Maximum Weight Clique Problem Research Articles

A signal transmission strategy driven by gap-regulated exonuclease hydrolysis for hierarchical molecular networks

Exonucleases serve as efficient tools for signal processing and play an important role in biochemical reactions. Here, we identify the mechanism of cooperative exonuclease hydrolysis, offering a method to regulate the cooperative hydrolysis driven by exonucleases through the modulation of the number of bases in gap region. A signal transmission strategy capable of producing amplified orthogonal DNA signal is proposed to resolve the polarity of signals and byproducts, which provides a solution to overcome the signal attenuation. The gap-regulated mechanism combined with DNA strand displacement (DSD) reduces the unpredictable secondary structures, allowing for the coexistence of similar structures in hierarchical molecular networks. For the application of the strategy, a molecular computing model is constructed to solve the maximum weight clique problems (MWCP). This work enhances for our knowledge of these important enzymes and promises application prospects in molecular computing, signal detection, and nanomachines.

A new parallel tabu search algorithm for the optimization of the maximum vertex weight clique problem

SummaryThe efficiency of metaheuristic algorithms depends significantly on the number of fitness value evaluations performed on candidate solutions. In addition to various intelligent techniques used to obtain better results, parallelization of calculations can substantially improve the solutions in cases where the problem is NP‐hard and requires many evaluations. This study proposes a new parallel tabu search method for solving the Maximum Vertex Weight Clique Problem (MVWCP) on the Non‐Uniform Memory Access (NUMA) architectures using the OpenMP parallel programming paradigm. Achieving scalability in the NUMA architectures presents significant challenges due to the high complexity of their memory systems, which can lead to performance loss. However, our proposed Tabu‐NUMA algorithm provides up to speed‐up with 64 cores for ten basic problem instances in DIMACS‐W and BHOSLIB‐W benchmarks. And it improves the performance of the serial Multi Neighborhood Tabu Search (MN/TS) algorithm for 38 problem instances in DIMACS‐W and BHOSLIB‐W benchmarks. We further evaluate our algorithm on larger datasets with thousands of edges and vertices from Network Data Repository benchmark problem instances, and we report significant improvements in terms of speed up. Our results confirm that the Tabu‐NUMA algorithm is among the best recent algorithms for solving MVWCP on the NUMA architectures.

Robust Loop Closure Selection Based on Inter-Robot and Intra-Robot Consistency for Multi-Robot Map Fusion

In multi-robot simultaneous localization and mapping (SLAM) systems, the system must create a consistent global map with multiple local maps and loop closures between robot poses. However, false-positive loop closures caused by perceptual aliasing can severely distort the global map, especially in GNSS-denied areas, where a good prior of relative poses between robots is unavailable. In addition, the performance of the consistency metric in existing map fusion methods relies on accurate odometry from each robot. However, in practice, cumulative noise is inevitably present in robot trajectories, which leads to poor map fusion with existing methods. Thus, in this paper, we propose a robust consistency-based inter-robot and intra-robot loop closure selection algorithm for map fusion. We consider both pairwise-loop consistency and loop-odometry consistency to improve robustness against false-positive loop closures and accumulative noise in the odometry. Specifically, we select a reliable inter-robot loop closure measurement with a consistency-based strategy to provide an initial prior of relative pose between two robot trajectories and update the pose variables of the robot trajectories. The loop closure selection problem is formulated as a maximum edge weight clique problem in graph theory. A performance evaluation of the proposed method was conducted on the ManhattanOlson3500, modified CSAIL and Bicocca datasets, and the experimental results demonstrate that the proposed method outperforms the pairwise consistency measurement set maximization method (PCM) under severe accumulative noise and can be integrated with M-estimation methods.

A Bio-Inspired Technique for the Maximum Weighted Clique Problem

A Bio-Inspired Technique for the Maximum Weighted Clique Problem

A new exact algorithm for the maximum-weight clique problem based on a heuristic vertex-coloring and a backtrack search

In this paper we present an exact algorithm for the maximum-weight clique problem on arbitrary undirected graphs. The algorithm based on a fact that vertices from the same independent set couldn’t be included into the same maximum clique. Those independent sets are obtained from a heuristic vertex coloring where each of them is a color class. Color classes and a backtrack search are used for pruning branches of the maximum-weight clique search tree. Those pruning strategies together result in a very effective algorithm for the maximum-weight clique finding. Computational experiments with random graphs show that the new algorithm works faster than earlier published algorithms; especially on dense graphs.

Improving the performance of stochastic local search for maximum vertex weight clique problem using programming by optimization

The maximum vertex weight clique problem (MVWCP) is an important generalization of the maximum clique problem (MCP) that has a wide range of real-world applications. In situations where rigorous guarantees regarding the optimality of solutions are not required, MVWCP is usually solved using stochastic local search (SLS) algorithms, which also define the state of the art for solving this problem. However, there is no single SLS algorithm that gives the best performance across all classes of MVWCP instances, and it is challenging to effectively identify the most suitable algorithm for each class of MVWCP instances. In this work, we follow the paradigm of Programming by Optimization (PbO) to develop a new, flexible and highly parametric SLS framework for solving MVWCP, combining, for the first time, a broad range of effective heuristic mechanisms. By automatically configuring this PbO-MWC framework, we achieve substantial advances in the state of the art in solving MVWCP over a broad range of prominent benchmarks, including two derived from real-world applications in transplantation medicine (kidney exchange) and assessment of research excellence.

Throughput Maximization in Cloud-Radio Access Networks Using Cross-Layer Network Coding

Cloud radio access networks (C-RANs) are promising paradigms for the fifth-generation (5G) networks due to their interference management capabilities. In a C-RAN, a central processor (CP) is responsible for coordinating multiple Remote Radio Heads (RRHs) and scheduling users to their radio resource blocks (RRBs). In this paper, we develop a novel <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">cross-layer network coding (CLNC)</i> approach that proposes to optimize RRH’s transmit powers and user’s rates in making the coding decisions. As such, cross-layer throughput of the network is maximized. The joint user scheduling, file encoding, and power adaptation problem is solved by designing a subgraph for each RRB, in which each vertex represents potential user-RRH associations, encoded files, transmission rates, and power levels (PLs) for one RRB. It is then shown that the C-RAN throughput maximization problem is equivalent to a maximum-weight clique problem over the union of all such subgraphs, called herein the CRAN-CLNC graph. Numerical results revealed that the proposed joint and iterative schemes offer improved throughput performances as compared to the existing algorithms in the literature. Compared to our proposed joint scheme, our proposed iterative scheme has a certain degradation, roughly in the range of 9%–14%. This small degradation in the throughput performance of the iterative scheme comes at the achieved low computational complexity as compared to the high complexity of the joint scheme.

Avoidable vertices and edges in graphs: Existence, characterization, and applications

A vertex in a graph is simplicial if its neighborhood forms a clique. We consider three generalizations of the concept of simplicial vertices: avoidable vertices (also known as OCF-vertices), simplicial paths, and their common generalization avoidable paths. In an earlier version of this paper, published as an extended abstract in the proceedings of the 16th International Symposium on Algorithms and Data Structures (WADS 2019), we presented a general conjecture on the existence of avoidable paths. The conjecture was recently proved by Bonamy et al. (2020). The conjecture—now a theorem—implies a result due to Ohtsuki, Cheung, and Fujisawa from 1976 on the existence of avoidable vertices and a result due to Chvátal, Sritharan, and Rusu from 2002 on the existence of simplicial paths in graphs excluding all sufficiently long induced cycles. In turn, both of these results generalize Dirac’s classical result on the existence of simplicial vertices in chordal graphs.We prove that every graph with at least two edges has two avoidable edges, which strengthens the statement of the theorem of Bonamy et al. for the case of paths of length one. We point out a close relationship between avoidable vertices in a graph and its minimal triangulations, and identify new algorithmic uses of avoidable vertices, leading to new polynomially solvable cases of the maximum weight clique problem in two classes of graphs simultaneously generalizing chordal graphs and circular-arc graphs. Finally, we observe that the results about the existence of avoidable vertices and edges have interesting consequences for highly symmetric graphs: in a vertex-transitive graph every induced two-edge path closes to an induced cycle, while in an edge-transitive graph every three-edge path closes to a cycle and every induced three-edge path closes to an induced cycle.

A Semi-exact Algorithm for Quickly Computing A Maximum Weight Clique in Large Sparse Graphs

This paper explores techniques to quickly solve the maximum weight clique problem (MWCP) in very large scale sparse graphs. Due to their size, and the hardness of MWCP, it is infeasible to solve many of these graphs with exact algorithms. Although recent heuristic algorithms make progress in solving MWCP in large graphs, they still need considerable time to get a high-quality solution. In this work, we focus on solving MWCP for large sparse graphs within a short time limit. We propose a new method for MWCP which interleaves clique finding with data reduction rules. We propose novel ideas to make this process efficient, and develop an algorithm called FastWClq. Experiments on a broad range of large sparse graphs show that FastWClq finds better solutions than state-of-the-art algorithms while the running time of FastWClq is much shorter than the competitors for most instances. Further, FastWClq proves the optimality of its solutions for roughly half of the graphs, all with at least 105 vertices, with an average time of 21 seconds.

Parallelization of a branch-and-bound algorithm for the maximum weight clique problem

In this paper, parallelization techniques are proposed for the branch-and-bound algorithm OTClique for the maximum weight clique problem. OTClique consists of the precomputation phase and the branch-and-bound phase. The proposed algorithm parallelizes both of them. In the precomputation phase, the construction of optimal tables is parallelized. In the branch-and-bound phase, the proposed algorithm generates small subproblems and assigns them to threads. A technique to share lower and upper bounds is also proposed. Experiments using some benchmarks show that the proposed parallelization techniques improve the performance of OTClique. With an 8-core CPU, the computation time of OTClique becomes 6.91 times shorter on random graphs and 5.38 times on DIMACS benchmarks on average.

Using Statistical Measures and Machine Learning for Graph Reduction to Solve Maximum Weight Clique Problems.

In this article, we investigate problem reduction techniques using stochastic sampling and machine learning to tackle large-scale optimization problems. These techniques heuristically remove decision variables from the problem instance, that are not expected to be part of an optimal solution. First we investigate the use of statistical measures computed from stochastic sampling of feasible solutions compared with features computed directly from the instance data. Two measures are particularly useful for this: 1) a ranking-based measure, favoring decision variables that frequently appear in high-quality solutions; and 2) a correlation-based measure, favoring decision variables that are highly correlated with the objective values. To take this further we develop a machine learning approach, called Machine Learning for Problem Reduction (MLPR), that trains a supervised learning model on easy problem instances for which the optimal solution is known. This gives us a combination of features enabling us to better predict the decision variables that belong to the optimal solution for a given hard problem. We evaluate our approaches using a typical optimization problem on graphs-the maximum weight clique problem. The experimental results show our problem reduction techniques are very effective and can be used to boost the performance of existing solution methods.

Completion Time Minimization in Fog-RANs Using D2D Communications and Rate-Aware Network Coding

The device-to-device communication-aided fog radio access network, referred to as <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D2D-aided</i> F-RAN, takes advantage of caching at enhanced remote radio heads (eRRHs) and D2D proximity for improved system performance. For D2D-aided F-RAN, we develop a framework that exploits the cached contents at eRRHs, their transmission rates/powers, and previously received contents by different users to deliver the requesting contents to users with a minimum completion time. Given the intractability of the completion time minimization problem, we formulate it at each transmission by approximating the completion time and decoupling it into two subproblems. In the first subproblem, we minimize the possible completion time in eRRH downlink transmissions, while in the second subproblem, we maximize the number of users to be scheduled on D2D links. We design two theoretical graphs, namely <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">interference-aware</i> instantly decodable network coding (IA-IDNC) and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D2D conflict</i> graphs to reformulate two subproblems as maximum weight clique and maximum independent set problems, respectively. Using these graphs, we heuristically develop joint and coordinated scheduling approaches. Simulation results show that the proposed two approaches achieve a considerable performance gain in terms of the completion time minimization.

Robust ENF Estimation Based on Harmonic Enhancement and Maximum Weight Clique

We present a framework for robust electric network frequency (ENF) extraction from real-world audio recordings, featuring multi-tone ENF harmonic enhancement and graph-based optimal harmonic selection. Specifically, We first extend the recently developed single-tone ENF signal enhancement method to the multi-tone scenario and propose a harmonic robust filtering algorithm (HRFA). It can respectively enhance each harmonic component without cross-component interference, thus further alleviating the effects of unwanted noise and audio content on the much weaker ENF signal. In addition, considering the fact that some harmonic components could be severely corrupted even after enhancement, disturbing rather than facilitating ENF estimation, we propose a graph-based harmonic selection algorithm (GHSA), which finds the optimal combination of harmonic components for more accurate ENF estimation. Noticeably, the harmonic selection problem is equivalently formulated as a maximum weight clique (MWC) problem in graph theory, and the Bron-Kerbosch algorithm (BKA) is adopted in the GHSA. With the enhanced and optimally selected harmonic components, both the existing maximum likelihood estimator (MLE) and weighted MLE (WMLE) are incorporated to yield the final ENF estimation results. The proposed framework is extensively evaluated using both synthetic signals and our ENF-WHU dataset consisting of $130$ real-world audio recordings, demonstrating substantially improved capability of extracting the ENF from realistically noisy observations over the existing single- and multi-tone competitors. This work further improves the applicability of the ENF as a forensic criterion in real-world situations.

A Lagrangian Bound on the Clique Number and an Exact Algorithm for the Maximum Edge Weight Clique Problem

This paper explores the connections between the classical maximum clique problem and its edge-weighted generalization, the maximum edge weight clique (MEWC) problem. As a result, a new analytic upper bound on the clique number of a graph is obtained and an exact algorithm for solving the MEWC problem is developed. The bound on the clique number is derived using a Lagrangian relaxation of an integer (linear) programming formulation of the MEWC problem. Furthermore, coloring-based bounds on the clique number are used in a novel upper-bounding scheme for the MEWC problem. This scheme is employed within a combinatorial branch-and-bound framework, yielding an exact algorithm for the MEWC problem. Results of computational experiments demonstrate a superior performance of the proposed algorithm compared with existing approaches.

다중 로봇 시스템 지도통합을 위한 최대 가중 클릭 기반 강인한 루프 폐쇄 기법

For efficient collaborations of multi-robot systems during missions, robots must estimate their poses and map the surrounding environment, which can be achieved through multi-robot SLAM (Simultaneous Localization and Mapping). Depending on the mission, the relative poses between the local coordinates of the robots, which must be inferred to generate a global map, may be unknown. The inference is made using the measurement constraints between the robot trajectories, which are often perception-derived measurements that rely on the similarity of two instances of sensor data. Due to this dependence, perceptual aliasing, which is a phenomenon of wrongly identifying two different places as the same location, may occur and produce false loop closures that lead to a catastrophic failure of the SLAM system. This study proposes a robust inter-robot loop closure selection method that reject outlier measurements by checking both consistency of the loop closure and the similarity between the sensor data associated with the loop closure. By considering these two properties, the correct loop closures can be found despite the lack of prior knowledge regarding the relative poses among robots. We demonstrate herein how this problem can be formulated as a maximum weight clique problem, in which the degree of data similarity associated with the loop closures is considered in the form of weight in the objective function. A simulation was executed to validate the method performance and the results showed that the proposed method outperforms existing methods.

An efficient local search algorithm for solving maximum edge weight clique problem in large graphs

Maximum vertex weight clique problem (MVWCP) and maximum edge weight clique problem (MEWCP) are two significant generalizations of maximum clique problem (MCP), and can be widely used in many real-world applications including molecular biology, broadband network design and pattern recognition. Recently, breakthroughs have been made for solving MVWCP in large graphs, resulting in several state-of-the-art algorithms, such as WLMC, FastWClq and LSCC + BMS. However, less attention has been paid to solving MEWCP in large graphs. In this paper, we present an efficient Stochastic Local Search (SLS) algorithm for MEWCP by combining clique construction, local search and graph reduction, resulting in a new algorithm named ReConSLS. We also propose a new upper bound function for edge weighted graphs which is essential for graph reduction. Extensive experiments on a wide range of large graphs demonstrate that ReConSLS surpasses state-of-the-art SLS competitors on the majority of testing graphs.

SCCWalk: An efficient local search algorithm and its improvements for maximum weight clique problem

The maximum weight clique problem (MWCP) is an important generalization of the maximum clique problem with wide applications. In this study, we develop two efficient local search algorithms for MWCP, namely SCCWalk and SCCWalk4L, where SCCWalk4L is improved from SCCWalk for large graphs. There are two main ideas in SCCWalk, including strong configuration checking (SCC) and walk perturbation. SCC is a new variant of a powerful strategy called configuration checking for local search. The walk perturbation procedure is used to lead the algorithm to leave the current area and come into a new area of feasible solution space. Moreover, to improve the performance on massive graphs, we apply a low-complexity heuristic called best from multiple selection to select the swapping vertex pair quickly and effectively, resulting in the SCCWalk4L algorithm. In addition, SCCWalk4L uses two recent reduction rules to decrease the scale of massive graphs.We carry out experiments to evaluate our algorithms on several popular benchmarks, which are divided into two groups, including classical benchmarks of small graphs namely DIMACS, BHOSLIB, winner determination problem, and graphs derived from clustering aggregation, as well as massive graphs, including a suite of massive real-world graphs and large-scale FRB graphs. Experiments show that, compared to state-of-the-art heuristic algorithms and exact algorithm, the proposed algorithms perform better on classical benchmarks, and obtain the best solutions for most massive graphs.

The class of (P7,C4,C5)‐free graphs: Decomposition, algorithms, and χ‐boundedness

AbstractAs usual, () denotes the path on vertices, and () denotes the cycle on vertices. For a family of graphs, we say that a graph is ‐free if no induced subgraph of is isomorphic to any graph in . We present a decomposition theorem for the class of ‐free graphs; in fact, we give a complete structural characterization of ‐free graphs that do not admit a clique‐cutset. We use this decomposition theorem to show that the class of ‐free graphs is ‐bounded by a linear function (more precisely, every ‐free graph satisfies ). We also use the decomposition theorem to construct an algorithm for the minimum coloring problem, an algorithm for the maximum weight stable set problem, and an algorithm for the maximum weight clique problem for this class, where denotes the number of vertices and the number of edges of the input graph.

Joint Mode Selection and Resource Allocation for D2D-Enabled NOMA Cellular Networks

The 5G cellular network employs non-orthogonal multiple access (NOMA) to enhance network connectivity and capacity, and device-to-device (D2D) communications to improve spectrum efficiency. However, the underlay D2D communications may destroy the execution condition for the successive interference cancellation (SIC) decoding of NOMA cellular networks by introducing the extra interference, which degrades the cellular transmission reliability. Thus, we develop the interlay mode as a special D2D mode for NOMA system, which enables the power domain multiplexing of the D2D pair and cellular users to eliminate the strong interference between them by the SIC decoding. When D2D pair conducts the selection between the interlay mode and underlay mode, the SIC decoding constraint should be satisfied at both D2D receiver and NOMA base station. In order to maximize the system sum rate while meeting the SIC decoding constraint, we propose a joint D2D mode selection and resource allocation scheme with interlay mode, which can be formulated as a combinatorial optimization problem. To tackle the combinatorial nature of mode selection and spectrum assignment, we first prove that the original problem can be reformulated as a maximum weight clique problem, and then propose a graph-based algorithm by applying branch-and-bound method to obtain its optimal solution. Finally, simulation results are provided to demonstrate that the interlay mode along with the proposed algorithms can coordinate D2D communications and NOMA cellular network to significantly improve the system sum rate and the D2D access rate.

A novel parallel local search algorithm for the maximum vertex weight clique problem in large graphs

This study proposes a new parallel local search algorithm (Par-LS) for solving the maximum vertex weight clique problem (MVWCP) in large graphs. Solving the MVWCP in a large graph with millions of edges and vertices is an intractable problem. Parallel local search methods are powerful tools to deal with such problems with their high-performance computation capability. The Par-LS algorithm is developed on a distributed memory environment by using message passing interface libraries and employs a different exploration strategy at each processor. The Par-LS introduces new operators parallel(\(\omega \),1)-swap and parallel(1,2)-swap, for searching the neighboring solutions while improving the current solution through iterations. During our experiments, 172 of 173 benchmark problem instances from the DIMACS, BHOSLIB and Network Data Repository graph libraries are solved optimally with respect to the best/optimal reported results. A new best solution for the largest problem instance of the BHOSLIB benchmark (frb100-40) is discovered. The Par-LS algorithm is reported as one of the best performing algorithms in the literature for the solution of the MVWCP in large graphs.