arXiv subjects

Kaiqiang Yu

Publications and source records attributed to Kaiqiang Yu.

16 recordsLinked to original sources

arXiv2026

Revisiting the Maximum Defective Clique Problem: Faster Branching and a Tighter Upper Bound

The $k$-defective clique model relaxes the strict completeness constraint of the traditional clique by allowing up to $k$ missing edges, providing a robust formulation for detecting cohesive structures in noisy graphs. Consequently, the maximum $k$-defective clique problem has attracted significant attention. State-of-the-art exact algorithms predominantly adopt the branch-and-bound framework, which recursively partitions the current problem instance (or branch) into two sub-problems via a branching procedure, until each sub-problem becomes trivially solvable. However, this strategy often leads to excessive branching by overlooking intermediate sub-problems that are non-trivial yet efficiently solvable. While recent studies have attempted to refine branching procedures, they fail to address this structural redundancy. To address this, we propose BBRes, a framework that incorporates a novel early termination strategy into the recursive branching process. By employing a specialized polynomial-time solver to identify and resolve tractable sub-instances, BBRes effectively avoids redundant branching steps. Additionally, we design a tailored branching strategy that synergizes with this termination mechanism. As a result, BBRes achieves an improved theoretical worst-case time complexity. To enhance practical performance, we propose a tighter upper bound based on a novel double graph coloring method integrated with max-flow techniques, which is orthogonal to the branching framework. Extensive experiments show that BBRes achieves at least 2X speedup over state-of-the-art methods on a substantial fraction of the datasets.

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arXiv2026

Maximal Biclique Enumeration with Improved Worst-Case Time Complexity Guarantee: A Partition-Oriented Strategy

The maximal biclique enumeration problem in bipartite graphs is fundamental and has numerous applications in E-commerce and transaction networks. Most existing studies adopt a branch-and-bound framework, which recursively expands a partial biclique with a vertex until no further vertices can be added. Equipped with a basic pivot selection strategy, all state-of-the-art methods have a worst-case time complexity no better than $O(m\cdot (\sqrt{2})^n)$}, where $m$ and $n$ are the number of edges and vertices in the graph, respectively. In this paper, we introduce a new branch-and-bound (BB) algorithm \texttt{IPS}. In \texttt{IPS}, we relax the strict stopping criterion of existing methods by allowing termination when all maximal bicliques within the current branch can be outputted in the time proportional to the number of maximal bicliques inside, reducing the total number of branches required. Second, to fully unleash the power of the new termination condition, we propose an improved pivot selection strategy, which well aligns with the new termination condition to achieve better theoretical and practical performance. Formally, \texttt{IPS} improves the worst-case time complexity to $O(m\cdot \alpha ^n + n\cdot \beta)$, where $\alpha (\approx 1.3954)$ is the largest positive root of $x^4-2x-1=0$ and $\beta$ represents the number of maximal bicliques in the graph, respectively. This result surpasses that of all existing algorithms given that $\alpha$ is strictly smaller than $\sqrt{2}$ and $\beta$ is at most $(\sqrt{2})^n-2$ theoretically. Furthermore, we apply an inclusion-exclusion-based framework to boost the performance of \texttt{IPS}, improving the worst-case time complexity to $O(n\cdot \gamma^2\cdot\alpha^\gamma + \gamma\cdot \beta)$ for large sparse graphs ($\gamma$ is a parameter satisfying $\gamma \ll n$ for sparse graphs).

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arXiv2026

Efficient Computation of Maximum Flexi-Clique in Networks

Discovering large cohesive subgraphs is a key task for graph mining. Existing models, such as clique, k-plex, and {\gamma}-quasi-clique, use fixed density thresholds that overlook the natural decay of connectivity as the subgraph size increases. The Flexi-clique model overcomes this limitation by imposing a degree constraint that grows sub-linearly with subgraph size. We provide the algorithmic study of Flexi-clique, proving its NP-hardness and analysing its non-hereditary properties. To address its computational challenge, we propose the Flexi-Prune Algorithm FPA, a fast heuristic using core-based seeding and connectivity-aware pruning, and the Efficient Branch-and-Bound Algorithm EBA, an exact framework enhanced with multiple pruning rules. Experiments on large real-world and synthetic networks demonstrate that FPA achieves near-optimal quality at much lower cost, while EBA efficiently computes exact solutions. Flexi-clique thus provides a practical and scalable model for discovering large, meaningful subgraphs in complex networks.

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arXiv2025

Temporal $k$-Core Query, Revisited

Querying cohesive subgraphs in temporal graphs is essential for understanding the dynamic structure of real-world networks, such as evolving communities in social platforms, shifting hyperlink structures on the Web, and transient communication patterns in call networks. Recently, research has focused on the temporal $k$-core query, which aims to identify all $k$-cores across all possible time sub-intervals within a given query interval. The state-of-the-art algorithm OTCD mitigates redundant computations over overlapping sub-intervals by exploiting inclusion relationships among $k$-cores in different time intervals. Nevertheless, OTCD remains limited in scalability due to the combinatorial growth in interval enumeration and repeated processing. In this paper, we revisit the temporal $k$-core query problem and introduce a novel algorithm CoreT, which dynamically records the earliest timestamp at which each vertex or edge enters a $k$-core. This strategy enables substantial pruning of redundant computations. As a result, CoreT requires only a single pass over the query interval and achieves improved time complexity, which is linear in both the number of temporal edges within the query interval and the duration of the interval, making it highly scalable for long-term temporal analysis. Experimental results on large real-world datasets show that CoreT achieves up to four orders of magnitude speedup compared to the existing state-of-the-art OTCD, demonstrating its effectiveness and scalability for temporal $k$-core analysis.

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arXiv2025

Maximum Degree-Based Quasi-Clique Search via an Iterative Framework

Cohesive subgraph mining is a fundamental problem in graph theory with numerous real-world applications, such as social network analysis and protein-protein interaction modeling. Among various cohesive subgraphs, the $\gamma$-quasi-clique is widely studied for its flexibility in requiring each vertex to connect to at least a $\gamma$ proportion of other vertices in the subgraph. However, solving the maximum $\gamma$-quasi-clique problem is NP-hard and further complicated by the lack of the hereditary property, which makes designing efficient pruning strategies challenging. Existing algorithms, such as DDA and FastQC, either struggle with scalability or exhibit significant performance declines for small values of $\gamma$. In this paper, we propose a novel algorithm, IterQC, which reformulates the maximum $\gamma$-quasi-clique problem as a series of $k$-plex problems that possess the hereditary property. IterQC introduces a non-trivial iterative framework and incorporates two key optimization techniques: (1) the pseudo lower bound (pseudo LB) technique, which leverages information across iterations to improve the efficiency of branch-and-bound searches, and (2) the preprocessing technique that reduces problem size and unnecessary iterations. Extensive experiments demonstrate that IterQC achieves up to four orders of magnitude speedup and solves significantly more graph instances compared to state-of-the-art algorithms DDA and FastQC.

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arXiv2025

Fast Maximum Common Subgraph Search: A Redundancy-Reduced Backtracking Approach

Given two input graphs, finding the largest subgraph that occurs in both, i.e., finding the maximum common subgraph, is a fundamental operator for evaluating the similarity between two graphs in graph data analysis. Existing works for solving the problem are of either theoretical or practical interest, but not both. Specifically, the algorithms with a theoretical guarantee on the running time are known to be not practically efficient; algorithms following the recently proposed backtracking framework called McSplit, run fast in practice but do not have any theoretical guarantees. In this paper, we propose a new backtracking algorithm called RRSplit, which at once achieves better practical efficiency and provides a non-trivial theoretical guarantee on the worst-case running time. To achieve the former, we develop a series of reductions and upper bounds for reducing redundant computations, i.e., the time for exploring some unpromising branches of exploration that hold no maximum common subgraph. To achieve the latter, we formally prove that RRSplit incurs a worst-case time complexity which matches the best-known complexity for the problem. Finally, we conduct extensive experiments on four benchmark graph collections, and the results demonstrate that our algorithm outperforms the practical state-of-the-art by several orders of magnitude.

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arXiv2024

Maximal Clique Enumeration with Hybrid Branching and Early Termination

Maximal clique enumeration (MCE) is crucial for tasks like community detection and biological network analysis. Existing algorithms typically adopt the branch-and-bound framework with the vertex-oriented Bron-Kerbosch (BK) branching strategy, which forms the sub-branches by expanding the partial clique with a vertex. In this paper, we present a novel approach called HBBMC, a hybrid framework combining vertex-oriented BK branching and edge-oriented BK branching, where the latter adopts a branch-and-bound framework which forms the sub-branches by expanding the partial clique with an edge. This hybrid strategy enables more effective pruning and helps achieve a worst-case time complexity better than the best known one under a condition that holds for the majority of real-world graphs. To further enhance efficiency, we introduce an early termination technique, which leverages the topological information of the graphs and constructs the maximal cliques directly without branching. Our early termination technique is applicable to all branch-and-bound frameworks. Extensive experiments demonstrate the superior performance of our techniques.

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arXiv2024

Maximum $k$-Plex Search: An Alternated Reduction-and-Bound Method

$k$-plexes relax cliques by allowing each vertex to disconnect to at most $k$ vertices. Finding a maximum $k$-plex in a graph is a fundamental operator in graph mining and has been receiving significant attention from various domains. The state-of-the-art algorithms all adopt the branch-reduction-and-bound (BRB) framework where a key step, called reduction-and-bound (RB), is used for narrowing down the search space. A common practice of RB in existing works is SeqRB, which sequentially conducts the reduction process followed by the bounding process once at a branch. However, these algorithms suffer from the efficiency issues. In this paper, we propose a new alternated reduction-and-bound method AltRB for conducting RB. AltRB first partitions a branch into two parts and then alternatively and iteratively conducts the reduction process and the bounding process at each part of a branch. With newly-designed reduction rules and bounding methods, AltRB is superior to SeqRB in effectively narrowing down the search space in both theory and practice. Further, to boost the performance of BRB algorithms, we develop efficient and effective pre-processing methods which reduce the size of the input graph and heuristically compute a large $k$-plex as the lower bound. We conduct extensive experiments on 664 real and synthetic graphs. The experimental results show that our proposed algorithm kPEX with AltRB and novel pre-processing techniques runs up to two orders of magnitude faster and solves more instances than state-of-the-art algorithms.

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arXiv2023

Efficient $k$-Clique Listing: An Edge-Oriented Branching Strategy

$k$-clique listing is a vital graph mining operator with diverse applications in various networks. The state-of-the-art algorithms all adopt a branch-and-bound (BB) framework with a vertex-oriented branching strategy (called VBBkC), which forms a sub-branch by expanding a partial $k$-clique with a vertex. These algorithms have the time complexity of $O(k m (\delta/2)^{k-2})$, where $m$ is the number of edges in the graph and $\delta$ is the degeneracy of the graph. In this paper, we propose a BB framework with a new edge-oriented branching (called EBBkC), which forms a sub-branch by expanding a partial $k$-clique with two vertices that connect each other (which correspond to an edge). We explore various edge orderings for EBBkC such that it achieves a time complexity of $O(\delta m + k m (\tau/2)^{k-2})$, where $\tau$ is an integer related to the maximum truss number of the graph and we have $\tau < \delta$. The time complexity of EBBkC is better than that of VBBkC algorithms for $k>3$ since both $O(\delta m)$ and $O(k m (\tau/2)^{k-2})$ are bounded by $O(k m (\delta/2)^{k-2})$. Furthermore, we develop specialized algorithms for sub-branches on dense graphs so that we can early-terminate them and apply the specialized algorithms. We conduct extensive experiments on 19 real graphs, and the results show that our newly developed EBBkC-based algorithms with the early termination technique consistently and largely outperform the state-of-the-art (VBBkC-based) algorithms.

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arXiv2023

Fast Maximal Quasi-clique Enumeration: A Pruning and Branching Co-Design Approach

Mining cohesive subgraphs from a graph is a fundamental problem in graph data analysis. One notable cohesive structure is $\gamma$-quasi-clique (QC), where each vertex connects at least a fraction $\gamma$ of the other vertices inside. Enumerating maximal $\gamma$-quasi-cliques (MQCs) of a graph has been widely studied. One common practice of finding all MQCs is to (1) find a set of QCs containing all MQCs and then (2) filter out non-maximal QCs. While quite a few algorithms have been developed (which are branch-and-bound algorithms) for finding a set of QCs that contains all MQCs, all focus on sharpening the pruning techniques and devote little effort to improving the branching part. As a result, they provide no guarantee on pruning branches and all have the worst-case time complexity of $O^*(2^n)$, where $O^*$ suppresses the polynomials and $n$ is the number of vertices in the graph. In this paper, we focus on the problem of finding a set of QCs containing all MQCs but deviate from further sharpening the pruning techniques as existing methods do. We pay attention to both the pruning and branching parts and develop new pruning techniques and branching methods that would suit each other better towards pruning more branches both theoretically and practically. Specifically, we develop a new branch-and-bound algorithm called FastQC based on newly developed pruning techniques and branching methods, which improves the worst-case time complexity to $O^*(\alpha_k^n)$, where $\alpha_k$ is a positive real number strictly smaller than 2. Furthermore, we develop a divide-and-conquer strategy for boosting the performance of FastQC. Finally, we conduct extensive experiments on both real and synthetic datasets, and the results show that our algorithms are up to two orders of magnitude faster than the state-of-the-art on real datasets.

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arXiv2022

Maximum $k$-Biplex Search on Bipartite Graphs: A Symmetric-BK Branching Approach

Enumerating maximal $k$-biplexes (MBPs) of a bipartite graph has been used for applications such as fraud detection. Nevertheless, there usually exists an exponential number of MBPs, which brings up two issues when enumerating MBPs, namely the effectiveness issue (many MBPs are of low values) and the efficiency issue (enumerating all MBPs is not affordable on large graphs). Existing proposals of tackling this problem impose constraints on the number of vertices of each MBP to be enumerated, yet they are still not sufficient (e.g., they require to specify the constraints, which is often not user-friendly, and cannot control the number of MBPs to be enumerated directly). Therefore, in this paper, we study the problem of finding $K$ MBPs with the most edges called MaxBPs, where $K$ is a positive integral user parameter. The new proposal well avoids the drawbacks of existing proposals. We formally prove the NP-hardness of the problem. We then design two branch-and-bound algorithms, among which, the better one called FastBB improves the worst-case time complexity to $O^*(\gamma_k^ n)$, where $O^*$ suppresses the polynomials, $\gamma_k$ is a real number that relies on $k$ and is strictly smaller than 2, and $n$ is the number of vertices in the graph. For example, for $k=1$, $\gamma_k$ is equal to $1.754$. We further introduce three techniques for boosting the performance of the branch-and-bound algorithms, among which, the best one called PBIE can further improve the time complexity to $O^*(\gamma_k^{d^3})$ for large sparse graphs, where $d$ is the maximum degree of the graph. We conduct extensive experiments on both real and synthetic datasets, and the results show that our algorithm is up to four orders of magnitude faster than all baselines and finding MaxBPs works better than finding all MBPs for a fraud detection application.

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arXiv2021

Efficient Algorithms for Maximal k-Biplex Enumeration

Mining maximal subgraphs with cohesive structures from a bipartite graph has been widely studied. One important cohesive structure on bipartite graphs is k-biplex, where each vertex on one side disconnects at most k vertices on the other side. In this paper, we study the maximal k-biplex enumeration problem which enumerates all maximal k-biplexes. Existing methods suffer from efficiency and/or scalability issues and have the time of waiting for the next output exponential w.r.t. the size of the input bipartite graph (i.e., an exponential delay). In this paper, we adopt a reverse search framework called bTraversal, which corresponds to a depth-first search (DFS) procedure on an implicit solution graph on top of all maximal k-biplexes. We then develop a series of techniques for improving and implementing this framework including (1) carefully selecting an initial solution to start DFS, (2) pruning the vast majority of links from the solution graph of bTraversal, and (3) implementing abstract procedures of the framework. The resulting algorithm is called iTraversal, which has its underlying solution graph significantly sparser than (around 0.1% of) that of bTraversal. Besides, iTraversal provides a guarantee of polynomial delay. Our experimental results on real and synthetic graphs, where the largest one contains more than one billion edges, show that our algorithm is up to four orders of magnitude faster than existing algorithms.

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arXiv2020

Reconfigurable Intelligent Surface Assisted Massive MIMO with Antenna Selection

Antenna selection is capable of reducing the hardware complexity of massive multiple-input multiple-output (MIMO) networks at the cost of certain performance degradation. Reconfigurable intelligent surface (RIS) has emerged as a cost-effective technique that can enhance the spectrum-efficiency of wireless networks by reconfiguring the propagation environment. By employing RIS to compensate the performance loss due to antenna selection, in this paper we propose a new network architecture, i.e., RIS-assisted massive MIMO system with antenna selection, to enhance the system performance while enjoying a low hardware cost. This is achieved by maximizing the channel capacity via joint antenna selection and passive beamforming while taking into account the cardinality constraint of active antennas and the unit-modulus constraints of all RIS elements. However, the formulated problem turns out to be highly intractable due to the non-convex constraints and coupled optimization variables, for which an alternating optimization framework is provided, yielding antenna selection and passive beamforming subproblems. The computationally efficient submodular optimization algorithms are developed to solve the antenna selection subproblem under different channel state information assumptions. The iterative algorithms based on block coordinate descent are further proposed for the passive beamforming design by exploiting the unique problem structures. Experimental results will demonstrate the algorithmic advantages and desirable performance of the proposed algorithms for RIS- assisted massive MIMO systems with antenna selection.

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arXiv2020

Reconfigurable Intelligent Surface Enhanced Cognitive Radio Networks

The cognitive radio (CR) network is a promising network architecture that meets the requirement of enhancing scarce radio spectrum utilization. Meanwhile, reconfigurable intelligent surfaces (RIS) is a promising solution to enhance the energy and spectrum efficiency of wireless networks by properly altering the signal propagation via tuning a large number of passive reflecting units. In this paper, we investigate the downlink transmit power minimization problem for the RIS-enhanced single-cell cognitive radio (CR) network coexisting with a single-cell primary radio (PR) network by jointly optimizing the transmit beamformers at the secondary user (SU) transmitter and the phase shift matrix at the RIS. The investigated problem is a highly intractable due to the coupled optimization variables and unit modulus constraint, for which an alternative minimization framework is presented. Furthermore, a novel difference-of-convex (DC) algorithm is developed to solve the resulting non-convex quadratic program by lifting it into a low-rank matrix optimization problem. We then represent non-convex rank-one constraint as a DC function by exploiting the difference between trace norm and spectral norm. The simulation results validate that our proposed algorithm outperforms the existing state-of-the-art methods.

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arXiv2020

Coordinated Passive Beamforming for Distributed Intelligent Reflecting Surfaces Network

Intelligent reflecting surface (IRS) is a proposing technology in 6G to enhance the performance of wireless networks by smartly reconfiguring the propagation environment with a large number of passive reflecting elements. However, current works mainly focus on single IRS-empowered wireless networks, where the channel rank deficiency problem has emerged. In this paper, we propose a distributed IRS-empowered communication network architecture, where multiple source-destination pairs communicate through multiple distributed IRSs. We further contribute to maximize the achievable sum-rates in this network via jointly optimizing the transmit power vector at the sources and the phase shift matrix with passive beamforming at all distributed IRSs. Unfortunately, this problem turns out to be non-convex and highly intractable, for which an alternating approach is developed via solving the resulting fractional programming problems alternatively. In particular, the closed-form expressions are proposed for coordinated passive beamforming at IRSs. The numerical results will demonstrate the algorithmic advantages and desirable performances of the distributed IRS-empowered communication network.

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arXiv2019

Efficient Algorithms for Densest Subgraph Discovery

Densest subgraph discovery (DSD) is a fundamental problem in graph mining. It has been studied for decades, and is widely used in various areas, including network science, biological analysis, and graph databases. Given a graph G, DSD aims to find a subgraph D of G with the highest density (e.g., the number of edges over the number of vertices in D). Because DSD is difficult to solve, we propose a new solution paradigm in this paper. Our main observation is that a densest subgraph can be accurately found through a k-core (a kind of dense subgraph of G), with theoretical guarantees. Based on this intuition, we develop efficient exact and approximation solutions for DSD. Moreover, our solutions are able to find the densest subgraphs for a wide range of graph density definitions, including clique-based and general pattern-based density. We have performed extensive experimental evaluation on eleven real datasets. Our results show that our algorithms are up to four orders of magnitude faster than existing approaches.

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