An adaptive multistart tabu search approach to solve the maximum clique problem

In this paper, we give a comprehensive overview of the development of clique algorithms and their use for drug design based on the search for cliques in protein product graphs. The maximum clique problem is a computational problem of finding largest subsets of vertices in a graph that are all pairwise adjacent. A related problem is the maximum weight clique problem and the highest weight k-clique problem, which both extend the algorithm to weighted graphs. The review covers our developed algorithms, starting with our improved branch-and-bound algorithm for finding maximum cliques in undirected graphs from 2007 up to the recent developments of algorithms for weighted graphs in 2024. We show the application of these algorithms to early stages of drug discovery, in particular to protein binding site detection based on protein similarity search in large protein databases and to protein-ligand molecular docking.

The intent of maximum clique problem is to discover a clique of largest size or a complete subgraph in a graph; for a graph clique is a subgraph which is complete i.e., every vertex of that subgraph are adjacent to each other. It belongs to the category of the NP-hard problems, for which many solutions exist and many more are continuously being proposed. We are recapitulating the core idea and search strategies of few of the branch and bound based recently proposed algorithms. Intent of this work is to give a comparative study of newly proposed solutions to maximum clique problem which will be helpful to decide which algorithm is efficient for a given graph. It also instigates to design more efficient solutions for maximum clique problem.

<title>Abstract</title> The maximum clique problem(MCP) is to find the largest complete subgraph in an undirected graph, that is, the subgraph in which there are edges between every two different vertices. It is an NP-Hard problem with wide applications, including bioinformatics, social networks, data mining, and other fields. This paper proposes an improved algorithm that dynamically tracks the maximum clique size by encoding prior constraints on the vertex count—derived from Turán's theorem and complete graph properties—into global variables through quantum circuit pre-detection. The algorithm further synergizes with Grover's search to optimize the solution space. Our auxiliary-qubit encoding scheme dynamically tracks clique sizes during quantum search, eliminating iterative measurements, achieving MCP solution with $O\left ( \sqrt{2^{n } } \right ) $ Grover iterations and $O\left ( 1 \right ) $ measurements. This represents an $\boldsymbol{n}$-fold improvement over state-of-the-art Grover-based methods, which require $O(n\sqrt{2^n})$ iterations and $O(n)$ measurements for $n$-vertex graphs. We validate algorithmic correctness through simulations on IBM's Qiskit platform and benchmark qubit/gate efficiency against existing Grover-based MCP solvers.

We study the equality generalized traveling salesman problem (E‐GTSP), which is a variant of the well‐known traveling salesman problem. We are given an undirected graph G = (V,E), with set of vertices V and set of edges E, each with an associated cost. The set of vertices is partitioned into clusters. E‐GTSP is to find an elementary cycle visiting exactly one vertex for each cluster and minimizing the sum of the costs of the traveled edges. We propose a multistart heuristic, which iteratively starts with a randomly chosen set of vertices and applies a decomposition approach combined with improvement procedures. The decomposition approach considers a first phase to determine the visiting order of the clusters and a second phase to find the corresponding minimum cost cycle. We show the effectiveness of the proposed approach on benchmark instances from the literature. On small instances, the heuristic always identifies the optimal solution rapidly and outperforms all known heuristics; on larger instances, the heuristic always improves, in comparable computing times, the best known solution values obtained by the genetic algorithm recently proposed by Silberholz and Golden. © 2011 Wiley Periodicals, Inc. NETWORKS, 2011

The maximum clique problem (MCP) finds the largest complete subgraph or clique of a given graph. The target here is to maximize the size of the clique. In this article, we have proposed a chemical reaction optimization (CRO)-based metaheuristic algorithm to solve the maximum clique problem. Two datasets have been used to measure the performance of the proposed method. The proposed method gives better results with less average errors in comparison to the state-of-the-art methods for the two datasets.KeywordsMetaheuristicMaximum clique problemChemical reaction optimizationNP-hard

A deep learning guided memetic framework for graph coloring problems

Let G(V, E) be a simple, undirected graph where V is the set of vertices and E is the set of edges. A b‐dimensional cube is a Cartesian product I1×I2×···×Ib, where each Ii is a closed interval of unit length on the real line. The cubicity of G, denoted by cub(G), is the minimum positive integer b such that the vertices in G can be mapped to axis parallel b‐dimensional cubes in such a way that two vertices are adjacent in G if and only if their assigned cubes intersect. An interval graph is a graph that can be represented as the intersection of intervals on the real line—i.e. the vertices of an interval graph can be mapped to intervals on the real line such that two vertices are adjacent if and only if their corresponding intervals overlap. Suppose S(m) denotes a star graph on m+1 nodes. We define claw number ψ(G) of the graph to be the largest positive integer m such that S(m) is an induced subgraph of G. It can be easily shown that the cubicity of any graph is at least ⌈log2ψ(G)⌉. In this article, we show that for an interval graph G ⌈log2ψ(G)⌉⩽cub(G)⩽⌈log2ψ(G)⌉+2. It is not clear whether the upper bound of ⌈log2ψ(G)⌉+2 is tight: till now we are unable to find any interval graph with cub(G)&gt;⌈log2ψ(G)⌉. We also show that for an interval graph G, cub(G)⩽⌈log2α⌉, where α is the independence number of G. Therefore, in the special case of ψ(G)=α, cub(G) is exactly ⌈log2α2⌉. The concept of cubicity can be generalized by considering boxes instead of cubes. A b‐dimensional box is a Cartesian product I1×I2×···×Ib, where each Ii is a closed interval on the real line. The boxicity of a graph, denoted box(G), is the minimum k such that G is the intersection graph of k‐dimensional boxes. It is clear that box(G)⩽cub(G). From the above result, it follows that for any graph G, cub(G)⩽box(G)⌈log2α⌉. © 2010 Wiley Periodicals, Inc. J Graph Theory 65: 323–333, 2010

The maximum clique problem involves finding the largest set of pairwise adjacent vertices in a graph. The problem is classic but still attracts much attention because of its hardness and its prominent applications. Our work is based on the existence of an order of all the vertices whereby those belonging to a maximum clique stay close enough to each other. Such an order can be identified via the extraction of a particular subgraph from the original graph. The problem can consequently be seen as a permutation problem that can be addressed efficiently by metaheuristics. We first design a memetic algorithm (MA) for this purpose. Computational experiments conducted on the DIMACS benchmark instances clearly show that our MA not only outperforms existing genetic approaches, but it also compares very well to state-of-the-art algorithms regarding the maximal clique size found after different runs. Furthermore, we show that a hybridization of MA with an iterated local search (ILS) improves the stability of the algorithm. This hybridization (MA-ILS) permits to find two distinct maximal cliques of size 79 and one of size 80 for the C2000.9 instance of the DIMACS benchmark.

The maximum weighted clique (MWC) problem, as a typical NP-complete problem, is difficult to be solved by the electronic computer algorithm. The aim of the problem is to seek a vertex clique with maximal weight sum in a given undirected graph. It is an extremely important problem in the field of optimal engineering scheme and control with numerous practical applications. From the point of view of practice, we give a parallel biological algorithm to solve the MWC problem. For the maximum weighted clique problem withmedges andnvertices, we use fixed length DNA strands to represent different vertices and edges, fully conduct biochemical reaction, and find the solution to the MVC problem in certain length range withO(n2)time complexity, comparing to the exponential time level by previous computer algorithms. We expand the applied scope of parallel biological computation and reduce computational complexity of practical engineering problems. Meanwhile, we provide a meaningful reference for solving other complex problems.

The clique problem consists in determining whether an undirected graph G of order n contains a clique of order l. In this paper we are concerned with the decremental version of clique problem, where the property of containing an l-clique is dynamically checked during deletions of nodes. We provide an improved dynamic algorithm for this problem for every fixed value of l ≥ 3. Our algorithm naturally applies to filtering for the constraint satisfaction problem. In particular, we show how to speed up the filtering based on an important local consistency property: the inverse consistency.

Minimum face-spanning subgraphs of plane graphs

In this paper we compare, from a practical point of view, approximation algorithms for the problem MaxCut . For this problem, we are given an undirected graph G = (V;E) with vertex set V and edge set E, and we are looking for a partition V = V1 [ V2 with V1 \ V2 = 0 of the vertex set which maximizes the number of edges e 2 E which have one endpoint in V1 and the other in V2 . The investigated algorithms include semidefinite programming, a random strategy, genetic algorithms, two combinatorial algorithms and a divide-and-conquer strategy.

R is a commutative ring with unity. The associate ring graph AG(R) is the graph with the vertex set V = R • {0} and edge set E = {(a, b) / a, b are associates and a ≠ b}. Since the relation of being associate is on equivalence relation, this graph is an undirected graph and also each component is complete. In this paper, I present some of the interesting results majority of which are for the ring of integers modulo n, n is a positive integer.

Let G(V, E) be a finite, simple and undirected graph with a vertex set V and an edge set E. An edge irregular total k-labeling is a map f : V ⋃ E → {1, 2, …, k} such that for any two different edges xy and x′y′ in E, ω(xy) ≠ ω(x′y′) where ω(xy) = f(x) + f(y) + f(xy). The minimum k for which the graph G admits an edge irregular total k-labeling is called the total edge irregularity strength of G, denoted by tes(G). We have constructed the formula of an edge irregular total k-labelling and determined the total edge irregularity strength of book graphs and double book graphs. In this paper, we construct an edge irregular total k-labeling that can be used for book graphs, double book graphs, and triple book graphs. We also show the exact value of the total edge irregularity strength of triple book graphs.

Let G be a simple, connected and undirected graph with vertex set V and edge set E. A total k-labeling is defined as totally irregular total k-labeling if the weights of any two different both vertices and edges are distinct. The weight of vertex x is defined as , while the weight of edge xy is . A minimum k for which G has totally irregular total k-labeling is mentioned as total irregularity strength of G and denoted by ts(G). This paper contains investigation of totally irregular total k-labeling and determination of their total irregularity strengths for caterpillar graphs with each internal vertex between two stars has degree three. The results are and for n > 4: