Problem In Bipartite Graphs Research Articles

HybridGNN: A Self-Supervised Graph Neural Network for Efficient Maximum Matching in Bipartite Graphs

Solving maximum matching problems in bipartite graphs is critical in fields such as computational biology and social network analysis. This study introduces HybridGNN, a novel Graph Neural Network model designed to efficiently address complex matching problems at scale. HybridGNN leverages a combination of Graph Attention Networks (GATv2), Graph SAGE (SAGEConv), and Graph Isomorphism Networks (GIN) layers to enhance computational efficiency and model performance. Through extensive ablation experiments, we identify that while the SAGEConv layer demonstrates suboptimal performance in terms of accuracy and F1-score, configurations incorporating GATv2 and GIN layers show significant improvements. Specifically, in six-layer GNN architectures, the combinations of GATv2 and GIN layers with ratios of 4:2 and 5:1 yield superior accuracy and F1-score. Therefore, we name these GNN configurations HybridGNN1 and HybridGNN2. Additionally, techniques such as mixed precision training, gradient accumulation, and Jumping Knowledge networks are integrated to further optimize performance. Evaluations on an email communication dataset reveal that HybridGNNs outperform traditional algorithms such as the Hopcroft–Karp algorithm, the Hungarian algorithm, and the Blossom/Edmonds’ algorithm, particularly for large and complex graphs. These findings highlight HybridGNN’s robust capability to solve maximum matching problems in bipartite graphs, making it a powerful tool for analyzing large-scale and intricate graph data. Furthermore, our study aligns with the goals of the Symmetry and Asymmetry Study in Graph Theory special issue by exploring the role of symmetry in bipartite graph structures. By leveraging GNNs, we address the challenges related to symmetry and asymmetry in graph properties, thereby improving the reliability and fault tolerance of complex networks.

Extremal bipartite independence number and balanced coloring

In this paper, we establish a couple of results on extremal problems in bipartite graphs. Firstly, we show that every sufficiently large bipartite graph with average degree D and with n vertices on each side has a balanced independent set containing (1−ϵ)logDDn vertices from each side for small ϵ>0. Secondly, we prove that the vertex set of every sufficiently large balanced bipartite graph with maximum degree at most Δ can be partitioned into (1+ϵ)ΔlogΔ balanced independent sets. Both of these results are algorithmic and best possible up to a factor of 2, which might be hard to improve as evidenced by the phenomenon known as ‘algorithmic barrier’ in the literature. The first result improves a recent theorem of Axenovich, Sereni, Snyder, and Weber in a slightly more general setting. The second result improves a theorem of Feige and Kogan about coloring balanced bipartite graphs.

Research on Matching and Vertex Cover Problems in Bipartite Graphs using Simplex Method

This paper considers a bipartite graph where a perfect matching does not necessarily exist. Linear programming is used in this paper as a special case of the linear program is the assignment problem, which is another name for the weighted maximum matching problem. The objective is to show that linear programming, in particular the simplex algorithm, can be used to calculate maximum weight matchings and minimum weighted vertex covers. This study also calculates and shows the equivalence of the maximum matching and minimum vertex cover cardinalities, and uses linear programming duality to present its relevance to König's theorem. Finally, Hall’s marriage theorem is used to explicitly prove the absence of a perfect matching in particular graphs. There exist many algorithms developed over the years that are designed to solve maximum matching problems in polynomial time, but the simplex method is one of the oldest and simplest algorithms to understand in solving these problems.

Data-Driven Matching Protocol for Vehicle-to-Vehicle Energy Management Considering Privacy Preservation

To address the increasing concern on environmental issues, electric vehicles (EVs) have been attracting more attention. On the one hand, the high penetration level of EVs will bring potential risks to the electricity grids. On the other hand, EVs can enhance the flexibility of the grids via vehicle-to-grid (V2G) technology. With the advancement of digitalization and communication technologies, the concept of vehicle-to-vehicle (V2V) has emerged, which can provide an alternative charging way for EVs to relieve the charging overload problem in the power systems. In this article, we present a data-driven matching protocol for V2V energy management. First, deep reinforcement learning (DRL) is utilized to learn the long-term reward of the matching action based on the formulated Markov decision process (MDP) at an offline stage. To protect the private information of the EV owners, a federated learning framework is proposed where the cooperation between different EV aggregators can be realized without sharing sensitive information. At the online matching stage, a matching optimization model is established and is converted into a bipartite graph problem to enhance the computation efficiency. The proposed methodology is verified in case studies. Simulation results show that the proposed methodology can help EV owners save costs and increase revenues while hedging the trading risks. Besides, the local energy self-sufficiency is increased, indicating that less burden is brought to the electricity network.

On the eigenvalues of Laplacian ABC-matrix of graphs

For a simple graph G, the ABC-index is a degree based topological index and is defined as where dv is the degree of the vertex υ in G. Recently, the Laplacian ABC-matrix was introduced in [22] is defined by where is the diagonal matrix of ABC-degrees and Ã(G) is the ABC-matrix of G: The eigenvalues of the matrix are called the Laplacian ABC-eigenvalues of G. In the article, we consider the problem of characterization of connected graphs having exactly three distinct Laplacian ABC-eigenvalues. We solve this problem for bipartite graphs, multipartite graphs, unicyclic graphs, regular graphs and prove the non-existence of such graphs with diameter greater than 2. We introduce the concept of trace norm of the matrix called the Laplacian ABC-energy of G. We obtain some upper and lower bounds for the Laplacian ABC-energy and characterize the extremal graphs which attain these bounds.

Relating dissociation, independence, and matchings

A dissociation set in a graph is a set of vertices inducing a subgraph of maximum degree at most 1. Computing the dissociation number diss ( G ) of a given graph G , defined as the order of a maximum dissociation set in G , is algorithmically hard even when G is restricted to be bipartite. Recently, Hosseinian and Butenko proposed a simple 4 3 -approximation algorithm for the dissociation number problem in bipartite graphs. Their result relies on the inequality diss ( G ) ≤ 4 3 α ( G − M ) , where G is a bipartite graph, M is a maximum matching in G , and α ( G − M ) denotes the independence number of G − M . We show that the pairs ( G , M ) for which this inequality holds with equality can be recognized efficiently, and that a maximum dissociation set can be determined for them efficiently. The dissociation number of a graph G satisfies max { α ( G ) , 2 ν s ( G ) } ≤ diss ( G ) ≤ α ( G ) + ν s ( G ) ≤ 2 α ( G ) , where ν s ( G ) denotes the induced matching number of G . We show that deciding whether diss ( G ) equals any of the four terms α ( G ) , 2 ν s ( G ) , α ( G ) + ν s ( G ) , and 2 α ( G ) is NP-hard.

A Learning and Operation Planning Method for Uber Energy Storage System: Order Dispatch

With the increasing penetration of intermittent renewable energy resources, electricity distribution networks may face many challenges in terms of system security and reliability. In this context, mobile power sources can provide various distribution network services, including load leveling, peak shaving, voltage regulation, and emergency backup. Different from the stationary energy storage system (SESS), mobile power sources show advantages in mobility and flexibility. In the current literature, the dispatch of the mobile power sources relies on day-ahead scheduling based on optimization, which lacks the capability of dealing with emerging power problems. In order to realize that the mobile power sources can provide on-demand local service efficiently, an uber energy storage system (UESS) is presented based on a learning and planning integrated approach. First, each bus generates the UESS service order. Then, a centralized platform dispatches the orders to the UESSs through a planning problem. To solve the dispatch problem efficiently, we convert the optimization to a bipartite graph problem with low complexity. With the assistance of deep reinforcement learning, the value of each order dispatch action is learned, and the weight of each edge in the bipartite graph equals the corresponding action value. The proposed method is verified in case studies. Simulation results reveal that the daily cost savings and the finish rate of the proposed method are around <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\$ $ </tex-math></inline-formula> 1562 and 15% higher than that of the nearest rule, respectively. Compared with the case without UESS, the voltage violation and the power loss issues are alleviated, and the profit of the system operator can be increased by 16.7%. Compared with SESS, the load curtailment cost with UESS can be reduced by 21.9k under contingency, which indicates that the resilience of the system is enhanced. Compared with MESS, UESS can better realize load restoration under emergencies.

Multi-task dispatch of shared autonomous electric vehicles for Mobility-on-Demand services – combination of deep reinforcement learning and combinatorial optimization method

The Autonomous Mobility-on-Demand system is an emerging green and sustainable transportation system providing on-demand mobility services for urban residents. To achieve the best recharging, delivering, and repositioning task assignment decision-making process for shared autonomous electric vehicles, this paper formulates the fleet dynamic operating process into a multi-agent multi-task dynamic dispatching problem based on Markov Decision Process. Specifically, the decision-making process at each time step is divided into 3 sub-processes, among which recharging and delivery task assignment processes are transformed into a maximum weight matching problem of bipartite graph respectively, and the repositioning task assignment process is quantified as a maximum flow problem. Kuhn-Munkres Algorithm and Edmond-Karp Algorithm are adopted to solve the above two mathematical problems to achieve the optimal task allocation policy. To further improve the dispatching performance, a new instant reward function balancing order income with trip satisfaction is designed, and a state-value function estimated by Back Propagation-Deep Neural Network is defined as a matching degree between each shared autonomous electric vehicle and each delivery task. The numerical results show that: (i) a reward function focusing on income and satisfaction can increase total revenue by 33.2%, (ii) the introduction of task allocation repositioning increases total revenue by 50.0%, (iii) a re-estimated state value function leads to a 2.8% increase in total revenue, (iv) the combination of charging and task repositioning can reduce user waiting time and significantly improve user satisfaction with the trip.

Connectivity keeping paths in k-connected bipartite graphs

In 2010, Mader ((2010) [10]) proved that every k-connected graph G with minimum degree at least ⌊3k2⌋+m−1 contains a path P of order m such that G−V(P) is still k-connected. In this paper, we consider similar problem for bipartite graphs, and prove that every k-connected bipartite graph G with minimum degree at least k+m contains a path P of order m such that G−V(P) is still k-connected.

Генерический полиномиальный алгоритм для проблемы изоморфизма двудольных графов

The isomorphism problem for bipartite graphs is as computationally hard as the well-known isomorphism problem for arbitrary finite graphs, for which no polynomial algorithms are known. The paper proposes a polynomial generic algorithm for the bipartite graph isomorphism problem, which gives the correct answer on the set of almost all inputs, and gives an undefined answer on the rare remaining inputs.

Bipartite Independence Number in Graphs with Bounded Maximum Degree

We consider a natural, yet seemingly not much studied, extremal problem in bipartite graphs. A bi-hole of size $t$ in a bipartite graph $G$ with a fixed bipartition is an independent set with exact...

From Independent Sets and Vertex Colorings to Isotropic Spaces and Isotropic Decompositions: Another Bridge between Graphs and Alternating Matrix Spaces

In the 1970s, Lovász built a bridge between graphs and alternating matrix spaces, in the context of perfect matchings [Proceedings of FCT, 1979, pp. 565--574]. A similar connection between bipartite graphs and matrix spaces plays a key role in the recent resolutions of the noncommutative rank problem [A. Garg et al., Proceedings of FOCS, 2016, pp. 109--117; G. Ivanyos, Y. Qiao, and K. V. Subrahmanyam, Comput. Complexity, 26 (2017), pp. 717--763]. In this paper, we lay the foundation for another bridge between graphs and alternating matrix spaces, in the context of independent sets and vertex colorings. The corresponding structures in alternating matrix spaces are isotropic spaces and isotropic decompositions, both useful structures in group theory and manifold theory. We first show that the maximum independent set problem and the vertex $c$-coloring problem reduce to the maximum isotropic space problem and the isotropic $c$-decomposition problem, respectively. Next, we show that several topics and results about independent sets and vertex colorings have natural correspondences for isotropic spaces and decompositions. These include algorithmic problems, such as the maximum independent set problem for bipartite graphs, and exact exponential-time algorithms for the chromatic number, as well as mathematical questions, such as the number of maximal independent sets, and the relation between the maximum degree and the chromatic number. These connections lead to new interactions between graph theory and algebra. Some results have concrete applications to group theory and manifold theory, and we initiate a variant of these structures in the context of quantum information theory. Finally, we propose several open questions for further exploration.

Designing low‐diameter interconnection networks with multi‐ported host‐switch graphs

SummaryA host‐switch graph was originally proposed as a graph that represents a network topology of a computer systems with 1‐port host computers and ‐port switches. It has been studied from both theoretical and practical aspects in terms of the diameter, the average shortest path length, and the performance of real applications. In recent high‐performance computing systems, however, a host computer is connected to multiple switches by using InfiniBand, NVSwitch, or Omni‐Path, and consequently they provide high bandwidths. Since a host‐switch graph cannot represent such systems, this article extends a host‐switch graph so that it can represent such systems. As a result, a host‐switch graph can include multi‐ported hosts. Furthermore, we propose to use multi‐port hosts for reducing the diameter. We show that the diameter minimization is equivalent to solving the degree diameter problem for bipartite graphs of diameter three. Our experimental results show that we can drastically reduce the diameter as well as increasing the bandwidth and improves performance of MPI applications by up to 162% as compared with networks with single‐ported hosts.

Approximability of the independent feedback vertex set problem for bipartite graphs

Given an undirected graph G with n vertices, the independent feedback vertex set problem is to find a vertex subset F of G with the minimum number of vertices such that F is both an independent set and a feedback vertex set of G, if it exists. This problem is known to be NP-hard for bipartite planar graphs of maximum degree four. In this paper, we study the approximability of the problem. We first show that, for any fixed ε>0, unless P=NP, there exists no polynomial-time n1−ε-approximation algorithm even for bipartite planar graphs. We then give an α(Δ−1)/2-approximation algorithm for bipartite graphs G of maximum degree Δ, which runs in t(α,G)+O(Δn) time, under the assumption that there is an α-approximation algorithm for the original feedback vertex set problem on bipartite graphs which runs in t(α,G) time. This algorithmic result also yields a polynomial-time (exact) algorithm for the independent feedback vertex set problem on bipartite graphs of maximum degree three.

A user-knowledge crowdsourcing task assignment model and heuristic algorithm for Expert Knowledge Recommendation Systems

Expert Knowledge Recommendation System (EKRS) is a scientific research assistant system that actively provides experts with the latest domain knowledge according to their professional knowledge background. This paper applies the crowdsourcing task assignment method, taking experts as users and recommending corresponding professional knowledge as tasks. To solve the problems of inaccurate user-knowledge matching and low assignments in EKRS, a user-knowledge task assignment model is established. To maximize the number of global task assignments, the model first applies an improved greedy assignment algorithm to convert the user-knowledge task maximum assignment problem into the maximum weight problem in bipartite graphs. Based on the matching value between a task and a user, a task is assigned to the user with a high matching value. Then, the assigned tasks are sorted with the tree decomposition technique to obtain the optimal task scheduling scheme. The heuristic depth-first search algorithm (DFS+HA) is used to update the boundaries of the heuristic function quickly, and the assignment scheme of the optimal solution can be obtained efficiently through the upper and lower bounds of the search process. Finally, the algorithm was experimentally verified with artificial data sets and the real data extracted from EKRS. The experimental results indicated that the algorithm proposed in this paper can improve the amount of user-knowledge task assignment in EKRS, find the optimal assignment scheme to maximize the number of global task assignments, and improve the search efficiency.

Weighted bilateral K-means algorithm for fast co-clustering and fast spectral clustering

Bipartite spectral graph partition (BSGP) is a school of the most well-known algorithms designed for the bipartite graph partition problem. It is also a fundamental mathematical model widely used in the tasks of co-clustering and fast spectral clustering. In BSGP, the key is to find the minimal normalized cuts (Ncuts) of bipartite graph. However, the convolutional BSGP algorithms usually need to use the singular value decomposition (SVD) to find the minimal Ncuts, which is computational prohibitive. Under this circumstance, the application range of those methods would be limited when the volume of the dataset is huge or the dimension of features is high. To overcome this problem, this paper proposes a novel weighted bilateral k-means (WBKM) algorithm and applies it for co-clustering and fast spectral clustering. Specifically, WBKM is a relaxation of the problem of finding the minimal Ncuts of bipartite graph, so it can be seen as a new solution for the minimal-Ncuts problem in bipartite graph. Different from the conventional relaxation ways, WBKM relaxes the minimal-Ncuts problem to a Non-negative decomposition problem which can be solved by an efficient iterative method. Therefore, the running speed of the proposed method is much faster. Besides, as our model can directly output the clustering results without any help of post-procedures, its solution tends to be more close to the ideal solution of the minimal-Ncuts problem than that of the conventional BSGP algorithms. To demonstrate the effectiveness and efficiency of the proposed method, extensive experiments on various types of datasets are conducted. Compared with other state-of-the-art methods, the proposed WBKM not only has faster computational speed, but also achieves more promising clustering results.

Relationship of two formulations for shortest bibranchings

The shortest bibranching problem is a common generalization of the minimum-weight edge cover problem in bipartite graphs and the minimum-weight arborescence problem in directed graphs. For the shortest bibranching problem, an efficient primal-dual algorithm is given by Keijsper and Pendavingh (J Comb Theory Ser B 73:130–145, 1998), and the tractability of the problem is ascribed to total dual integrality in a linear programming formulation by Schrijver (Ann Discret Math 16:261–280, 1982). Another view on the tractability of this problem is afforded by a valuated matroid intersection formulation by Takazawa (Jpn J Ind Appl Math 29:561–573, 2012). In the present paper, we discuss the relationship between these two formulations for the shortest bibranching problem. We first demonstrate that the valuated matroid intersection formulation can be derived from the linear programming formulation through the Benders decomposition, where integrality is preserved in the decomposition process and the resulting convex programming is endowed with discrete convexity. We then show how a pair of primal and dual optimal solutions of one formulation is constructed from that of the other formulation, thereby providing a connection between polyhedral combinatorics and discrete convex analysis.

Algorithmic aspects of k-part degree restricted domination in graphs

The concept of network is predominantly used in several applications of computer communication networks. It is also a fact that the dominating set acts as a virtual backbone in a communication network. These networks are vulnerable to breakdown due to various causes, including traffic congestion. In such an environment, it is necessary to regulate the traffic so that these vulnerabilities could be reasonably controlled. Motivated by this, [Formula: see text]-part degree restricted domination is defined as follows. For a positive integer [Formula: see text], a dominating set [Formula: see text] of a graph [Formula: see text] is said to be a [Formula: see text]-part degree restricted dominating set ([Formula: see text]-DRD set) if for all [Formula: see text], there exists a set [Formula: see text] such that [Formula: see text] and [Formula: see text]. The minimum cardinality of a [Formula: see text]-DRD set of a graph [Formula: see text] is called the [Formula: see text]-part degree restricted domination number of [Formula: see text] and is denoted by [Formula: see text]. In this paper, we present a polynomial time reduction that proves the NP -completeness of the [Formula: see text]-part degree restricted domination problem for bipartite graphs, chordal graphs, undirected path graphs, chordal bipartite graphs, circle graphs, planar graphs and split graphs. We propose a polynomial time algorithm to compute a minimum [Formula: see text]-DRD set of a tree and minimal [Formula: see text]-DRD set of a graph.

Generalized and Sub-Optimal Bipartite Constraints for Conflict-Based Search

The main idea of conflict-based search (CBS), a popular, state-of-the-art algorithm for multi-agent pathfinding is to resolve conflicts between agents by systematically adding constraints to agents. Recently, CBS has been adapted for new domains and variants, including non-unit costs and continuous time settings. These adaptations require new types of constraints. This paper introduces a new automatic constraint generation technique called bipartite reduction (BR). BR converts the constraint generation step of CBS to a surrogate bipartite graph problem. The properties of BR guarantee completeness and optimality for CBS. Also, BR's properties may be relaxed to obtain suboptimal solutions. Empirical results show that BR yields significant speedups in 2k connected grids over the previous state-of-the-art for both optimal and suboptimal search.

Many-to-Many Recruitment and Scheduling in Spatial Mobile Crowdsourcing

Spatial mobile crowdsourcing (SMCS) enables a task requester to commission workers to physically travel to specific locations to perform a set of spatial assignments (i.e., tasks are related to a specific geographical location besides time). To efficiently perform such tasks and guarantee the best possible quality of returned results, optimizing the worker recruitment and task assignment processes must be conducted. Because both workers and task requesters impose certain criteria, this procedure is not obvious. To tackle this issue, we propose a novel formulation of the SMCS recruitment where task matching and worker scheduling are jointly optimized. A Mixed Integer Linear Program (MILP) is first developed to optimally maximize the quality of matching measured as a weighted score function of different recruitment metrics while determining the trajectory of each selected worker executing tasks. To cope with NP-hardness, we propose a heuristic SMCS recruitment approach allowing the achievement of sub-optimal matching and recruitment solution by iteratively solving a weighted bipartite graph problem. Simulation results illustrate the performance of the SMCS framework for selected scenarios and show that our proposed SMCS recruitment algorithm outperforms an existing greedy recruitment approach. Moreover, compared to the optimal MILP solution, the proposed SMCS recruitment approach achieves close results with significant computational time saving.