A new upper bound for the maximum weight clique problem

Optimal parallel time bounds for the maximum clique problem on intervals

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

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.

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.

We describe an exact branch-and-bound algorithm for the maximum weight clique problem (MWC), called WLMC, that is especially suited for large vertex-weighted graphs. WLMC incorporates two original contributions: a preprocessing to derive an initial vertex ordering and to reduce the size of the graph, and incremental vertex-weight splitting to reduce the number of branches in the search space. Experiments on representative large graphs from real-world applications show that WLMC greatly outperforms relevant exact and heuristic MWC algorithms, and refute the prevailing hypothesis that exact MWC algorithms are less adequate for large graphs than heuristic algorithms.

The faces in a 2D fine drawing of an object provide important information for the reconstruction of its 3D geometry. In this paper, a graph-based optimization method is proposed for identifying the faces is a line drawing. The face identification is formulated as a maximum weight clique problem. This formulation is proven to be equivalent to the formulation proposed by Shpitalni and Upson (1996). The advantage of our formulation is that it enables one to develop a much faster algorithm to find the faces in a drawing. The significant improvement in speed is derived from two algorithms provided: the depth-first graph search for quickly generating possible faces from a drawing; and the maximum weight clique finding for obtaining the optimal face configurations of the drawing. The experimental results shown that our algorithm generates the same results of face identification as Shpitalni and Lipson's method, but is much faster when dealing with objects of more than 20 faces.

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 radio-frequency (RF) technology is a scalable solution for the backhaul planning. However, its performance is limited in terms of data rate and latency. Free Space Optical (FSO) backhaul, on the other hand, offers a higher data rate but is sensitive to weather conditions. To combine the advantages of RF and FSO backhauls, this paper proposes a cost-efficient backhaul network using the hybrid RF/FSO technology. To ensure a resilient backhaul, the paper imposes a given degree of redundancy by connecting each node through K link-disjoint paths so as to cope with potential link failures. Hence, the network planning problem considered in this paper is the one of minimizing the total deployment cost by choosing the appropriate link type, i.e., either hybrid RF/FSO or optical fiber (OF), between each couple of base-stations while guaranteeing K link-disjoint connections, a data rate target, and a reliability threshold. The paper solves the problem using graph theory techniques. It reformulates the problem as a maximum weight clique problem in the planning graph, under a specified realistic assumption about the cost of OF and hybrid RF/FSO links. Simulation results show the cost of the different planning and suggest that the proposed heuristic solution has a close-to-optimal performance for a significant gain in computation complexity.

Graphs and Algorithms A graph class has few cliques if there is a polynomial bound on the number of maximal cliques contained in any member of the class. This restriction is equivalent to the requirement that any graph in the class has a polynomial sized intersection representation that satisfies the Helly property. On any such class of graphs, some problems that are NP-complete on general graphs, such as the maximum clique problem and the maximum weighted clique problem, admit polynomial time algorithms. Other problems, such as the vertex clique cover and edge clique cover problems remain NP-complete on these classes. Several classes of graphs which have few cliques are discussed, and the complexity of some partitioning and covering problems are determined for the class of all graphs which have fewer cliques than a given polynomial bound.

A Lagrangian relaxation approach to combinatorial exchange in freight logistics

The maximum clique problem ( <i>MCP</i> ) has various applications to reveal the structure and function of graphs. Graphs are constantly updated in the real life. However, no algorithm is specifically designed for dynamic graph. Although <i>MCP</i> in dynamic graphs can be solved by simply invoking a state-of-the-art static approach, such as <i>PMC</i> , when the graph is updated, such an approach of simply re-calculating from scratch is inefficient. The key issue with <i>MCP</i> algorithm is to find a large clique, namely a <i>seed</i> , as fast as possible. Thus, search space can be pruned based on the seed. Size of the seed greedily found by <i>PMC</i> cannot be guaranteed, as it fluctuates considerably. Moreover, the time required to find a seed under <i>PMC</i> is up to <inline-formula><tex-math notation="LaTeX">$O(| E| \cdot \Delta (G))$</tex-math></inline-formula> , where <inline-formula><tex-math notation="LaTeX">$\Delta (G)$</tex-math></inline-formula> is the highest degree in <i>G</i> . In this article, we intend to find a sizable seed by updating the previous maximum clique with the incident vertices of the inserted/deleted edge. Size of the seed now is guaranteed to be no less than <inline-formula><tex-math notation="LaTeX">$\omega (G^{\prime})\; - \;1$</tex-math></inline-formula> , where <inline-formula><tex-math notation="LaTeX">$\omega (G^{\prime})$</tex-math></inline-formula> is the size of the maximum clique on the updated graph. Moreover, the seed can be found in a time complexity of <inline-formula><tex-math notation="LaTeX">$O(\Delta (G)^{2})$</tex-math></inline-formula> . Two other crucial issues related to the <i>MCP</i> in dynamic graphs are refreshing rate and refreshing overhead. After a tight upper bound is imposed on <inline-formula><tex-math notation="LaTeX">$\omega (G^{\prime})$</tex-math></inline-formula> , the necessity of refreshing is evaluated by comparing the seed with its largest challenger, then unnecessary refreshing is wiped out effectively. The size of the largest challenger is judiciously estimated using a lazy growth strategy. Subsequently, the search space in refreshing is confined on a much smaller subgraph using a local refreshing strategy. Extensive experiments indicate that the proposed approach outperforms the baseline algorithm by approximately one order of magnitude.

In communication networks such as social networks, wireless networks and biology networks, it is of importance to find all cliques which can help understand the network topology. The clique structures can also be utilized in facilitating message forwarding in wireless networks. For instance, using a set of cliques of maximal and disjoint, one of the nodes in each clique can be elected to forward messages, by which the duplicate message transmissions can be efficiently reduced. Further, it is well known that the maximum clique problem (MCP) is closely related to the maximum independent set and the vertex cover problems. In recent years, the fundamental problem of finding maximal cliques or maximum cliques has attracted lots of attentions. However, few of these works focus on distributed solutions in wireless networks. In this paper, we pay our attention to this missing corner of research. Specifically, we first give a distributed algorithm which can compute all maximal cliques in a wireless network represented by a graph. The algorithm takes O(n) time and uses O(mn) messages, where n is the number of nodes and m the number of edges. Then, with the proposed algorithms MCP (maximum clique problem) and UMCP (unique MCP), we show that a unique maximum clique can be selected from all maximal cliques in O(n) rounds and using O(mn) messages. To the best of our knowledge, our algorithms are the first deterministic distributed solutions for MCP in wireless networks.

For real time applications, time slacks of a preliminary task schedule may be exploited to conserve energy. This can be accomplished by leveraging the dynamic voltage scaling (DVS) technique to slow down clock frequency of certain cores as long as the deadline is met. In this chapter, the task of fine-tuning an existing task assignment and schedule and using DVS to lower the overall energy consumption is formulated as a graph-theoretic maximum weight clique (MWC) problem. An efficient heuristic algorithm is proposed to systematically solve this problem. A unique feature of our approach is concurrently applying DVS to slow down the execution of multiple tasks to achieve better energy savings. Extensive simulations are performed to compare this proposed algorithm against leading energy-aware task scheduling algorithm and DVS algorithm. Our algorithm exhibits 22% more energy savings than the Energy Aware Scheduling (EAS) algorithm. As for energy saving in DVS process, our MWC-based method provides a 97% saving improvement over the PV-DVS algorithm.

This paper considers a family of cutting planes, recently developed for mixed 0–1 polynomial programs and shows that they define facets for the maximum edge-weighted clique problem. There exists a polynomial time exact separation algorithm for these inequalities. The result of this paper may contribute to the development of more efficient algorithms for the maximum edge-weighted clique problem that use cutting planes.

This paper studies the semidefinite programming SDP problem, i.e., the optimization problem of a linear function of a symmetric matrix subject to linear equality constraints and the additional condition that the matrix be positive semidefinite. First the classical cone duality is reviewed as it is specialized to SDP is reviewed. Next an interior point algorithm is presented that converges to the optimal solution in polynomial time. The approach is a direct extension of Ye’s projective method for linear programming. It is also argued that many known interior point methods for linear programs can be transformed in a mechanical way to algorithms for SDP with proofs of convergence and polynomial time complexity carrying over in a similar fashion. Finally, the significance of these results is studied in a variety of combinatorial optimization problems including the general 0-1 integer programs, the maximum clique and maximum stable set problems in perfect graphs, the maximum k-partite subgraph problem in graph...