Augmentation Problem Research Articles

2-node-connectivity network design

We consider 2-connectivity network design problems in which we are given a graph and seek a min-size 2-connected subgraph that satisfies a prescribed property.•In the 1-Connectivity Augmentation problem the goal is to augment a connected graph by a minimum size edge subset of a specified edge set such that the augmented graph is 2-connected. We breach the natural approximation ratio 2 for this problem, and also for the more general Crossing Family Cover problem.•In the 2-Connected Dominating Set problem, we seek a minimum size 2-connected subgraph that dominates all nodes. We give the first non-trivial approximation algorithm with expected approximation ratio O(σ(n)⋅log3⁡n), where σ(n)=O(log⁡n⋅log⁡log⁡n⋅(log⁡log⁡log⁡n)3).The unifying technique of both results is a reduction to the Subset Steiner Connected Dominating Set problem. Such a reduction was known for edge-connectivity, and we extend it to 2-node connectivity problems. We show that the same method can be used to obtain easily polylogarithmic approximation ratios that are not too far from the best known ones for several other problems.

Scarcity-GAN: Scarce data augmentation for defect detection via generative adversarial nets

Data augmentation is a crucial and challenging task for improving defect detection with limited data. Many generative models have been proposed and shown promising performance on this task. However, existing models are unable to capture the fine features of defects when training data is scarce, resulting in the inability to synthesize defects and a lack of diversity in the synthesized defects. Additionally, most models do not consider the location of synthesized defects in the image, thus limiting the ability for augmenting defect data through data generation. In this paper, we propose a new augmentation model named Scarce Data Augmentation Generative Adversarial Nets (Scarcity-GAN) to address the scarce data augmentation problem. Firstly, we design a new clustering module which selects data containing similar features to the target defect from extra datasets, in order to help the GAN learn the features of the target defect. Secondly, we modify the vanilla generator with an Encoder–Decoder model. The generator takes two inputs: one is the defect-free images, which are encoded by the Encoder to obtain defect-free features, and the other is the extra defect feature maps in the target defect set after clustering. Next, we design a Fusion Patch-Embedding module to merge the two different features, ensuring that the synthesized defects are located on the object accurately. We also design a new loss function for the generator, and then prove that it makes our model get converged. Last, we conduct extensive experiments to demonstrate the significant performance improvement and generalizability of Scarcity-GAN on two scarce datasets: industrial O-ring and Metal Iron Sheet datasets; and one general dataset: the public CODEBRIM dataset. The experimental results show that our Scarcity-GAN outperforms the SOTA augmentation models on different scarce datasets.

Better-than-43\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\frac{4}{3}$$\\end{document}-approximations for leaf-to-leaf tree and connectivity augmentation

The Connectivity Augmentation Problem (CAP) together with a well-known special case thereof known as the Tree Augmentation Problem (TAP) are among the most basic Network Design problems. There has been a surge of interest recently to find approximation algorithms with guarantees below 2 for both TAP and CAP, culminating in the currently best approximation factor for both problems of 1.393 through quite sophisticated techniques. We present a new and arguably simple matching-based method for the well-known special case of leaf-to-leaf instances. Combining our work with prior techniques, we readily obtain a (4/3+ε)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(4/3+\\varepsilon )$$\\end{document}-approximation for Leaf-to-Leaf CAP by returning the better of our solution and one of an existing method. Prior to our work, a 4/3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$4/3$$\\end{document}-guarantee was only known for Leaf-to-Leaf TAP instances on trees of height 2. Moreover, when combining our technique with a recently introduced stack analysis approach, which is part of the above-mentioned 1.393-approximation, we can further improve the approximation factor to 1.29, obtaining for the first time a factor below 43\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\frac{4}{3}$$\\end{document} for a nontrivial class of TAP/CAP instances.

Bridging the Gap Between Tree and Connectivity Augmentation: Unified and Stronger Approaches

We consider the Connectivity Augmentation Problem (CAP), a classical problem in the area of Survivable Network Design. It is about increasing the edge-connectivity of a graph by one unit in the cheapest possible way. More precisely, given a $k$-edge-connected graph $G=(V,E)$ and a set of extra edges, the task is to find a minimum cardinality subset of extra edges whose addition to $G$ makes the graph $(k+1)$-edge-connected. If $k$ is odd, the problem is known to reduce to the Tree Augmentation Problem (TAP) -- i.e., $G$ is a spanning tree -- for which significant progress has been achieved recently, leading to approximation factors below $1.5$ (the currently best factor is $1.458$). However, advances on TAP did not carry over to CAP so far. Indeed, only very recently, Byrka, Grandoni, and Ameli (STOC 2020) managed to obtain the first approximation factor below $2$ for CAP by presenting a $1.91$-approximation algorithm based on a method that is disjoint from recent advances for TAP. We first bridge the gap between TAP and CAP, by presenting techniques that allow for leveraging insights and methods from TAP to approach CAP. We then introduce a new way to get approximation factors below $1.5$, based on a new analysis technique. Through these ingredients, we obtain a $1.393$-approximation algorithm for CAP, and therefore also TAP. This leads to the currently best approximation result for both problems in a unified way, by significantly improving on the above-mentioned $1.91$-approximation for CAP and also the previously best approximation factor of $1.458$ for TAP by Grandoni, Kalaitzis, and Zenklusen (STOC 2018). Additionally, a feature we inherit from recent TAP advances is that our approach can deal with the weighted setting when the ratio between the largest to smallest cost on extra links is bounded, in which case we obtain approximation factors below $1.5$.

EID-GAN: Generative Adversarial Nets for Extremely Imbalanced Data Augmentation

Imbalanced data cause deep neural networks to output biased results, and it becomes more serious when facing extremely imbalanced data regarding the outliers with tiny size (the ratio of the outlier size to the image size is around 0.05%). Many data argumentation models are proposed to supplement imbalanced data to alleviate biased results. However, the existing augmentation models cannot synthesize tiny outliers, which make the generated data unavailable. In this article, we propose a new augmentation model named extremely imbalanced data augmentation generative adversarial nets (EID-GANs) to address the extremely imbalanced data augmentation problem. First, we design a new penalty function by subtracting the outliers from the cropped region of generated instance to guide the generator to learn the features of outliers. After this, we combine the output value of the penalty function with the generator loss to jointly update the generator’s parameters with backpropagation. Second, we propose a new evaluation approach that adopts two outlier detectors with <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> -fold cross-validation to assess the availability of generated instances. We conduct extensive experiments to demonstrate the significant performance improvement of EID-GAN on two extremely imbalanced datasets, which are the industrial Piston and the Fabric datasets, and one general imbalanced dataset, i.e., the public DAGM dataset. The experimental results show that our EID-GAN outperforms the state-of-the-art (SOTA) augmentation models on different imbalanced datasets.

An Improved Approximation Algorithm for the Matching Augmentation Problem

We present a -approximation algorithm for the matching augmentation problem (MAP): given a multigraph with edges of cost either zero or one such that the edges of cost zero form a matching, find a 2-edge connected spanning subgraph (2-ECSS) of minimum cost. A -approximation algorithm for the same problem was presented recently; see Cheriyan et al. [Math. Program., 182 (2020), pp. 315–354]. Our improvement is based on new algorithmic techniques, and some of these may lead to advances on related problems.

Network augmentation for disaster‐resilience against geographically correlated failure

AbstractWe introduce a formal framework for the study of augmenting networks in the plane for disaster‐resilience, where a disaster is modeled by a straight‐line segment. We generalize various graph structures from classical 2‐edge‐connectivity, including minimal cuts and blocks. The key concept that we introduce is that of an ‐leaf, which builds on the fundamental “leaf‐block” concept from classical augmentation. We present a number of algorithms for constructing the above‐mentioned graph structures, including a sweep‐line algorithm that finds all edge‐cuts that can be destroyed by a single disaster. We also present an algorithm which optimally adds a single edge between a pair of ‐leaves or blocks while avoiding certain disaster regions. Finally, we present a number of heuristic schemes for solving the disaster‐resilient network augmentation problem and perform extensive experiments to demonstrate the power of the ‐leaf concept within heuristic design.

Coloring down: 3/2-approximation for special cases of the weighted tree augmentation problem

In this paper, we investigate the weighted tree augmentation problem (TAP), where the goal is to augment a tree with a minimum cost set of edges such that the graph becomes two edge connected. First we show that in weighted TAP, we can restrict our attention to trees which are binary and where all the non-tree edges go between two leaves of the tree. We then give a top-down coloring algorithm that differs from known techniques for approximating TAP.The algorithm we describe always gives a 2-approximation starting from any feasible fractional solution to the natural tree cut covering LP. When the structure of the fractional solution is such that all the edges with non-zero weight are at least α, then this algorithm achieves a 21+α-approximation.We also investigate a variant of TAP where every tree edge must belong to a cycle of length three (triangle) in the solution. We give a Θ(log⁡n)-approximation algorithm for this problem in the weighted case in n-node graphs and a 4-approximation in the unweighted case.

On small-depth tree augmentations

We study the Weighted Tree Augmentation Problem for general link costs. We show that the integrality gap of the odd-LP relaxation for the (weighted) Tree Augmentation Problem for a k-level tree instance is at most 2−12k−1. For 2- and 3-level trees, these ratios are 32 and 74 respectively. Our proofs are constructive and yield polynomial-time approximation algorithms with matching guarantees.

Fault‐tolerance based on augmenting approach in wireless sensor networks

SummaryA critical node is a sensor whose failure causes loss of connectivity and network fragmentation. In wireless sensor networks, the failure of a critical node, like the failure of all sensor nodes, can be caused by energy depletion or physical failure. To overcome the problem of failure of such nodes, this article proposes a fault‐tolerant strategy that allows a routing protocol to tolerate the failure of a critical node. The proposed strategy is carried out in two phases. In the first phase, a hybrid genetic algorithm with a local search heuristic called genetic algorithm‐critical node problem (GA‐CNP) is used to select the critical nodes and in the second phase, an algorithm called Aug‐CNP is involved to deal with the augmentation problem by deploying additional wireless edges to preserve network connectivity in case of critical node failure. Our proposal has been developed using the OMNET++ simulator, evaluated, and compared to the AODV protocol. The simulation results showed that the GA‐CNP algorithm selects the critical nodes whose failure can degrade the network lifetime with a rate of 40%. Moreover, the Aug‐CNP algorithm applied to the AODV protocol brings improvements in terms of network lifetime which reaches 22% compared to the traditional AODV protocol.

Optimal Augmentation of Distribution Networks for Improved Reliability

Reliability is a crucial factor influencing electricity network planning and has gained considerable attention in the past decade due to the increasing number of weather-related outage events. This article explores connections between network topology and reliability, and proposes a graph-based approach to enhance the reliability of distribution systems by installing new tie-lines. The problem is cast here as an edge-connectivity augmentation problem, with 2-edge connectivity being desired to ensure service restoration under the outage of any single line in the network. However, in practice, such a target may not be economical or feasible. Therefore, a systematic optimization framework is presented in this article to model partial augmentations to the system for improving the reliability at a select subset of nodes. This approach allows for selectively strengthening parts of the network based on connectivity requirements. The article also proposes the use of graph-based metrics to characterize the reliability of distribution networks. Applications of the approach to realistic distribution test systems demonstrate its effectiveness in identifying line additions that can favorably impact system reliability. Numerical results also quantify the reliability improvement achievable through partial augmentation.

Edge Augmentation With Controllability Constraints in Directed Laplacian Networks

In this letter, we study the maximum edge augmentation problem in directed Laplacian networks to improve their robustness while preserving lower bounds on their strong structural controllability (SSC). Since adding edges could adversely impact network controllability, the main objective is to maximally densify a given network by selectively adding missing edges while ensuring that SSC of the network does not deteriorate beyond certain levels specified by the SSC bounds. We consider two widely used bounds: first is based on the notion of zero forcing (ZF), and the second relies on the distances between nodes in a graph. We provide an edge augmentation algorithm that adds the maximum number of edges in a graph while preserving the ZF-based bound, and also derive a closed-form expression for the exact number of edges added to the graph. Then, we examine the edge augmentation while preserving the distance-based bound and present a randomized algorithm that guarantees an α-approximate solution with high probability. Finally, we numerically evaluate and compare these edge augmentation solutions.

Tight Bounds for Online Weighted Tree Augmentation

The Weighted Tree Augmentation problem (WTAP) is a fundamental problem in network design. In this paper, we consider this problem in the online setting. We are given an n-vertex tree \(T = (V,E)\) and an additional set \(L \subseteq {V \atopwithdelims ()2}\) of edges (called links) with costs. Then, terminal pairs arrive one-by-one and our task is to maintain a low-cost subset of links F such that every terminal pair that has arrived so far is 2-edge-connected in \(T \cup F\). This online problem was first studied by Gupta, Krishnaswamy and Ravi (SICOMP 2012) who used it as a subroutine for the online survivable network design problem. They gave a deterministic \(O(\log ^2 n)\)-competitive algorithm and showed an \(\varOmega (\log n)\) lower bound on the competitive ratio of randomized algorithms. The case when T is a path is also interesting: it is exactly the online interval set cover problem, which also captures as a special case the parking permit problem studied by Meyerson (FOCS 2005). The contribution of this paper is to give tight results for online weighted tree and path augmentation problems. The main result of this work is a deterministic \(O(\log n)\)-competitive algorithm for online WTAP, which is tight up to constant factors.

Sparse graphs and an augmentation problem

For two integers k>0 and ell , a graph G=(V,E) is called (k,ell )-tight if |E|=k|V|-ell and i_G(X)le k|X|-ell for each Xsubseteq V for which i_G(X)ge 1, where i_G(X) denotes the number of induced edges by X. G is called (k,ell )-redundant if G-e has a spanning (k,ell )-tight subgraph for all ein E. We consider the following augmentation problem. Given a graph G=(V,E) that has a (k,ell )-tight spanning subgraph, find a graph H=(V,F) with the minimum number of edges, such that Gcup H is (k,ell )-redundant. We give a polynomial algorithm and a min-max theorem for this augmentation problem when the input is (k,ell )-tight. For general inputs, we give a polynomial algorithm when kge ell and show the NP-hardness of the problem when k<ell . Since (k,ell )-tight graphs play an important role in rigidity theory, these algorithms can be used to make several types of rigid frameworks redundantly rigid by adding a smallest set of new bars.

Request Reliability Augmentation With Service Function Chain Requirements in Mobile Edge Computing

Provisioning reliable network services for mobile users in edge computing environments is the top priority of network service providers, as unreliable services will result in tremendous losses of revenues and customers. In this paper, we study a novel service reliability augmentation problem in a mobile edge computing (MEC) network, where mobile users request network services with service function chain (SFC) and reliability expectation requirements. To enhance the service reliability of user requests, it is a common practice to make use of redundant virtualized network function (VNF) instance placement in case the primary VNF instance fails. We aim to augment the service reliability of each admitted request to its specified reliability expectation, subject to computing capacity on each cloudlet. To this end, we first formulate a novel service reliability augmentation problem for each request with an SFC and a reliability expectation requirement, by augmenting its reliability through redundant VNF instance deployment. We then show that the problem is NP-hard, and provide an admission framework of user requests by placing primary VNF instances of network functions in the SFC to different cloudlets. We then deal with the service reliability augmentation problem of an admitted request under the assumption that all secondary VNF instances of each primary VNF instance must be placed into the cloudlets no more than <inline-formula><tex-math notation="LaTeX">$l$</tex-math></inline-formula> hops from the cloudlet of its primary VNF instance for a fixed <inline-formula><tex-math notation="LaTeX">$l$</tex-math></inline-formula> with <inline-formula><tex-math notation="LaTeX">$1\leq l \leq n-1$</tex-math></inline-formula> , where <inline-formula><tex-math notation="LaTeX">$n$</tex-math></inline-formula> is the number of cloudlets in the network, for which we formulate an integer linear program solution, and develop a randomized algorithm with a good approximation ratio and high probability, at the expense of moderate resource constraint violations. We also devise a deterministic heuristic for the problem without any resource violation. We third study the service reliability augmentation problem for a set of admitted requests by extending the proposed algorithm for the service reliability augmentation problem for a single request admission. We finally evaluate the performance of the proposed algorithms through experimental simulations. Experimental results demonstrate that the proposed algorithms are promising, and their empirical results are superior to their analytical counterparts.

The (2, k)-Connectivity Augmentation Problem: Algorithmic Aspects

Durand de Gevigney and Szigeti (J Gr Theory 91(4):305–325, 2019) have recently given a min–max theorem for the (2, k)-connectivity augmentation problem. This article provides an \(O(n^3(m+ n \text { }\log \text { }n))\) time algorithm to find an optimal solution for this problem.

Societal context-dependent multi-modal transportation network augmentation in Johannesburg, South Africa

In most developing countries, formal and informal transportation schemes coexist without effective and smart integration. In this paper, the authors show how to leverage opportunities offered by formal and informal transportation schemes to build an integrated multi-modal network. Precisely, the authors consider integration of rickshaws to a bus-train network, by taking into account accessibility and societal constraints. By modelling the respective networks with weighted graphs, a graph augmentation problem is solved with respect to a composite cost taking into account constraints on the use of rickshaws. The solution, is based on finding a minimum cost spanning tree of a merged graph. The method is applied in the South African context, in the city of Johannesburg where rickshaws are not yet a significant part of the transportation system. The implications of the study reveal that using non-motorised transportation services is a viable option of improving mobility in the city. The composite cost introduced herein could be used for new routing algorithm including societal, environmental, architectural contexts and commuter experiences through rating.

On the Cycle Augmentation Problem: Hardness and Approximation Algorithms

In the k-Connectivity Augmentation Problem we are given a k-edge-connected graph and a set of additional edges called links. Our goal is to find a set of links of minimum size whose addition to the graph makes it (k + 1)-edge-connected. There is an approximation preserving reduction from the mentioned problem to the case k = 1 (a.k.a. the Tree Augmentation Problem or TAP) or k = 2 (a.k.a. the Cactus Augmentation Problem or CacAP). While several better-than-2 approximation algorithms are known for TAP, for CacAP only recently this barrier was breached (hence for k-Connectivity Augmentation in general). As a first step towards better approximation algorithms for CacAP, we consider the special case where the input cactus consists of a single cycle, the Cycle Augmentation Problem (CycAP). This apparently simple special case retains part of the hardness of the general case. In particular, we are able to show that it is APX-hard. In this paper we present a combinatorial left (frac {3}{2}+varepsilon right )-approximation for CycAP, for any constant ε > 0. We also present an LP formulation with a matching integrality gap: this might be useful to address the general case of the problem.

On the Tree Augmentation Problem

In the Tree Augmentation problem we are given a tree $$T=(V,F)$$ and a set $$E \subseteq V \times V$$ of edges with positive integer costs $$\{c_e:e \in E\}$$ . The goal is to augment T by a minimum cost edge set $$J \subseteq E$$ such that $$T \cup J$$ is 2-edge-connected. We obtain the following results.

Plane augmentation of plane graphs to meet parity constraints

A plane topological graph G=(V,E) is a graph drawn in the plane whose vertices are points in the plane and whose edges are simple curves that do not intersect, except at their endpoints. Given a plane topological graph G=(V,E) and a set CG of parity constraints, in which every vertex has assigned a parity constraint on its degree, either even or odd, we say that G is topologically augmentable to meet CG if there exists a set E′ of new edges, disjoint with E, such that G′=(V,E∪E′) is noncrossing and meets all parity constraints.In this paper, we prove that the problem of deciding if a plane topological graph is topologically augmentable to meet parity constraints is NP-complete, even if the set of vertices that must change their parities is V or the set of vertices with odd degree. In particular, deciding if a plane topological graph can be augmented to a Eulerian plane topological graph is NP-complete. Analogous complexity results are obtained, when the augmentation must be done by a plane topological perfect matching between the vertices not meeting their parities.We extend these hardness results to planar graphs, when the augmented graph must be planar, and to plane geometric graphs (plane topological graphs whose edges are straight-line segments). In addition, when it is required that the augmentation is made by a plane geometric perfect matching between the vertices not meeting their parities, we also prove that this augmentation problem is NP-complete for plane geometric paths.For the particular family of maximal outerplane graphs, we characterize maximal outerplane graphs that are topological augmentable to satisfy a set of parity constraints. We also provide a polynomial time algorithm that decides if a maximal outerplane graph is topologically augmentable to meet parity constraints, and if so, produces a set of edges with minimum cardinality.