (NT) and no-tillage systems (NTS) are widely used conservation agriculture practices in Brazil, and in the state of Paraná, nearly 80% of annual crops are cultivated using these methods. Compared with NT, NTS sites permanently include: minimum soil disturbance, soil cover (straw and living vegetation) and crop rotation and diversification with cover crops. These practices often increase earthworm populations, that can be used to indicate soil health. Herein, we review soil health classification of NT sites, and the species found in Paraná state. We compiled information from 130 sites with NT or NTS, located in 29 counties, of which 93 had biomass and 91 had species richness data, aiming to compare NT with NTS sites, and the effect of the age of these practices on earthworm populations. Overall, 29 earthworm species were recorded, of which 17 were native, including many new to science. Mean abundance and biomass in NT + NTS were 104 ind m−2 and 2.5 g m−2, respectively, and richness 2.5 species per site. Abundance was significantly higher in NT than NTS in the initial phase of adoption, and both abundance and biomass decreased with increasing age of NT. Earthworm abundance and species richness were positively correlated with clayey soils and phosphorus content, while higher biomass was associated with soil carbon content and pH. A new classification system was proposed, with the categories poor (<50 ind. m−2, 1 species), moderate (≥50 to <100 ind m−2, 2 species), good (≥100 to <150 ind m−2, 3 species), very good (≥150 to 200 ind m−2, 4 species) and excellent (≥200 ind m−2, ≥5 species) soil health, using earthworm abundance and species richness, respectively. Most of the sites sampled showed poor or moderate soil health, with few displaying very good or excellent health, indicating the need for improved management practices, in order to promote earthworm populations and their potential benefits to soil fertility and plant production.

Parameterized complexity of broadcasting in graphs

The task of the broadcast problem is, given a graph G and a source vertex s, to compute the minimum number of rounds required to disseminate a piece of information from s to all vertices in the graph. It is assumed that, at each round, an informed vertex can transmit the information to at most one of its neighbors. The broadcast problem is known to NP-hard. We show that the problem is FPT when parametrized by the size k of a feedback edge set, or by the size k of a vertex cover, or by k=n−t, where t is the input deadline for the broadcast protocol to complete.

Packing arc-disjoint cycles in oriented graphs

Arc-Disjoint Cycle Packing is a classical NP-complete problem and we study it from two perspectives: (1) by restricting the cycles in the packing to be of a fixed length, and (2) by restricting the inputs to bipartite tournaments. Focusing first on Arc-Disjointr-Cycle Packing (where the cycles in the packing are required to be of length r), we show NP-completeness in oriented graphs with girth r for each r≥3 and study the parameterized complexity of the problem with respect to two parameterizations (solution size and vertex cover size) for r=4 in oriented graphs. Moving on to Arc-Disjoint Cycle Packing in bipartite tournaments, we show that every bipartite tournament either contains k arc-disjoint cycles or has a feedback arc set of size at most 7(k−1). This result adds to the set of Erdös-Pósa-type results known in the combinatorics literature for packing and covering problems.

Semidefinite Approximations for Bicliques and Bi-Independent Pairs

We investigate some graph parameters dealing with bi-independent pairs (A, B) in a bipartite graph [Formula: see text], that is, pairs (A, B) where [Formula: see text], and [Formula: see text] are independent. These parameters also allow us to study bicliques in general graphs. When maximizing the cardinality [Formula: see text], one finds the stability number [Formula: see text], well-known to be polynomial-time computable. When maximizing the product [Formula: see text], one finds the parameter g(G), shown to be NP-hard by Peeters in 2003, and when maximizing the ratio [Formula: see text], one finds h(G), introduced by Vallentin in 2020 for bounding product-free sets in finite groups. We show that h(G) is an NP-hard parameter and, as a crucial ingredient, that it is NP-complete to decide whether a bipartite graph G has a balanced maximum independent set. These hardness results motivate introducing semidefinite programming (SDP) bounds for g(G), h(G), and [Formula: see text] (the maximum cardinality of a balanced independent set). We show that these bounds can be seen as natural variations of the Lovász ϑ-number, a well-known semidefinite bound on [Formula: see text]. In addition, we formulate closed-form eigenvalue bounds, and we show relationships among them as well as with earlier spectral parameters by Hoffman and Haemers in 2001 and Vallentin in 2020. Funding: This work was supported by H2020 Marie Skłodowska-Curie Actions [Grant 813211 (POEMA)].

Circumventing superexponential runtimes for hard instances of quantum adiabatic optimization

Classical optimization problems can be solved by adiabatically preparing the ground state of a quantum Hamiltonian that encodes the problem. The performance of this approach is determined by the smallest gap encountered during the evolution. Here, we consider the maximum independent set problem, which can be efficiently encoded in the Hamiltonian describing a Rydberg atom array. We present a general construction of instances of the problem for which the minimum gap decays superexponentially with system size, implying a superexponentially large time to solution via adiabatic evolution. The small gap arises from locally independent choices which cause the system to initially evolve and localize into a configuration far from the solution in terms of Hamming distance. We investigate remedies to this problem. Specifically, we show that quantum quenches in these models can exhibit signatures of quantum many-body scars, which in turn, can circumvent the superexponential gaps. By quenching from a suboptimal configuration, states with a larger ground-state overlap can be prepared, illustrating the utility of quantum quenches as an algorithmic tool. Published by the American Physical Society 2024

Finding Optimal Solutions with Neighborly Help

AbstractCan we efficiently compute optimal solutions to instances of a hard problem from optimal solutions to neighbor instances, that is, instances with one local modification? For example, can we efficiently compute an optimal coloring for a graph from optimal colorings for all one-edge-deleted subgraphs? Studying such questions not only gives detailed insight into the structure of the problem itself, but also into the complexity of related problems, most notably, graph theory’s core notion of critical graphs (e.g., graphs whose chromatic number decreases under deletion of an arbitrary edge) and the complexity-theoretic notion of minimality problems (also called criticality problems, e.g., recognizing graphs that become 3-colorable when an arbitrary edge is deleted). We focus on two prototypical graph problems, colorability and vertex cover. For example, we show that it is $$\text {NP}$$ NP -hard to compute an optimal coloring for a graph from optimal colorings for all its one-vertex-deleted subgraphs, and that this remains true even when optimal solutions for all one-edge-deleted subgraphs are given. In contrast, computing an optimal coloring from all (or even just two) one-edge-added supergraphs is in $$\text {P}$$ P . We observe that vertex cover exhibits a remarkably different behavior, demonstrating the power of our model to delineate problems from each other more precisely on a structural level. Moreover, we provide a number of new complexity results for minimality and criticality problems. For example, we prove that Minimal-3-UnColorability is complete for $$\text {DP}$$ DP (differences of $$\text {NP}$$ NP sets), which was previously known only for the more amenable case of deleting vertices rather than edges. For vertex cover, we show that recognizing $$\beta $$ β -vertex-critical graphs is complete for $$\Theta _2^\text {p}$$ Θ 2 p (parallel access to $$\text {NP}$$ NP ), obtaining the first completeness result for a criticality problem for this class.

Contribution of offshore platforms and surrounding habitats to fish production in the Bass Strait, south-east Australia

Information on the contribution of offshore oil and gas (O&G) platforms to fish productivity is required to contribute to the decision-making process to remove, partially remove, or retain these structures during decommissioning after petroleum production ceases. The present study assesses the biomass and fish production of one common and abundant fish species (Caesioperca lepidoptera – butterfly perch) and two commercially fished species (Helicolenus percoides – reef ocean perch; Nemadactylus macropterus – jackass morwong) on eight O&G platforms and in surrounding natural habitats in the Bass Strait, south-east Australia, where options for decommissioning are being assessed. High-definition stereo-video imagery was collected by remotely operated vehicle (ROV) from eight platform facilities, their immediate benthic surrounds, reference areas reflective of the likely pre-installation seabed state (sand-dominated) and a nearby natural ‘reef’ area referred to as south-east reef (some limestone foundation). The biomass of all three species was low in the benthic surrounds of platforms, at reference locations and at south-east reef where minimal cover by benthic organisms was recorded and, as such, there was little to no fish production for the three study species in these areas. We observed a total fish biomass of 2.85 tonnes across the eight platforms for the three fish species surveyed, with high variability across platforms. Total production (P) across all platforms was estimated at 1244 kg/year for the three species, with a mean fish production density of 82 g/m2/year. Approximately 79% of total production is considered ‘new’ production (984 kg/year i.e., the production attributed to the presence of the platforms; with a mean production density of 64 g/m2/year). The remaining 21% could be retained or redistributed into the surrounding area if platforms were removed. C. lepidoptera accounted for the majority (90%) of biomass and of total production for all three species across all locations surveyed. Despite only accounting for a small proportion of platform surface area, the bottom 5 m sections of platforms had 41% of the total biomass observed and accounted for 46% of total production of these three study species. Production measures for platforms surveyed here are relatively high compared to other artificial reefs and habitats around the world. Total removal of these platforms will likely result in a reduction of fish biomass and fish productivity (incl. several fishery species) in the immediate vicinity.

Space limited linear-time graph algorithms on big data

We study algorithms for graph problems in which the graphs are of extremely large size N so that super-linear time ω(N) or linear space Θ(N) would become impractical. We use a parameter k to characterize the computational power of a normal computer that can provide additional time and space bounded by polynomials of k in dealing with the large graphs. In particular, we are interested in strict linear-time algorithms using space O(kO(1)). In our case studies, as examples, we present (1) a randomized greedy algorithm of time O(N) and space O(k2) for a parameterized version of the Maximal Matching problem; and (2) randomized kernelization algorithms of time O(N) and space O(kO(1)) for a number of well-known NP-hard problems. Our kernelization algorithms have their kernel sizes match the best kernel sizes by known polynomial-time kernelization algorithms with no space constraints for the problems. We also study the relationship between our proposed model and the streaming model. This study motivates a new streaming kernelization algorithm for the famous Vertex Cover problem that has an optimal update time complexity while matches the best known space complexity of streaming algorithms for the problem.

The Realizability of Theta Graphs as Reconfiguration Graphs of Minimum Independent Dominating Sets

Abstract The independent domination number i(G) of a graph G is the minimum cardinality of a maximal independent set of G, also called an i(G)-set. The i-graph of G, denoted ℐ (G), is the graph whose vertices correspond to the i(G)-sets, and where two i(G)-sets are adjacent if and only if they differ by two adjacent vertices. Not all graphs are i-graph realizable, that is, given a target graph H, there does not necessarily exist a source graph G such that H = ℐ (G). We consider a class of graphs called “theta graphs”: a theta graph is the union of three internally disjoint nontrivial paths with the same two distinct end vertices. We characterize theta graphs that are i-graph realizable, showing that there are only finitely many that are not. We also characterize those line graphs and claw-free graphs that are i-graphs, and show that all 3-connected cubic bipartite planar graphs are i-graphs.

An Attention-Based Method for the Minimum Vertex Cover Problem on Complex Networks

Solving combinatorial problems on complex networks represents a primary issue which, on a large scale, requires the use of heuristics and approximate algorithms. Recently, neural methods have been proposed in this context to find feasible solutions for relevant computational problems over graphs. However, such methods have some drawbacks: (1) they use the same neural architecture for different combinatorial problems without introducing customizations that reflects the specificity of each problem; (2) they only use a nodes local information to compute the solution; (3) they do not take advantage of common heuristics or exact algorithms. Following this interest, in this research we address these three main points by designing a customized attention-based mechanism that uses both local and global information from the adjacency matrix to find approximate solutions for the Minimum Vertex Cover Problem. We evaluate our proposal with respect to a fast two-factor approximation algorithm and a widely adopted state-of-the-art heuristic both on synthetically generated instances and on benchmark graphs with different scales. Experimental results demonstrate that, on the one hand, the proposed methodology is able to outperform both the two-factor approximation algorithm and the heuristic on the test datasets, scaling even better than the heuristic with harder instances and, on the other hand, is able to provide a representation of the nodes which reflects the combinatorial structure of the problem.

Implementing Graph-Theoretic Feature Selection by Quantum Approximate Optimization Algorithm.

Feature selection plays a significant role in computer science; nevertheless, this task is intractable since its search space scales exponentially with the number of dimensions. Motivated by the potential advantages of near-term quantum computing, three graph-theoretic feature selection (GTFS) methods, including minimum cut (MinCut)-based, densest k -subgraph (DkS)-based, and maximal-independent set/minimal vertex cover (MIS/MVC)-based, are investigated in this article, where the original graph-theoretic problems are naturally formulated as the quadratic problems in binary variables and then solved using the quantum approximate optimization algorithm (QAOA). Specifically, three separate graphs are created from the raw feature set, where the vertex set consists of individual features and pairwise measure describes the edge. The corresponding feature subset is generated by deriving a subgraph from the established graph using QAOA. For the above three GTFS approaches, the solving procedure and quantum circuit for the corresponding graph-theoretic problems are formulated with the framework of QAOA. In addition, those proposals could be employed as a local solver and integrated with the Tabu search algorithm for solving large-scale GTFS problems utilizing limited quantum bit resource. Finally, extensive numerical experiments are conducted with 20 publicly available datasets and the results demonstrate that each model is superior to its classical scheme. In addition, the complexity of each model is only O(p n2) even in the worst cases, where p is the number of layers in QAOA and n is the number of features.

Vertex Cover Hop Dominating Sets in Graphs

Let $G$ be a graph. Then a subset $C$ of vertices of $G$ is called a vertex cover hop dominating if $C$ is both a vertex cover and a hop dominating of $G$. The vertex cover hop domination number of $G$, denoted by $\gamma_{vch}(G)$, is the minimum cardinality among all vertex cover hop dominating sets in $G$. In this paper, we initiate the study of vertex cover hop domination in a graph and we determine its relations with other parameters in graph theory. We characterize the vertex cover hop dominating sets in some special graphs, join, and corona of two graphs and we finally obtain the exact values or bounds of the parameters of these graphs.

Quantum computing dataset of maximum independent set problem on king lattice of over hundred Rydberg atoms

Finding the maximum independent set (MIS) of a large-size graph is a nondeterministic polynomial-time (NP)-complete problem not efficiently solvable with classical computations. Here, we present a set of quantum adiabatic computing data of Rydberg-atom experiments performed to solve the MIS problem of up to 141 atoms randomly arranged on the king lattice. A total of 582,916 events of Rydberg-atom measurements are collected for experimental MIS solutions of 733,853 different graphs. We provide the raw image data along with the entire binary determinations of the measured many-body ground states and the classified graph data, to offer bench-mark testing and advanced data-driven analyses for validation of the performance and system improvements of the Rydberg-atom approach.

A telepítési sűrűség és a lombkorona-borítottság vizsgálata budapesti szabadtereken

Trees in public open spaces are an important part of the urban green space system, which has to adapt to increasingly difficult conditions due to urbanisation, climate change and human activity. Green spaces are a major factor in improving the microclimate, helping to protect biodiversity and playing an important aesthetic role, so they need to be preserved and enhanced. In our country, since the 2000s, a large number of studies and civil initiatives have highlighted the importance of green spaces, but typically from a quantitative approach. However, improving the quality of planting is an equally important task. Our research assesses the canopy cover of trees in open spaces in Budapest in terms of planting density, with a focus on the planting distance of trees in certain squares and streets and the resulting cover values. In our studies, we have considered the quantitative and qualitative variation in crown canopy cover at the time of planting and in maturity, so we can formulate ideal planting suggestions based on the environment, requirements and characteristics (crown shape, growth vigour) of each taxa. In the series of case studies, we first studied the woody species of one of the most important intermodal nodes of Budapest, Móricz Zsigmond Square and Bartók Béla Street. The canopy cover of the square reaches the ideally defined minimum canopy cover of 25-30% in 5-7 years. Considering canopy cover and planting density, it was found that in 2023 the canopy cover of the trees in the Móricz Zsigmond square averages 13%, while at maturity it will cover more than 50%. This means that about half of the tree crowns will be able to have the highest potential impact in terms of ecosystem services.

Evaluation of Chloride Ion Attack in Self-Compacting Concrete Using Recycled Construction and Demolition Waste Aggregates

Construction and demolition waste (CDW) destined for recycling companies has great potential for use in civil construction, since it gives rise to recycled aggregates of different particle sizes that can be used in concrete. However, there is a lack of studies on the durability of concrete produced with recycled aggregates from CDW. This study analyzed the influence of incorporating recycled aggregates from CDW, sand, and gravel on the durability parameters of SCC mixtures, with and without the addition of metakaolin (MK), when subjected to two exposure conditions: outdoors and in cycles of attack by chloride ions. Five mixtures were produced: reference SCC, with natural sand and gravel; SCC with recycled sand and gravel; SCC with recycled sand and gravel and the addition of 10% MK; SAC with recycled sand, natural gravel, and the addition of 10% MK; and SCC with natural sand, recycled gravel and the addition of 10% MK. The water/binder ratio was kept constant for all mixtures and the additive dosage was adjusted according to the variation in the use of aggregates. The mechanical and durability properties were assessed using axial compressive strength, ultrasonic pulse velocity, chloride penetration, chloride ion diffusion, and electrical resistivity tests. The results showed the feasibility of using recycled aggregates from CDW in SCC. The addition of MK significantly improved the performance of SCC using these aggregates. The mixtures with added MK showed a low risk of corrosion and high resistance to chloride ion penetration, and, under highly aggressive attack conditions, it was observed that the chloride ions did not exceed the minimum cover thickness recommended for reinforced concrete structures. The addition of MK to the mix with recycled aggregates caused an 84.6% reduction in the Cl− diffusion coefficient, there was also a 40.3% reduction in Cl− penetration and an increase of up to 156.14% in electrical resistivity compared to the mix with recycled aggregates without the addition of MK. The SCC mix with recycled sand and metakaolin stood out positively compared to the others, achieving an axial compressive strength similar to the reference mix (55.10 MPa). We, therefore, conclude that it is possible to produce such a mix with acceptable performance and ensure good behavior under aggressive environmental conditions.

Three remarks on W2 graphs

Let k≥1. A graph G is Wk if for any k pairwise disjoint independent vertex subsets A1,…,Ak in G, there exist k pairwise disjoint maximum independent sets S1,…,Sk in G such that Ai⊆Si for i∈[k]. Recognizing W1 graphs is coNP-hard, as shown by Chvátal and Slater (1993) and, independently, by Sankaranarayana and Stewart (1992). Extending this result and answering a recent question of Levit and Tankus, we show that recognizing Wk graphs is coNP-hard for k≥2. On the positive side, we show that recognizing Wk graphs is, for each k≥2, FPT parameterized by clique-width and by tree-width. Finally, we construct graphs G that are not W2 such that, for every vertex v in G and every maximal independent set S in G−N[v], the largest independent set in N(v)∖S consists of a single vertex, thereby refuting a conjecture of Levit and Tankus.

Offensive alliances in graphs

The Offensive Alliance problem has been studied extensively during the last twenty years. A set S⊆V of vertices is an offensive alliance in an undirected graph G=(V,E) if each v∈N(S) has at least as many neighbors in S as it has neighbors (including itself) not in S. We study the classical and parameterized complexity of the Offensive Alliance problem, where the aim is to find a minimum size offensive alliance. We enhance our understanding of the problem from the viewpoint of parameterized complexity by showing that (1) the problem is W[1]-hard parameterized by a wide range of fairly restrictive structural parameters such as the feedback vertex set number, treewidth, pathwidth, and treedepth of the input graph; this puts it among the few problems that are FPT when parameterized by solution size but not when parameterized by treewidth (unless FPT=W[1]), (2) the problem cannot be solved in time O⁎(2o(klog⁡k)) where k is the solution size, unless ETH fails, (3) it does not admit a polynomial kernel parameterized by solution size and vertex cover of the input graph. On the positive side we prove that (4) it can be solved in time O⁎(vc(G)O(vc(G))) where vc(G) is the vertex cover number of the input graph. (5) it admits an FPT algorithm when parameterized by vertex integrity of input graph. In terms of classical complexity, we prove that (6) the problem cannot be solved in time 2o(n) even when restricted to bipartite graphs, unless ETH fails, (7) it cannot be solved in time 2o(n) even when restricted to apex graphs, unless ETH fails. We also prove that (8) it is NP-complete even when restricted to bipartite, chordal, split and circle graphs.