Clique Problem Research Articles

Reinforcement learning based coarse-to-fine search for the maximum [formula omitted]-plex problem

The maximum k-plex problem is a computationally challenging problem, which emerged from graph-theoretic social network studies in the context of community detection. As a relaxation of the traditional maximum clique problem, intended to identify cohesive structures within a graph that are not necessarily fully connected, the aim of this problem is to find the largest set of vertices in a graph in which all vertices are not adjacent to at most k vertices within this set. This paper introduces an effective hybrid local search algorithm for solving the maximum k-plex problem that combines a stochastic local search method and a reinforcement learning strategy. The proposed approach includes a number of distinguishing features, including: a new coarse-to-fine strategy to balance the search process, a distance-and-quality reward for actions, and a parameter control mechanism based on reinforcement learning. We conduct a computational analysis of the key components of the proposed algorithm, assessing their impact on the overall performance. Extensive experiments for the maximum k-plex problem (k=2,3,4,5) on 80 benchmark instances from the second DIMACS Challenge demonstrate that the proposed approach competes favourably with the state-of-the-art algorithms from the literature.

The complexity of quantum circuit mapping with fixed parameters

A quantum circuit must be preprocessed before implementing on NISQ devices due to the connectivity constraint. Quantum circuit mapping (QCM) transforms the circuit into an equivalent one that is compliant with the NISQ device’s architecture constraint by adding SWAP gates. The QCM problem asks the minimal number of auxiliary SWAP gates, and is NP-complete. The complexity of QCM with fixed parameters is studied in the paper. We give an exact algorithm for QCM, and show that the algorithm runs in polynomial time if the NISQ device’s architecture is fixed. If the number of qubits of the quantum circuit is fixed, we show that the QCM problem is NL-complete by a reduction from the undirected shortest path problem. Moreover, the fixed-parameter complexity of QCM is W[1]-hard when parameterized by the number of qubits of the quantum circuit. We prove the result by a reduction from the clique problem. If taking the depth of the quantum circuits and the coupling graphs as parameters, we show that the QCM problem is still NP-complete over shallow quantum circuits, and planar, bipartite and degree bounded coupling graphs.

CliSAT: A new exact algorithm for hard maximum clique problems

Given a graph, the maximum clique problem (MCP) asks for determining a complete subgraph with the largest possible number of vertices. We propose a new exact algorithm, called CliSAT, to solve the MCP to proven optimality. This problem is of fundamental importance in graph theory and combinatorial optimization due to its practical relevance for a wide range of applications. The newly developed exact approach is a combinatorial branch-and-bound algorithm that exploits the state-of-the-art branching scheme enhanced by two new bounding techniques with the goal of reducing the branching tree. The first one is based on graph colouring procedures and partial maximum satisfiability problems arising in the branching scheme. The second one is a filtering phase based on constraint programming and domain propagation techniques. CliSAT is designed for structured MCP instances which are computationally difficult to solve since they are dense and contain many interconnected large cliques. Extensive experiments on hard benchmark instances, as well as new hard instances arising from different applications, show that CliSAT outperforms the state-of-the-art MCP algorithms, in some cases by several orders of magnitude.

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.

On designing networks resilient to clique blockers

Robustness and vulnerability analysis of networked systems is often performed using the concept of vertex blockers. In particular, in the minimum cost vertex blocker clique problem, we seek a subset of vertices with the minimum total blocking cost such that the weight of any remaining clique in the interdicted graph (after the vertices are blocked) is upper bounded by some pre-defined parameter. Loosely speaking, we aim at disrupting the network with the minimum possible cost in order to guarantee that the network does not contain cohesive (e.g., closely related) groups of its structural elements with large weights; such groups are modeled as weighted cliques. In this paper, our focus is on designing networks that are resilient to clique blockers. Specifically, we construct additional connections (edges) in the network and our goal is to ensure (at the minimum possible cost of newly added edges) that the adversarial decision-maker (or the worst-case realization of random failures) cannot disrupt the network (namely, the weight of its cohesive groups) at some sufficiently low cost. The proposed approach is useful for modeling effective formation and preservation of influential clusters in networked systems. We first explore structural properties of our problem. Then, we develop several exact solution schemes based on integer programming and combinatorial branch-and-bound techniques. Finally, the performance of our approaches is explored in a computational study with randomly-generated and real-life network instances.

A Hierarchy of Standard Polynomial Programming Formulations for the Maximum Clique Problem

The maximum clique problem is a classical optimization problem which finds many important applications. Some of the most theoretically interesting and computationally successful approaches to this problem are based on the Motzkin--Straus formulation, expressing the clique number in terms of the maximal value of a standard quadratic program. This intriguing relationship has also motivated significant developments in quadratic optimization, copositive programming, and complexity in nonlinear optimization. Inspired by such developments, this paper establishes a hierarchy of polynomial programming formulations (\textbfP$^k$), $k\in\{2,\ldots, \omega\}$ for the maximum clique problem, relating the clique number $\omega$ of a given graph to the global maximal value of a multilinear polynomial function $f_k(x)$ of degree $k$ over the standard simplex in $\mathbb{R}^{|V|}$. The case of $k=2$ corresponds to the Motzkin--Straus formulation, and the hierarchical feature is in the fact that the set of local maxima of (\textbfP$^{k+1}$) is a subset of the set of local maxima of (\textbfP$^{k}$), $k\in\{2,\ldots, \omega-1\}$. In particular, every local maximum of (\textbfP$^{\omega}$) is global.

A local dense decision-making method for solving the maximum clique problem in large-scale undirected graphs

The maximum clique problem is a classic difficult problem. Most traditional solutions are branch and bound-based exact algorithms that perform well in terms of both accuracy and time at small legend sizes. However, in recent years, the scale of legend models in practical applications has expanded rapidly, and the traditional undirected graphical model solving methods are no longer applicable. At larger scales, a heuristic decision-making method for local density is proposed, a density index function is constructed, and a decision-making inference for finding maximum cliques is established. Our algorithm avoids random and disorderly traversal solutions, and it ensures the accuracy of the solution process. At the same time, a fast search threshold is added to the algorithm to improve the solution efficiency and local optimization ability. The experimental comparison shows that this algorithm and its improvements have better solution accuracy and solution time compared with the maximum clique accuracy algorithm.

The multicolored graph realization problem

We introduce the multicolored graph realization problem (MGR). The input to this problem is a colored graph (G,φ), i.e., a graph G together with a coloring φ on its vertices. We associate each colored graph (G,φ) with a cluster graph (Gφ) in which, after collapsing all vertices with the same color to a node, we remove multiple edges and self-loops. A set of vertices S is multicolored when S has exactly one vertex from each color class. The MGR problem is to decide whether there is a multicolored set S so that, after identifying each vertex in S with its color class, G[S] coincides with Gφ.The MGR problem is related to the well-known class of generalized network problems, most of which are NP-hard, like the generalized Minimum Spanning Tree problem. The MGR is a generalization of the multicolored clique problem, which is known to be W[1]-hard when parameterized by the number of colors. Thus, MGR remains W[1]-hard, when parameterized by the size of the cluster graph. These results imply that the MGR problem is W[1]-hard when parameterized by any graph parameter on Gφ, among which lies treewidth. Consequently, we look at the instances of the problem in which both the number of color classes and the treewidth of Gφ are unbounded. We consider three natural such graph classes: chordal graphs, convex bipartite graphs and 2-dimensional grid graphs. We show that MGR is NP-complete when Gφ is either chordal, biconvex bipartite, complete bipartite or a 2-dimensional grid. Our reductions show that the problem remains hard even when the maximum number of vertices in a color class is 3. In the case of the grid, the hardness holds even for graphs with bounded degree. We provide a complexity dichotomy with respect to cluster size.

An Exact Algorithm with New Upper Bounds for the Maximum k-Defective Clique Problem in Massive Sparse Graphs

The Maximum k-Defective Clique Problem (MDCP), as a clique relaxation model, has been used to solve various problems. Because it is a hard computational task, previous works can hardly solve the MDCP for massive sparse graphs derived from real-world applications. In this work, we propose a novel branch-and-bound algorithm to solve the MDCP based on several new techniques. First, we propose two new upper bounds of the MDCP as well as corresponding reduction rules to remove redundant vertices and edges. The proposed reduction rules are particularly useful for massive graphs. Second, we present another new upper bound by counting missing edges between fixed vertices and an unfixed vertex for cutting branches. We perform extensive computational experiments to evaluate our algorithm. Experimental results show that our reduction rules are very effective for removing redundant vertices and edges so that graphs are reduced greatly. Also, our algorithm can solve benchmark instances efficiently, and it has significantly better performance than state-of-the-art algorithms.

An approximation lagrangian-based algorithm for the maximum clique problem via deterministic annealing neural network

The maximum clique problem is an NP hard combinatorial optimization problem, which is widely used in industrial and management processes. In order to approach the solution of the maximum clique problem, a deterministic annealing algorithm is proposed. When the barrier parameter is reduced from a large positive number to zero, the algorithm can track the minimum path of the obstacle problem to obtain a high-quality solution, which is a continuous method. The global convergence iteration of the Lagrange multiplier can be performed to obtain the minimum point of the obstacle problem in the feasible descent direction for any given positive value of the barrier parameter and has a desired property that it automatically satisfies the upper and lower bounds of variables if the steplength is a number that is between 0 and 1. Numerical results are provided to illustrate the efficiency of the proposed algorithm.

Fast Loop Closure Selection Method with Spatiotemporal Consistency for Multi-Robot Map Fusion

This paper presents a robust method based on graph topology to find the topologically correct and consistent subset of inter-robot relative pose measurements for multi-robot map fusion. However, the absence of good prior on relative pose gives a severe challenge to distinguish the inliers and outliers, and once the wrong inter-robot loop closures are used to optimize the pose graph, which can seriously corrupt the fused global map. Existing works mainly rely on the consistency of spatial dimension to select inter-robot measurements, while it does not always hold. In this paper, we propose a fast inter-robot loop closure selection method that integrates the consistency and topology relationship of inter-robot measurements, which both conform to the continuity characteristics of similar scenes and spatiotemporal consistency. Firstly, a clustering method integrating topology correctness of inter-robot loop closures is proposed to split the entire measurement set into multiple clusters. Then, our method decomposes the traditional high-dimensional consistency matrix into the sub-matrix blocks corresponding to the overlapping trajectory regions. Finally, we define the weight function to find the topologically correct and consistent subset with the maximum cardinality, then convert the weight function to the maximum clique problem from graph theory and solve it. We evaluate the performance of our method in a simulation and in a real-world experiment. Compared to state-of-the-art methods, the results show that our method can achieve competitive performance in accuracy while reducing computation time by 75%.

The Maximum Clique Problem for Permutation Hamming Graphs

This paper explores a new approach to reduce the maximum clique problem associated with permutation Hamming graphs to smaller clique problems. The vertices of a permutation Hamming graph are permutations of n integers and the edges connect pairs of vertices at a Hamming distance greater than or equal to a threshold d. The maximum clique problem for permutation Hamming graphs is a challenging task due to the size, density and regularity of the graphs. However, symmetry properties, which are still partly unexplored, can help to reduce the problems’ size and hardness. A property of edge transitivity with respect to automorphisms is proven and leads to a classification for cycle-equivalent edges. This property enables to reduce the full-size clique problem to a set of significantly smaller (and easier to solve) clique problems. The number of reduced problems can be expressed by means of the partition function of integer numbers. Computational experiments confirm that additional knowledge on the automorphism group leads to a more targeted and efficient solving method for the maximum clique problem associated with permutation Hamming graphs.

An Approach to Aid Decision-Making by Solving Complex Optimization Problems Using SQL Queries

In combinatorial optimization, the more complex a problem is, the more challenging it becomes, usually causing most research to focus on creating solvers for larger cases. However, real-life situations also contain small-sized instances that deserve a researcher’s attention. For example, within a web development context, a developer might face small combinatorial optimization cases that fall in the following situations to solve them: (1) the development of an ad hoc specialized strategy is not justified; (2) the developer could lack the time, or skills, to create the solution; (3) the efficiency of naive brute force strategies might be compromised due to the programming paradigm use. Similar situations in this context, combined with a recent increasing interest in optimization information from databases, open a research area to develop easy-to-implement strategies that compete with those naive approaches and do not require specialized knowledge. Therefore, this work revises Structured Query Language (SQL) approaches and proposes new methods to tackle combinatorial optimization problems such as the Portfolio Selection Problem, Maximum Clique Problem, and Graph Coloring Problem. The performance of the resulting queries is compared against naive approaches; its potential to extend to other optimization problems is studied. The presented examples demonstrate the simplicity and versatility of using a SQL approach to solve small optimization problem instances.

On maximum ratio clique relaxations

AbstractThis article introduces and studies two clique relaxation models with fractional objectives: the maximum ratio ‐plex problem and the maximum ratio ‐defective clique problem. The decision version of each problem is shown to be strongly ‐complete; we also discuss some related computational complexity issues. The base optimization models are single‐ratio fractional 0‐1 problems. We describe two general types of solution methods: the first one is based on mixed‐integer linear programs (MILPs) obtained by linearizing the original nonlinear 0‐1 models; the other one exploits parametric methods (namely, a binary search and Newton's method) that solve MILPs in an iterative manner. For the proposed MILPs, we derive valid inequalities that are shown to substantially improve the performance of an off‐the‐shelf MILP solver. Finally, the considered solution approaches are compared via extensive numerical experiments using a set of artificially generated and real‐life network instances.

Clique Finder

The Maximum Clique Problem (MCP) is a classical NP-hard problem that has gained considerable attention due to its numerous real-world applications and theoretical complexity. It is inherently computationally complex, and so exact methods may require prohibitive computing time. Nature-inspired meta-heuristics have proven their utility in solving many NP-hard problems. In this research, we propose a simulated annealing-based algorithm that we call Clique Finder algorithm to solve the MCP. Our algorithm uses a logarithmic cooling schedule and two moves that are selected in an adaptive manner. The objective (error) function is the total number of missing links in the clique, which is to be minimized. The proposed algorithm was evaluated using benchmark graphs from the open-source library DIMACS, and results show that the proposed algorithm had a high success rate.

Parallel quantum annealing

Quantum annealers of D-Wave Systems, Inc., offer an efficient way to compute high quality solutions of NP-hard problems. This is done by mapping a problem onto the physical qubits of the quantum chip, from which a solution is obtained after quantum annealing. However, since the connectivity of the physical qubits on the chip is limited, a minor embedding of the problem structure onto the chip is required. In this process, and especially for smaller problems, many qubits will stay unused. We propose a novel method, called parallel quantum annealing, to make better use of available qubits, wherein either the same or several independent problems are solved in the same annealing cycle of a quantum annealer, assuming enough physical qubits are available to embed more than one problem. Although the individual solution quality may be slightly decreased when solving several problems in parallel (as opposed to solving each problem separately), we demonstrate that our method may give dramatic speed-ups in terms of the Time-To-Solution (TTS) metric for solving instances of the Maximum Clique problem when compared to solving each problem sequentially on the quantum annealer. Additionally, we show that solving a single Maximum Clique problem using parallel quantum annealing reduces the TTS significantly.

Clique Search in Graphs of Special Class and Job Shop Scheduling

In this paper, we single out the following particular case of the clique search problem. The vertices of the given graph are legally colored with k colors and we are looking for a clique with k nodes in the graph. In other words, we want to decide if a given k-partite graph contains a clique with k nodes. The maximum clique problem asks for finding a maximum clique in a given finite simple graph. The problem of deciding if the given graph contains a clique with k vertices is called the k-clique problem. The first problem is NP-hard and the second one is NP-complete. The special clique search problem, we propose, is still an NP-complete problem. We will show that the k-clique problem in the special case of k-partite graphs is more tractable than in the general case. In order to illustrate the possible practical utility of this restricted type clique search problem we will show that the job shop scheduling problem can be reduced to such a clique search problem in a suitable constructed graph. We carry out numerical experiments to assess the efficiency of the approach. It is a common practice that before one embarks on a large scale clique search typically one attempts to simplify and tidy up the given graph. This procedure is commonly referred as preconditioning or kernelization of the given graph. Of course, the preconditioning or kernelization is meant with respect to the given type of clique search problem. The other main topic of the paper is to describe a number of kernelization methods tailored particularly to the proposed special k-clique problem. Some of these techniques works in connection with the generic k-clique problem. In these situations, we will see that they are more efficient in the case of k-partite graphs. Some other preconditioning methods applicable only to k-partite graphs. We illustrate how expedient these preconditioning methods can be by solving non-trivial scheduling problems to optimality employing only kernelization techniques dispensing with exhaustive clique search algorithms altogether.

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.

Maximum Decoding Clique Based Maximum Weight Vertex Search Algorithm for Buffered Instantly Decodable Network Codes

The traditional instantly decodable network coding (IDNC) schemes discard the non-instantly decodable packets (NIDPs), which results in the loss of useful information. In this letter, we consider a network coding scheme named as buffered instantly decodable network coding (B-IDNC) where each receiving node can cache the NIDPs containing two unreceived packets for subsequent network decoding, and give a decoding algorithm by utilizing the maximum decoding clique. The maximum decoding clique is defined as a subset of vertices in the IDNC graph, which can be decoded together. We approximate the average packet decoding delay minimization problem as the maximum weight encoding clique problem, and a heuristic algorithm named as maximum decoding clique based maximum weight vertex (MDC-MWV) searching algorithm is proposed to solve the approximated problem. Simulation results show that our proposed algorithm can reduce the average packet decoding delay compared with the conventional IDNC schemes.

Lexicographic Breadth First-Search and Branch and Bound Algorithms for the Maximum Clique Problem

Lexicographic Breadth First-Search and Branch and Bound Algorithms for the Maximum Clique Problem