Maximum Edge Weight Clique Problem Research Articles

A Fast Exact Solver with Theoretical Analysis for the Maximum Edge-Weighted Clique Problem

The maximum vertex-weighted clique problem (MVWCP) and the maximum edge-weighted clique problem (MEWCP) are two natural extensions of the fundamental maximum clique problem. In this paper, we systematically study MEWCP and make the following major contributions: (1) We show that MEWCP is NP-hard even when the minimum degree of the graph is n-2, in contrast to MVWCP which is polynomial-time solvable when the minimum degree of the graph is at least n-3. This result distinguishes the complexity of the two problems for the first time. (2) To address MEWCP, we develop an efficient branch-and-bound algorithm called MEWCat with both practical and theoretical performance guarantees. In practice, MEWCat utilizes a new upper bound tighter than existing ones, which allows for more efficient pruning of branches. In theory, we prove a running-time bound of O*(1.4423^n) for MEWCat, which breaks the trivial bound of O*(2^n) in the research line of practical exact MEWCP solvers for the first time. (3) Empirically, we evaluate the performance of MEWCat on various benchmark instances. The experiments demonstrate that MEWCat outperforms state-of-the-art exact solvers significantly. For instance, on 16 DIMACS graphs that the state-of-the-art solver BBEWC fails to solve within 7200 seconds, MEWCat solves all of them with an average time of less than 1000 seconds. On real-world graphs, MEWCat achieves an average speedup of over 36x.

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 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.

A maximum edge-weight clique extraction algorithm based on branch-and-bound

The maximum edge-weight clique problem is to find a clique whose sum of edge-weight is the maximum for a given edge-weighted undirected graph. The problem is NP-hard and some branch-and-bound algorithms have been proposed. In this paper, we propose a new exact algorithm based on the branch-and-bound method. It assigns edge-weights to vertices and calculates upper bounds using vertex coloring. By some computational experiments, we confirmed our algorithm works fine.

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.

HEEL: exploratory entity linking for heterogeneous information networks

A heterogeneous information network (HIN) is a ubiquitous data model, consisting of multiple types of entities and relations. Names of entities in HINs are inherently ambiguous, making it difficult to fully disambiguate a HIN. In this paper, we introduce the task of exploratory entity linking for HINs. Given a partially disambiguated HIN, we aim at linking ambiguous names to disambiguated entities in the HIN if their referent entities are present. We also try to “explore” other alternatives by discovering new entities and adding them to the HIN. A partial classification EM-based approach is proposed to address this task. We present a constrained probability propagation model to link surface names to entities in the HIN. New entity detection process is modeled as a maximum edge weight clique problem. Experiments illustrate that our method outperforms state-of-the-art methods for entity linking with HINs and author name disambiguation.

A new branch-and-bound algorithm for the maximum edge-weighted clique problem

Abstract We study the maximum edge-weighted clique problem, a problem related to the maximum (vertex-weighted) clique problem which asks for finding a complete subgraph (i.e., a clique) of maximum total weight on its edges. The problem appears in a wide range of applications, including bioinformatics, material science, computer vision, robotics, and many more. In this work, we propose a new combinatorial branch-and-bound algorithm for the problem which relies on a novel bounding procedure capable of pruning a very large amount of nodes of the branch-and-bound tree. Extensive computational experiments on random and structured graphs, encompassing standard benchmarks used in the literature as well as recently introduced real-world large-scale graphs, show that our new algorithm outperforms the state-of-the-art by several orders of magnitude on many instances.

A nonconvex quadratic optimization approach to the maximum edge weight clique problem

The maximum edge weight clique (MEWC) problem, defined on a simple edge-weighted graph, is to find a subset of vertices inducing a complete subgraph with the maximum total sum of edge weights. We propose a quadratic optimization formulation for the MEWC problem and study characteristics of this formulation in terms of local and global optimality. We establish the correspondence between local maxima of the proposed formulation and maximal cliques of the underlying graph, implying that the characteristic vector of a MEWC in the graph is a global optimizer of the continuous problem. In addition, we present an exact algorithm to solve the MEWC problem. The algorithm is a combinatorial branch-and-bound procedure that takes advantage of a new upper bound as well as an efficient construction heuristic based on the proposed quadratic formulation. Results of computational experiments on some benchmark instances are also presented.

An Efficient Local Search for the Maximum Edge Weighted Clique Problem

The maximum edge weighted clique problem (MEWCP), an extension of the classical maximum clique problem, is an important NP-hard combinatorial optimization problem. The problem has been widely used in various areas. The objective of this paper is to design an efficient local search algorithm to solve the MEWCP. First, the proposed scoring strategy is used to evaluate the benefit of adding and swapping operators. Second, the vertex weighting strategy is used to increase the diversity of solutions and the configuration checking strategy is used to avoid the cycling problem. By combining these three strategies, we propose multiple rules to select the added vertex or the swapped vertex pair. Based on the multiple rules, an efficient local search algorithm, namely, local search based on multiple rules (LSMR), is proposed. LSMR is compared with several representative algorithms on massive graph instances. The experimental results indicate that LSMR is superior to competitors in terms of solution quality and computational efficiency in most instances.

A new family of facet defining inequalities for the maximum edge-weighted clique problem

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.

Models for a Steiner multi-ring network design problem with revenues

The Steiner multi-ring network design problem with revenues consists of designing node-disjoint multiple rings connected by a specific node (hub) and passing through all the nodes with high priority of service and some of the nodes with low priority of service. The number of nodes in each ring has an upper bound to assure a certain level of service. Besides the usual arc link costs, we also consider revenues between each pair of nodes in the same ring, even when they are not connected by a direct link. The objective is to minimize the difference between the total connection cost and total revenue. The problem is a generalization of the problem studied in Gouveia and Pires (Eur J Oper Res 133:21–31, 2001a) and it can also be seen as a combination of variants of two NP-Hard problems, the vehicle routing problem and the maximum edge-weighted clique problem. We introduce and discuss two types of integer linear programming formulations and propose some valid inequalities to strengthen the linear programming relaxation. Computational results are presented to evaluate the quality of the linear programming relaxation bounds associated with these formulations as well as efficiency of the models to obtain the optimal integer solutions.

Solving the maximum edge-weight clique problem in sparse graphs with compact formulations

This paper studies the behavior of compact formulations for solving the maximum edge-weight clique (MEWC) problem in sparse graphs. The MEWC problem has long been discussed in the literature, but mostly addressing complete graphs, with or without a cardinality constraint on the clique. Yet, several real-world applications are defined on sparse graphs, where the missing edges are due to some threshold process or because they are not even supposed to be in the graph, at all. Such situations often arise in cell’s metabolic networks, where the amount of metabolites shared among reactions is an important issue to understand the cell’s prevalent elements. We propose new node-discretized formulations for the problem, which are more compact than other models known from the literature. Computational experiments on benchmark and real-world instances are conducted for discussing and comparing the models. These tests indicate that the node-discretized formulations are more efficient for solving large size sparse graphs. Additionally, we also address a new variant of the MEWC problem where the objective to be maximized includes the neighboring edges of the clique.

Solving the maximum edge weight clique problem via unconstrained quadratic programming

The unconstrained quadratic binary program (UQP) is proving to be a successful modeling and solution framework for a variety of combinatorial optimization problems. Experience reported in the literature with several problem classes has demonstrated that this approach works surprisingly well in terms of solution quality and computational times, often rivaling and sometimes surpassing more traditional methods. In this paper we report on the application of UQP to the maximum edge-weighted clique problem. Computational experience is reported illustrating the attractiveness of the approach.

Dynamic Programming and Clique Based Approaches for Protein Threading with Profiles and Constraints

The protein threading problem with profiles is known to be efficiently solvable using dynamic programming. In this paper, we consider a variant of the protein threading problem with profiles in which constraints on distances between residues are given. We prove that protein threading with profiles and constraints is NP-hard. Moreover, we show a strong hardness result on the approximation of an optimal threading satisfying all the constraints. On the other hand, we develop two practical algorithms: CLIQUETHREAD and BBDPTHREAD. CLIQUETHREAD reduces the threading problem to the maximum edge-weight clique problem, whereas BBDPTHREAD combines dynamic programming and branch-and-bound techniques. We perform computational experiments using protein structure data in PDB (Protein Data Bank) using simulated distance constraints. The results show that constraints are useful to improve the alignment accuracy of the target sequence and the template structure. Moreover, these results also show that BBDPTHREAD is in general faster than CLIQUETHREAD for larger size proteins whereas CLIQUETHREAD is useful if there does not exist a feasible threading.

The edge-weighted clique problem: Valid inequalities, facets and polyhedral computations

Let K n =( V, E) be the complete undirected graph with weights c e associated to the edges in E. We consider the problem of finding the subclique C=( U, F) of K n such that the sum of the weights of the edges in F is maximized and | U|⩽ b, for some b∈[1,…, n]. This problem is called the Maximum Edge-Weighted Clique Problem (MEWCP) and is NP-hard. In this paper we investigate the facial structure of the polytope associated to the MEWCP introducing new classes of facet defining inequalities. Computational experiments with a branch-and-cut algorithm are reported confirming the strength of these inequalities. All instances with up to 48 nodes could be solved without entering into the branching phase. Moreover, we show that some of these new inequalities also define facets of the Boolean Quadric Polytope and generalize many of the previously known inequalities for this well-studied polytope.