Vertex Cover Research Articles

Node and edge averaged complexities of local graph problems

We continue the recently started line of work on the distributed node-averaged complexity of distributed graph algorithms. The node-averaged complexity of a distributed algorithm running on a graph G=(V,E) is the average over the times at which the nodes V of G finish their computation and commit to their outputs. We study the node-averaged complexity for some of the central distributed symmetry breaking problems and provide the following results (among others). As our main result, we show that the randomized node-averaged complexity of computing a maximal independent set (MIS) in n-node graphs of maximum degree Delta is at least Omega big (min big {frac{log Delta }{log log Delta },sqrt{frac{log n}{log log n}}big }big ). This bound is obtained by a novel adaptation of the well-known lower bound by Kuhn, Moscibroda, and Wattenhofer [JACM’16]. As a side result, we obtain that the worst-case randomized round complexity for computing an MIS in trees is also Omega big (min big {frac{log Delta }{log log Delta },sqrt{frac{log n}{log log n}}big }big )—this essentially answers open problem 11.15 in the book by Barenboim and Elkin and resolves the complexity of MIS on trees up to an O(sqrt{log log n}) factor. We also show that, perhaps surprisingly, a minimal relaxation of MIS, which is the same as (2, 1)-ruling set, to the (2, 2)-ruling set problem drops the randomized node-averaged complexity to O(1). For maximal matching, we show that while the randomized node-averaged complexity is Omega big (min big {frac{log Delta }{log log Delta },sqrt{frac{log n}{log log n}}big }big ), the randomized edge-averaged complexity is O(1). Further, we show that the deterministic edge-averaged complexity of maximal matching is O(log ^2Delta + log ^* n) and the deterministic node-averaged complexity of maximal matching is O(log ^3Delta + log ^* n). Finally, we consider the problem of computing a sinkless orientation of a graph. The deterministic worst-case complexity of the problem is known to be Theta (log n), even on bounded-degree graphs. We show that the problem can be solved deterministically with node-averaged complexity O(log ^* n), while keeping the worst-case complexity in O(log n).

Pseudorandom sets in Grassmann graph have near-perfect expansion

We prove that pseudorandom sets in Grassmann graph have near-perfect expansion. This completes the proof of the $2$-to-$2$ Games Conjecture (albeit with imperfect completeness). Some implications of this new result are improved hardness results for Minimum Vertex Cover, improving on the work of Dinur and Safra [Ann. of Math. ${\bf 162}$ (2005), 439--485], and new hardness gaps for Unique-Games. The Grassmann graph ${\sf Gr}_{\sf{global}}$ contains induced subgraphs ${\sf Gr}_{\sf{local}}$ that are themselves isomorphic to Grassmann graphs of lower orders. A set is called pseudorandom if its density is $o(1)$ inside all subgraphs ${\sf Gr}_{\sf{local}}$ whose order is $O(1)$ lower than that of ${\sf Gr}_{\sf{global}}$. We prove that pseudorandom sets have expansion $1-o(1)$, greatly extending the results and techniques of a previous work of the authors with Dinur and Kindler.

Quantum speedup for solving the minimum vertex cover problem based on Grover search algorithm

Quantum speedup for solving the minimum vertex cover problem based on Grover search algorithm

Properties of symbolic powers of edge ideals of weighted oriented graphs

Let [Formula: see text] be a weighted oriented graph and [Formula: see text] be its edge ideal. We provide one method to find all the minimal generators of [Formula: see text], where [Formula: see text] is a maximal strong vertex cover of [Formula: see text] and [Formula: see text] is the intersections of irreducible ideals associated to the strong vertex covers contained in [Formula: see text]. If [Formula: see text] is an induced digraph of [Formula: see text], under a certain condition on the strong vertex covers of [Formula: see text] and [Formula: see text], we show that [Formula: see text] for some [Formula: see text] implies [Formula: see text]. We provide the necessary and sufficient condition for the equality of ordinary and symbolic powers of edge ideal of the union of two naturally oriented paths with a common sink vertex. We characterize all the maximal strong vertex covers of [Formula: see text] such that at most one edge is oriented into each of its vertices and [Formula: see text] if [Formula: see text] for all [Formula: see text]. Finally, if [Formula: see text] is a weighted rooted tree with the degree of root is [Formula: see text] and [Formula: see text] when [Formula: see text] for all [Formula: see text], we show that [Formula: see text] for all [Formula: see text].

Long-term durability of underground reinforced concrete pipes in natural chloride and carbonation environments

In this study, concrete cores extracted from different underground reinforced concrete pipelines exposed to chloride and carbonation environments were investigated. The age of the pipelines was ranging from 43 to 70 years. The main purpose of this study was to determine the minimum concrete cover to reinforcement allowing 100 years of service life with respect to the exposure conditions of the pipelines. Chloride content at the steel–concrete interface, the carbonation depth, the reinforcement corrosion and other concrete durability related properties were assessed. The service-life of the pipelines was thoroughly discussed based on both corrosion rates measured and corrosion induced concrete cracking observed. The high humidity in concrete pipes is suspected to greatly delay oxygen renewal at the steel–concrete interface. Concrete cover of 25 mm and 20 mm seem adequate for 100 years of service life in chloride and carbonation environments respectively.

Approximation Algorithms for Partial Vertex Covers in Trees

This paper is concerned with designing algorithms for and analyzing the computational complexity of the partial vertex cover problem in trees. Graphs (and trees) are frequently used to model risk management in various systems. In particular, Caskurlu et al. in [4] have considered a system which essentially represents a tripartite graph. The goal in this model is to reduce the risk in the system below a predefined risk threshold level. It can be shown that the main goal in this risk management system can be formulated as a Partial Vertex Cover problem on bipartite graphs. In this paper, we focus on a special case of the partial vertex cover problem, when the input graph is a tree. We consider four possible versions of this setting, depending on whether or not, the vertices and edges are weighted. Two of these versions, where edges are assumed to be unweighted, are known to be polynomial-time solvable. However, the computational complexity of this problem with weighted edges, and possibly with weighted vertices, remained open. The main contribution of this paper is to resolve these questions by fully characterizing which variants of partial vertex cover remain NP-hard in trees, and which can be solved in polynomial time. In the paper, we propose two pseudo-polynomial DP-based algorithms for the most general case in which weights are present on both the edges and the vertices of the tree. One of these algorithms leads to a polynomial-time procedure, when weights are confined to the edges of the tree. The insights used in this algorithm are combined with additional scaling ideas to derive an FPTAS for the general case. A secondary contribution of this work is to propose a novel way of using centroid decompositions in trees, which could be useful in other settings as well.

Efficiently Approximating Vertex Cover on Scale-Free Networks with Underlying Hyperbolic Geometry

Finding a minimum vertex cover in a network is a fundamental NP-complete graph problem. One way to deal with its computational hardness, is to trade the qualitative performance of an algorithm (allowing non-optimal outputs) for an improved running time. For the vertex cover problem, there is a gap between theory and practice when it comes to understanding this trade-off. On the one hand, it is known that it is NP-hard to approximate a minimum vertex cover within a factor of sqrt{2}. On the other hand, a simple greedy algorithm yields close to optimal approximations in practice. A promising approach towards understanding this discrepancy is to recognize the differences between theoretical worst-case instances and real-world networks. Following this direction, we narrow the gap between theory and practice by providing an algorithm that efficiently computes nearly optimal vertex cover approximations on hyperbolic random graphs; a network model that closely resembles real-world networks in terms of degree distribution, clustering, and the small-world property. More precisely, our algorithm computes a (1 + o(1))-approximation, asymptotically almost surely, and has a running time of {mathcal {O}}(m log (n)). The proposed algorithm is an adaptation of the successful greedy approach, enhanced with a procedure that improves on parts of the graph where greedy is not optimal. This makes it possible to introduce a parameter that can be used to tune the trade-off between approximation performance and running time. Our empirical evaluation on real-world networks shows that this allows for improving over the near-optimal results of the greedy approach.

Parameterized Algorithms for Colored Clustering

In the Colored Clustering problem, one is asked to cluster edge-colored (hyper-)graphs whose colors represent interaction types. More specifically, the goal is to select as many edges as possible without choosing two edges that share an endpoint and are colored differently. Equivalently, the goal can also be described as assigning colors to the vertices in a way that fits the edge-coloring as well as possible. As this problem is NP-hard, we build on previous work by studying its parameterized complexity. We give a 2ᴼ⁽ᵏ⁾·nᴼ⁽¹⁾-time algorithm where k is the number of edges to be selected and n the number of vertices. We also prove the existence of a problem kernel of size O(k⁵ᐟ²), resolving an open problem posed in the literature. We consider parameters that are smaller than k, the number of edges to be selected, and r, the number of edges that can be deleted. Such smaller parameters are obtained by considering the difference between k or r and some lower bound on these values. We give both algorithms and lower bounds for Colored Clustering with such parameterizations. Finally, we settle the parameterized complexity of Colored Clustering with respect to structural graph parameters by showing that it is W[1]-hard with respect to both vertex cover number and tree-cut width, but fixed-parameter tractable with respect to local feedback edge number.

The Parameterized Complexity of Network Microaggregation

Microaggregation is a classical statistical disclosure control technique which requires the input data to be partitioned into clusters while adhering to specified size constraints. We provide novel exact algorithms and lower bounds for the task of microaggregating a given network while considering both unrestricted and connected clusterings, and analyze these from the perspective of the parameterized complexity paradigm. Altogether, our results assemble a complete complexity-theoretic picture for the network microaggregation problem with respect to the most natural parameterizations of the problem, including input-specified parameters capturing the size and homogeneity of the clusters as well as the treewidth and vertex cover number of the network.

Correction to: Powers of the vertex cover ideals (Collect. Math. 65 (2014) 169–181)

We describe a combinatorial condition on a graphwhich guarantees that all powers of its vertex cover ideal are componentwise linear. Then motivated by Eagon and Reiner’s Theorem we study whether all powers of the vertex cover ideal of a Cohen-Macaulay graph have linear free resolutions. After giving a complete characterization of Cohen-Macaulay cactus graphs (i.e., connected graphs in which each edge belongs to at most one cycle) we show that all powers of their vertex cover ideals have linear resolutions.

A Distributed-GPU Deep Reinforcement Learning System for Solving Large Graph Optimization Problems

Graph optimization problems (such as minimum vertex cover, maximum cut, traveling salesman problems) appear in many fields including social sciences, power systems, chemistry, and bioinformatics. Recently, deep reinforcement learning (DRL) has shown success in automatically learning good heuristics to solve graph optimization problems. However, the existing RL systems either do not support graph RL environments or do not support multiple or many GPUs in a distributed setting. This has compromised the ability of reinforcement learning in solving large-scale graph optimization problems due to lack of parallelization and high scalability. To address the challenges of parallelization and scalability, we develop RL4GO , a high-performance distributed-GPU DRL framework for solving graph optimization problems. RL4GO focuses on a class of computationally demanding RL problems, where both the RL environment and policy model are highly computation intensive. Traditional reinforcement learning systems often assume either the RL environment is of low time complexity or the policy model is small. In this work, we distribute large-scale graphs across distributed GPUs and use the spatial parallelism and data parallelism to achieve scalable performance. We compare and analyze the performance of the spatial parallelism and data parallelism and show their differences. To support graph neural network (GNN) layers that take as input data samples partitioned across distributed GPUs, we design parallel mathematical kernels to perform operations on distributed 3D sparse and 3D dense tensors. To handle costly RL environments, we design a parallel graph environment to scale up all RL-environment-related operations. By combining the scalable GNN layers with the scalable RL environment, we are able to develop high-performance RL4GO training and inference algorithms in parallel. Furthermore, we propose two optimization techniques—replay buffer on-the-fly graph generation and adaptive multiple-node selection—to minimize the spatial cost and accelerate reinforcement learning. This work also conducts in-depth analyses of parallel efficiency and memory cost and shows that the designed RL4GO algorithms are scalable on numerous distributed GPUs. Evaluations on large-scale graphs show that (1) RL4GO training and inference can achieve good parallel efficiency on 192 GPUs, (2) its training time can be 18 times faster than the state-of-the-art Gorila distributed RL framework [ 34 ], and (3) its inference performance achieves a 26 times improvement over Gorila.

Untangling temporal graphs of bounded degree

In this contribution we consider a variant of the vertex cover problem in temporal graphs that has been recently introduced to summarize timeline activities in social networks. The problem is NP-hard, even when the time domain considered consists of two timestamps. We further analyze the complexity of this problem, focusing on temporal graphs of bounded degree. We prove that the problem is NP-hard when (1) each vertex has degree at most one in each timestamp and (2) each vertex is connected with at most three neighbors, has degree at most two in each timestamp and the time domain consists of three timestamps. On the other hand, we prove that the problem is in P when each vertex is connected with at most two neighbors. Then we present a fixed-parameter algorithm for the restriction where we bound the number of interactions in each timestamp and the length of the interval where a vertex has incident temporal edges.

Using PlanetScope images to investigate the evolution of small glaciers in the Alps

Small glaciers (having an area smaller than 1 km2) dominate in terms of total number in the Alps and it is important to study their evolution. In this study we assessed the potential of daily, high resolution (3 m pixel size) PlanetScope images for investigating small Alpine glaciers. We compared the evolution of five small glaciers, covering an area lower or slightly higher than 1 km2, located in different areas of the Alps and having different characteristics and meteorological conditions. The investigated glaciers are: Bors (0.645 km2, Piedmont, Italy), Seewjinen (1.339 km2, Valais, Switzerland), Pizzo Ferrè (0.442 km2, Lombardy, Italy), Gran Zebrù (0.579 km2, Lombardy, Italy) and Solda Orientale (0.829 km2, South Tirol, Italy). We manually mapped the glacier outline and the highest position of the transient snowline (TSL) through PlanetScope images from the end of the ablation season for each year from 2017 to 2021 and each glacier. The snow cover area ratio (SCAR) was retrieved using the TSLs.We found that the glaciers lost between 4 and 14% of their area between 2017 of 2021, with a mean area loss of −1.5% per year. The strongest shrinkage (14%) is recorded at the smallest glacier (Pizzo Ferrè). The highest TSL showed a variability from year to year. For two glaciers, a supervised classification of the surface facies (snow, ice and supraglacial debris) was performed on all the available cloud-free images from the 2017–2021 ablation seasons.PlanetScope images, with their high spatial and temporal resolution proved to be a valuable tool for studying small glaciers. They permitted to map small glaciers with small uncertainties (between 2.2 and 5.1% of the glacier area), much lower than with medium resolution satellite data (e.g. Landsat 15/30 m, Sentinel 10 m) and to increase the number of images available for analysis. The daily repeat cycle is well suited for increasing the probability to find images from the end of the ablation season with minimum snow cover, in order to identify the real end-of-summer snowline.

The Recycling of Carbon Components and the Reuse of Carbon Fibers for Concrete Reinforcements

Carbon fiber reinforced plastics are increasingly used in all areas of industry. With the increasing number of components and semi-finished products, more and more new carbon fibers will be produced. This also generates a greater number of end-of-life components. These end-of-life components can currently only be fed back, to a limited extent, for reuse, thus leading to a non-optimal, closed-material cycle of the carbon fiber. This article provides an overview of the recycling of carbon components, their further processing and their reuse in reinforcement elements made of carbon fibers. In addition, first results from recycled single fibers and yarn tensile tests from recycled carbon fibers (rCF) are presented. By demonstrating the reuse of carbon fibers in the construction sector, there is the potential to effectively close the carbon cycle. The utilization of carbon reinforcements also enables the reduction of concrete consumption, as the minimum concrete cover required to protect the reinforcement from corrosion is no longer needed.

Computing Solution Space Properties of Combinatorial Optimization Problems Via Generic Tensor Networks

We introduce a unified framework to compute the solution space properties of a broad class of combinatorial optimization problems. These properties include finding one of the optimum solutions, counting the number of solutions of a given size, and enumeration and sampling of solutions of a given size. Using the independent set problem as an example, we show how all these solution space properties can be computed in the unified approach of generic tensor networks. We demonstrate the versatility of this computational tool by applying it to several examples, including computing the entropy constant for hardcore lattice gases, studying the overlap gap properties, and analyzing the performance of quantum and classical algorithms for finding maximum independent sets.

A new robust approach to solve minimum vertex cover problem: Malatya vertex-cover algorithm

A new robust approach to solve minimum vertex cover problem: Malatya vertex-cover algorithm

Automatic generation of a hybrid algorithm for the maximum independent set problem using genetic programming

One of graph optimization’s fundamental and most challenging problems is determining the maximum set of unconnected vertices in a graph, called the maximum independent set problem. This problem consists of finding the largest independent set in a graph, where an independent set is a set of vertices such that no two vertices are adjacent. This paper presents a new artificially generated algorithm for the maximum independent set problem. The new algorithm is generated by the automatic generation of algorithms, a technique that allows the construction of new hybrid algorithms, taking advantage of existing algorithms. Thus, the automatic generation of algorithms combines basic heuristics for the problem, a tabu search method selected from the literature, and an exact method that solves the problem’s mathematical formulation working for a limited computational time. With these components, the space of possible algorithms is traversed by employing genetic programming. Algorithms of small sizes are generated to study their structure and discover new algorithmic combinations. Then, we select the algorithm that finds solutions with the best computational performance among all the generated algorithms. This best algorithm is compared with three state-of-the-art algorithms for the problem, presenting the best computational performance for the 131 instances in the literature.

Recognizing well-dominated graphs is coNP-complete

A graph G is well-covered if every minimal vertex cover of G is minimum, and it is well-dominated if every minimal dominating set of G is minimum. Studies on well-covered graphs were initiated in [Plummer, JCT 1970], and well-dominated graphs were first introduced in [Finbow, Hartnell and Nowakow, AC 1988]. Well-dominated graphs are well-covered, and both classes have been widely studied in the literature. The recognition of well-covered graphs has been proved to be coNP-complete by [Chvátal and Slater, AODM 1993] and [Sankaranarayana and Stewart, Networks 1992], but the complexity of recognizing well-dominated graphs has been left open since their introduction. We close this complexity gap by proving that recognizing well-dominated graphs is coNP-complete. This solves a well-known open question (cf. [Levit and Tankus, DM 2017] and [Gözüpek, Hujdurovic and Milanič, DMTCS 2017]), which was first asked in [Caro, Sebő and Tarsi, JAlg 1996]. Although the problem has been open for a long time, our proof is surprisingly simple. Finally, we show that recognizing well-totally-dominated graphs is coNP-complete, answering a question of [Bahadır, Ekim, and Gözüpek, AMC 2021].

On minimal coverings and pairwise generation of some primitive groups of wreath product type

The covering number of a finite group [Formula: see text], denoted [Formula: see text], is the smallest positive integer [Formula: see text] such that [Formula: see text] is a union of [Formula: see text] proper subgroups. We calculate [Formula: see text] for a family of primitive groups [Formula: see text] with a unique minimal normal subgroup [Formula: see text], isomorphic to [Formula: see text] with [Formula: see text] divisible by [Formula: see text] and [Formula: see text] cyclic. This is a generalization of a result of Swartz concerning the symmetric groups. We also prove an asymptotic result concerning pairwise generation.

Matroid-constrained vertex cover

In this paper, we introduce the Matroid-Constrained Vertex Cover problem: given a graph with weights on the edges and a matroid imposed on the vertices, our problem is to choose a subset of vertices that is independent in the matroid, with the objective of maximizing the total weight of covered edges. This problem is a generalization of the much studied maxk-vertex cover problem, in which the matroid is the simple uniform matroid, and it is also a special case of the problem of maximizing a monotone submodular function under a matroid constraint.In the first part of this work, we give a Fixed-Parameter Tractable Approximation Scheme (FPT-AS) when the given matroid is a partition matroid, a laminar matroid, or a transversal matroid. Precisely, if k is the rank of the matroid, we obtain (1−ε) approximation using (1/ε)O(k)nO(1) time for partition and laminar matroids and using (1/ε+k)O(k)nO(1) time for transversal matroids. This extends a result of Manurangsi for uniform matroids [33]. We also show that these ideas can be applied in the context of (single-pass) streaming algorithms. Besides, our FPT-AS introduces a new technique based on matroid union, which may be of independent interest in extremal combinatorics.In the second part, we consider general matroids. We propose a simple local search algorithm that guarantees 2/3≈0.66 approximation. For the more general problem where two matroids are imposed on the vertices and a feasible solution must be a common independent set, we show that a local search algorithm gives a 2/3⋅(1−1/(p+1)) approximation in nO(p) time, for any integer p. We also provide some evidence to show that with the constraint of one or two matroids, the approximation ratio of 2/3 is likely the best possible, using the currently known techniques of local search.