Subset Of Edges Research Articles

Attack Graphs & Subset Sabotage Games

We consider an extended version of sabotage games played over Attack Graphs. Such games are two-player zero-sum reachability games between an Attacker and a Defender. This latter player can erase particular subsets of edges of the Attack Graph. To reason about such games we introduce a variant of Sabotage Modal Logic (that we call Subset Sabotage Modal Logic) in which one modality quantifies over non-empty subset of edges. We show that we can characterize the existence of winning Attacker strategies by formulas of Subset Sabotage Modal Logic.

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 $1.5$.

Anti-Kekulé number of the {(3, 4), 4}-fullerene.

A {(3,4),4}-fullerene graph G is a 4-regular plane graph with exactly eight triangular faces and other quadrangular faces. An edge subset S of G is called an anti-Kekulé set, if G - S is a connected subgraph without perfect matchings. The anti-Kekulé number of G is the smallest cardinality of anti-Kekulé sets and is denoted by . In this paper, we show that ; at the same time, we determine that the {(3, 4), 4}-fullerene graph with anti-Kekulé number 4 consists of two kinds of graphs: one of which is the graph consisting of the tubular graph , where Q n is composed of concentric layers of quadrangles, capped on each end by a cap formed by four triangles which share a common vertex (see Figure 2 for the graph Q n ); and the other is the graph , which contains four diamonds D 1, D 2, D 3, and D 4, where each diamond consists of two adjacent triangles with a common edge such that four edges e 1, e 2, e 3, and e 4 form a matching (see Figure 7D for the four diamonds D 1 - D 4). As a consequence, we prove that if , then ; moreover, if , we give the condition to judge that the anti-Kekulé number of graph G is 4 or 5.

The generalized measure of edge fault tolerance in exchanged 3-ary n-cube

The exchanged 3-ary n-cube , proposed by Lv et al. in 2021, is obtained by removing edges from a 3-ary n-cube , where r + s + t + 1 = n. The topological interconnection network of a multiprocessor system can be modeled as a connected graph. Analyzing the fault tolerance of its topological structure is critical in the course of design and maintenance of it. Given a connected graph G, let F be an edge subset of G. F is called an h-edge-cut of G, if G−F is disconnected and each remaining component has the minimum degree of at least h. The h-edge-connectivity is the minimum cardinality of all h-edge-cuts of G. For and , in this paper, we determine the -edge-connectivity of exchanged 3-ary n-cubes, , and prove the exact values .

Delving into Crispness: Guided Label Refinement for Crisp Edge Detection.

Learning-based edge detection usually suffers from predicting thick edges. Through extensive quantitative study with a new edge crispness measure, we find that noisy human-labeled edges are the main cause of thick predictions. Based on this observation, we advocate that more attention should be paid on label quality than on model design to achieve crisp edge detection. To this end, we propose an effective Canny-guided refinement of human-labeled edges whose result can be used to train crisp edge detectors. Essentially, it seeks for a subset of over-detected Canny edges that best align human labels. We show that several existing edge detectors can be turned into a crisp edge detector through training on our refined edge maps. Experiments demonstrate that deep models trained with refined edges achieve significant performance boost of crispness from 17.4% to 30.6%. With the PiDiNet backbone, our method improves ODS and OIS by 12.2% and 12.6% on the Multicue dataset, respectively, without relying on non-maximal suppression. We further conduct experiments and show the superiority of our crisp edge detection for optical flow estimation and image segmentation.

Stackelberg packing games

Stackelberg pricing games are pricing problems over a set of items. One player, the leader, sets prices for the items and the second player, the follower, buys a subset of items at minimal total cost subject to feasibility constraints. The constraints are determined by an optimization problem. For example, the Stackelberg shortest path game is used to optimize the income from road tolls (cf. [21]). The items are edges in a network graph and the follower buys a subset of edges forming a path.The Stackelberg pricing games studied in the literature are formulated on top of covering problems. In this paper, we introduce pricing games based on packing problems. We are interested in the complexity of computing leader-optimal prices depending on different types of follower-constraints. We show that optimal prices can be computed in polynomial time if the follower-constraints are determined by the well-known interval scheduling problem. This problem is equivalent to the independent set problem on interval graphs. In case the follower-constraints are based on the independent set problem on perfect graphs or on the bipartite matching problem, we prove APX-hardness for the pricing problem.On a more general note, we prove Σ2p-completeness if the follower-constraints are given by a packing problem that is NP-complete, i.e., the leader's pricing problem is hard even if she has an NP-oracle at hand.

Effective resistance against pandemics: Mobility network sparsification for high-fidelity epidemic simulations.

Network science has increasingly become central to the field of epidemiology and our ability to respond to infectious disease threats. However, many networks derived from modern datasets are not just large, but dense, with a high ratio of edges to nodes. This includes human mobility networks where most locations have a large number of links to many other locations. Simulating large-scale epidemics requires substantial computational resources and in many cases is practically infeasible. One way to reduce the computational cost of simulating epidemics on these networks is sparsification, where a representative subset of edges is selected based on some measure of their importance. We test several sparsification strategies, ranging from naive thresholding to random sampling of edges, on mobility data from the U.S. Following recent work in computer science, we find that the most accurate approach uses the effective resistances of edges, which prioritizes edges that are the only efficient way to travel between their endpoints. The resulting sparse network preserves many aspects of the behavior of an SIR model, including both global quantities, like the epidemic size, and local details of stochastic events, including the probability each node becomes infected and its distribution of arrival times. This holds even when the sparse network preserves fewer than 10% of the edges of the original network. In addition to its practical utility, this method helps illuminate which links of a weighted, undirected network are most important to disease spread.

An LP-based approximation algorithm for the generalized traveling salesman path problem

The traveling salesman problem (TSP) is one of the classic research topics in the field of operations research, graph theory and computer science. In this paper, we propose a generalized model of traveling salesman problem, denoted by generalized traveling salesman path problem. Let G=(V,E,c) be a weighted complete graph, in which c is a nonnegative metric cost function on edge set E, i.e., c:E→R+. The traveling salesman path problem aims to find a Hamiltonian path in G with minimum cost. Compared to the traveling salesman path problem, we are given extra vertex subset V′ and edge subset E′ in the problem proposed in this paper; its goal is to construct a path which traverses all the edges in E′ while only needs to visit each vertex in V′ exactly once. Based on integer programming, we give a mathematical model of the problem, and design a 1+52-approximation algorithm for the problem by combining linear programming rounding strategy and a special graph structure.

A Lower Bound on Cycle-Finding in Sparse Digraphs

We consider the problem of finding a cycle in a sparse directed graph G that is promised to be far from acyclic, meaning that the smallest feedback arc set , i.e., a subset of edges whose deletion results in an acyclic graph, in G is large. We prove an information-theoretic lower bound, showing that for N -vertex graphs with constant outdegree, any algorithm for this problem must make Ω̄(N 5/9 ) queries to an adjacency list representation of G . In the language of property testing, our result is an Ω̄(N 5/9) lower bound on the query complexity of one-sided algorithms for testing whether sparse digraphs with constant outdegree are far from acyclic. This is the first improvement on the Ω (√ N ) lower bound, implicit in the work of Bender and Ron, which follows from a simple birthday paradox argument.

NetTDP: permutation-based true discovery proportions for differential co-expression network analysis.

Existing methods for differential network analysis could only infer whether two networks of interest have differences between two groups of samples, but could not quantify and localize network differences. In this work, a novel method, permutation-based Network True Discovery Proportions (NetTDP), is proposed to quantify the number of edges (correlations) or nodes (genes) for which the co-expression networks are different. In the NetTDP method, we propose an edge-level statistic and a node-level statistic, and detect true discoveries of edges and nodes in the sense of differential co-expression network, respectively, by the permutation-based sumSome method. Furthermore, the NetTDP method could further localize the differences by inferring the TDPs for edge or gene subsets of interest, which can be selected post hoc. Our NetTDP method allows inference on data-driven modules or biology-driven gene sets, and remains valid even when these sub-networks are optimized using the same data. Experimental results on both simulation data sets and five real data sets show the effectiveness of the proposed method in inferring the quantification and localization of differential co-expression networks. The R code is available at https://github.com/LiminLi-xjtu/NetTDP.

Solving maximum weighted matching on large graphs with deep reinforcement learning

Maximum weighted matching (MWM), which finds a subset of vertex-disjoint edges with maximum weight, is a fundamental topic with a wide spectrum of applications in various fields, such as biotechnology, social analysis, web services, etc. However, traditional methods cannot achieve a good balance between cost and quality. Inspired by the reinforcement learning techniques, we propose an MWM framework, L2M, with a Deep Reinforcement Learning (DRL) model, which can efficiently and effectively solve the MWM problem on large-scale general graphs. First, in contrast to traditional DRL methods that define the Markov Decision Process (MDP) without considering the efficiency issue, we represent MWM as an MDP that is carefully designed to accelerate computation. In particular, this MDP supports selecting multiple edges for each action and uses a pruning method to reduce search space efficiently. Second, since none of the existing DRL methods can support edge selection on large-scale graphs efficiently, we propose an edge-message-passing network EEN to generate edge embedding. To the best of our knowledge, this is the first attempt that uses DRL model for solving the MWM problem. Experimental results show that L2M outperforms state-of-the-art algorithms and generalizes well on graphs of different sizes and distributions.

Anti-Forcing Spectra of Convex Hexagonal Systems

For any perfect matching M of a graph AG, the anti-forcing number of M af(G,M) is the cardinality of a minimum edge subset S⊆E(G)\M such that the graph G−S has only one perfect matching. The anti-forcing numbers of all perfect matchings of G form its anti-forcing spectrum, denoted by Specaf(G). For a convex hexagonal system O(n1,n2,n3) with n1≤n2≤n3, denoted by H, it has the minimum anti-forcing number n1. In this paper, we derive a formula for its maximum anti-forcing number Af(H), i.e., the Fries number. Next, we prove that [n1,c]∪{c+2,c+4,…,Af(H)−2,Af(H)}⊆Specaf(H) for the specific integer c with the same parity as Af(H). In particular, we obtain that if n1+n2−n3≤1, then c=Af(H), which implies that Specaf(H)=[n1,Af(H)] is an integer interval. Finally, we also give some non-continuous situations: Specaf(O(2,n,n))=[2,4n−2]\{4n−3} for n≥2; the anti-forcing spectrum of H has a gap Af(H)−1 for n1=n2≥2 and n3 even, or n2=n3 and n1≥2 even.

Not all edges are peers: Accurate structure-aware graph pooling networks

Graph Neural Networks (GNNs) have achieved state-of-the-art performance in graph-related tasks. For graph classification task, an elaborated pooling operator is vital for learning graph-level representations. Most pooling operators derived from existing GNNs generate a coarsen graph through ordering the nodes and selecting some top-ranked ones. However, these methods fail to explore the fundamental elements other than nodes in graphs, which may not efficiently utilize the structure information. Besides, all edges attached to the low-ranked nodes are discarded, which destroys graphs’ connectivity and loses information. Moreover, the selected nodes tend to concentrate on some substructures while overlooking information in others. To address these challenges, we propose a novel pooling operator called Accurate Structure-Aware Graph Pooling (ASPool), which can be integrated into various GNNs to learn graph-level representation. Specifically, ASPool adaptively retains a subset of edges to calibrate the graph structure and learns the abstracted representations, wherein all the edges are viewed as non-peers instead of simply connecting nodes. To preserve the graph’s connectivity, we further introduce the selection strategy considering both top-ranked nodes and dropped edges. Additionally, ASPool performs a two-stage calculation process to promise that the sampled nodes are distributed throughout the graph. Experiment results on 9 widely used benchmarks show that ASPool achieves superior performance over the state-of-the-art graph representation learning methods.

Maximizing fair content spread via edge suggestion in social networks

Content spread inequity is a potential unfairness issue in online social networks, disparately impacting minority groups. In this paper, we view friendship suggestion, a common feature in social network platforms, as an opportunity to achieve an equitable spread of content. In particular, we propose to suggest a subset of potential edges (currently not existing in the network but likely to be accepted) that maximizes content spread while achieving fairness. Instead of re-engineering the existing systems, our proposal builds a fairness wrapper on top of the existing friendship suggestion components. We prove the problem is NP-hard and inapproximable in polynomial time unless P = NP. Therefore, allowing relaxation of the fairness constraint, we propose an algorithm based on LP-relaxation and randomized rounding with fixed approximation ratios on fairness and content spread. We provide multiple optimizations, further improving the performance of our algorithm in practice. Besides, we propose a scalable algorithm that dynamically adds subsets of nodes, chosen via iterative sampling, and solves smaller problems corresponding to these nodes. Besides theoretical analysis, we conduct comprehensive experiments on real and synthetic data sets. Across different settings, our algorithms found solutions with near-zero unfairness while significantly increasing the content spread. Our scalable algorithm could process a graph with half a million nodes on a single machine, reducing the unfairness to around 0.0004 while lifting content spread by 43%.

Critical node/edge detection problems on trees

We consider the problem of removing a limited subset of nodes and/or edges from a graph in order to minimize the so-called pairwise connectivity of the residual graph, which is defined as the total cost of the pairs of nodes still connected by a path. This is a well-studied version of a family of problems known as critical node or edge detection problems. However, while most of the literature focuses on deleting nodes or edges separately, we allow the simultaneous removal of nodes and edges. We consider both the case in which the nodes and edges removed must satisfy a joint weight limit, and the case in which two separate weight limits are given for nodes and edges. We study the complexity of several problems of this type when the given graph is a tree, providing NP-hardness results or polynomial-time algorithms for the different cases that we analyze.

Proof-labeling schemes: Broadcast, unicast and in between

We study the effect of limiting the number of different messages a node can transmit simultaneously on the verification complexity of proof-labeling schemes (PLS). In a PLS, each node is given a label, and the goal is to verify, by exchanging messages over each link in each direction, that a certain global predicate is satisfied by the system configuration. We consider a single parameter r that bounds the number of distinct messages that can be sent concurrently by any node: in the case r=1, each node may only send the same message to all its neighbors (the broadcast model), in the case r≥Δ, where Δ is the largest node degree in the system, each neighbor may be sent a distinct message (the unicast model), and in general, for 1≤r≤Δ, each of the r messages is destined to a subset of the neighbors.We show that message compression linear in r is possible for verifying fundamental problems such as the agreement between edge endpoints on the edge state. Some problems, including verification of maximal matching, exhibit a large gap in complexity between r=1 and r>1. For some other important predicates, the verification complexity is insensitive to r, e.g., the question whether a subset of edges constitutes a spanning-tree. We also consider the congested clique model. We show that the crossing technique [1] for proving lower bounds on the verification complexity can be applied in the case of congested clique only if r=1. Together with a new upper bound, this allows us to determine the verification complexity of MST in the broadcast clique. Finally, we establish a general connection between the deterministic and randomized verification complexity for any given number r.

Optimal Frameworks for Detecting Anomalies in Sensor-Intensive Heterogeneous Networks

Many network/graph structures are continuously monitored by various sensors that are placed at a subset of nodes and edges. The multidimensional data collected from these sensors over time create large-scale graph data in which the data points are highly dependent. Monitoring large-scale attributed networks with thousands of nodes and heterogeneous sensor data to detect anomalies and unusual events is a complex and computationally expensive process. This paper introduces a new generic approach inspired by state-space models for network anomaly detection that can utilize the information from the network topology, the node attributes (sensor data), and the anomaly propagation sets in an integrated manner to analyze the entire network all at once. This article presents how heterogeneous network sensor data can be analyzed to locate the sources of anomalies as well as the anomalous regions in a network, which can be impacted by one or multiple anomalies at any time instance. Experimental results demonstrate the superior performance of our proposed framework in detecting anomalies in attributed graphs. Summary of Contribution: With the increasing availability of large-scale network sensors and rapid advances in artificial intelligence methods, fundamentally new analytical tools are needed that can integrate data collected from sensors across the networks for decision making while taking into account the stochastic and topological dependencies between nodes, sensors, and anomalies. This paper develops a framework to intelligently and efficiently analyze complex and highly dependent data collected from disparate sensors across large-scale network/graph structures to detect anomalies and abnormal behavior in real time. Unlike general purpose (often black-box) machine learning models, this paper proposes a unique framework for network/graph structures that incorporates the complexities of networks and interdependencies between network entities and sensors. Because of the multidisciplinary nature of the paper that involves optimization, machine learning, and system monitoring and control, it can help researchers in both operations research and computer science domains to develop new network-specific computing tools and machine learning frameworks to efficiently manage large-scale network data.

Super Edge Connectivity Number of an Arithmetic Graph

An edge subset F of a connected graph G is a super edge cut if G − F is disconnected and every component of G−F has atleast two vertices. The minimum cardinality of super edge cut is called super edge connectivity number and it is denoted by λ'(G). Every arithmetic graph G = Vn, n not equal to p1 × p2 has super edge cut. In this paper, the authors study super edge connectivity number of an arithmetic graphs G = Vn, n = p_1^a_1 × p_2^a_2 , a1 > 1, a2 ≥ 1, and G = Vn, n = p_1^a_1 × p_2^a_2 × · · · ×p_r^a_r , r > 2, ai ≥ 1, 1 ≤ i ≤ r.

The anti-forcing spectra of [formula omitted]-fullerenes

For an edge subset S of connected graph G, if G−S has only one perfect matching M, then S is called an anti-forcing set of M. The number of edges in a smallest anti-forcing set of M is the anti-forcing number of M, generally indicated by the symbol af(G,M). For a graph G, its anti-forcing spectrum is defined as the integer set Specaf(G):={af(G,M):M is a perfect matching of G}. In this paper, we show that for a (4,6)-fullerene graph Tn with cyclic edge-connectivity 3, Specaf(Tn)=[n+3,2n+4]. Moreover, we show that for any perfect matching M of a (4,6)-fullerene graph G, a minimum anti-forcing set S of M and each M-alternating facial boundary share exactly one edge. Applying this conclusion, we prove that the minimum anti-forcing number of a lantern structure (4,6)-fullerene of order n is ⌊n8⌋+2, and Specaf(Bn)=[⌊n8⌋+2,⌊n3⌋+2].

Leray numbers of complexes of graphs with bounded matching number

Given a graph G on the vertex set V, the non-matching complex of G, denoted by NMk(G), is the family of subgraphs G′⊂G whose matching number ν(G′) is strictly less than k. As an attempt to extend the result by Linusson, Shareshian and Welker on the homotopy types of NMk(Kn) and NMk(Kr,s) to arbitrary graphs G, we show that (i) NMk(G) is (3k−3)-Leray, and (ii) if G is bipartite, then NMk(G) is (2k−2)-Leray. This result is obtained by analyzing the homology of the links of non-empty faces of the complex NMk(G), which vanishes in all dimensions d≥3k−4, and all dimensions d≥2k−3 when G is bipartite. As a corollary, we have the following rainbow matching theorem which generalizes a result by Aharoni, Berger, Chudnovsky, Howard and Seymour: Let E1,…,E3k−2 be non-empty edge subsets of a graph and suppose that ν(Ei∪Ej)≥k for every i≠j. Then E=⋃Ei has a rainbow matching of size k. Furthermore, the number of edge sets Ei can be reduced to 2k−1 when E is the edge set of a bipartite graph.