Maximum Weight Clique Research Articles

A Two-Stage MaxSAT Reasoning Approach for the Maximum Weight Clique Problem

MaxSAT reasoning is an effective technology used in modern branch-and-bound (BnB) algorithms for the Maximum Weight Clique problem (MWC) to reduce the search space. However, the current MaxSAT reasoning approach for MWC is carried out in a blind manner and is not guided by any relevant strategy. In this paper, we describe a new BnB algorithm for MWC that incorporates a novel two-stage MaxSAT reasoning approach. In each stage, the MaxSAT reasoning is specialised and guided for different tasks. Experiments on an extensive set of graphs show that the new algorithm implementing this approach significantly outperforms relevant exact and heuristic MWC algorithms in both small/medium and massive real-world graphs.

A new upper bound for the maximum weight clique problem

The maximum weight clique problem (MWCP) for a vertex-weighted graph is to find a complete subgraph in which the sum of vertex weights is maximum. The main goal of this paper is to develop an efficient branch-and-bound algorithm to solve the MWCP. As a crucial aspect of branch-and-bound MWCP algorithms is the incorporation of a tight upper bound, we first define a new upper bound for the MWCP, called UBWC, that is based on a novel notion called weight cover. The idea of a weight cover is to compute a set of independent sets of the graph and define a weight function for each independent set so that the weight of each vertex of the graph is covered by such weight functions. We then propose a new branch-and-bound MWCP algorithm called WC-MWC that uses UBWC to reduce the number of branches of the search space that must be traversed by incrementally constructing a weight cover for the graph. Finally, we present experimental results that show that UBWC reduces the search space much more than previous upper bounds, and the new algorithm WC-MWC outperforms some of the best performing exact and heuristic MWCP algorithms on both small/medium graphs and real-world massive graphs.

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.

A robust and cooperative parallel tabu search algorithm for the maximum vertex weight clique problem

Abstract The maximum vertex weight clique problem (MVWCP) is a challenging NP-Hard combinatorial optimization problem that searches for a clique with maximum total sum of vertices’ weights. In this study, we propose a robust and cooperative parallel tabu search algorithm (PTC) for the MVWCP. Our proposed algorithm uses a dedicated tabu search algorithm with a multistart strategy for the diversification of search space on a parallel computation environment. An effective seeding mechanism is developed with respect to the rank of the processors to choose diversified starting points for a better exploration of the search space. Classical add, swap and drop operators of tabu search are improved with parallel computation and a combined neighborhood approach. The PTC algorithm is evaluated on a set of 120 problem instances from DIMACS-W and BHOSLIB-W benchmarks. Computational results show that the PTC algorithm competes with state-of-the-art heuristic algorithms by reporting average best (optimal) result hit ratios up to 99.0%.

A BPPE Algorithm for Instantly Decodable Network Coding in Wireless Broadcasting

Instantly decodable network coding (IDNC) is a low-cost network coding technique for recovering unattained packets by multiple receivers. Due to its NP-hard nature, finding the optimal IDNC coded packets is intractable. In the well-known graph-based method proposed by Sorour et al. , a maximum weight clique selection problem is formulated as an approximation, and a heuristic algorithm called maximum weight clique (MWC) is proposed in maximizing the clique weight. In this letter, a base policy-based partial enumeration (BPPE) algorithm is proposed in addressing the maximum weight clique selection problem. In particular: 1) the necessity of constructing the IDNC graph as in MWC is eliminated and 2) different enumeration lengths could be chosen, so that a consistent performance-run time tradeoff could be achieved. Simulation results testify the effectiveness of BPPE in addressing the maximum weight clique selection problem over existing algorithms in terms of performance-run time tradeoff.

Clique cutsets beyond chordal graphs

Truemper configurations (thetas, pyramids, prisms, and wheels) have played an important role in the study of complex hereditary graph classes (e.g. the class of perfect graphs and the class of even-hole-free graphs), appearing both as excluded configurations, and as configurations around which graphs can be decomposed. In this paper, we study the structure of graphs that contain (as induced subgraphs) no Truemper configurations other than (possibly) universal wheels and twin wheels. We also study several subclasses of this class. We use our structural results to analyze the complexity of the recognition, maximum weight clique, maximum weight stable set, and optimal vertex coloring problems for these classes. We also obtain polynomial χ-bounding functions for these classes.

A CPU-GPU local search heuristic for the maximum weight clique problem on massive graphs

Given an undirected graph with positive weights on the vertices, the Maximum Weight Clique Problem (MWCP) consists in finding a clique with maximum total weight. In this paper, we present a CPU-GPU local search heuristic for solving the MWCP on massive graphs. The heuristic is based on two new neighborhood structures for the problem. These neighborhoods are explored using an efficient procedure that is suitable to be mapped onto a GPU-based massively parallel architecture. We test the proposed heuristic on real-world massive graphs with millions of edges and vertices. The results indicate that, even when the heuristic is executed on a CPU-only architecture, it is able to outperform the best-known heuristics for the MWCP. Moreover, the hybrid CPU-GPU implementation obtained an average speedup of up to 12 times over the CPU-only implementation.

Connectivity Assessment of Random Directed Graphs with Application to Underwater Sensor Networks

In this brief, the problem of connectivity assessment for a random network is investigated. The weighted vertex connectivity (WVC) is introduced as a metric to evaluate the connectivity of the weighted expected graph of a random sensor network, where the elements of the weight matrix characterize the operational probability of their corresponding communication links. The WVC measure extends the notion of vertex connectivity (VC) for random graphs by taking into account the joint effects of path reliability and network robustness to node failure. The problem of computing the WVC measure is transformed into a sequence of iterative deepening depth-first search and maximum weight clique problems. An algorithm is developed accordingly to find the proposed connectivity metric. The approximate WVC measure is defined subsequently as a lower bound on the introduced connectivity metric which can be found by applying a polynomial-time shortest path algorithm in a sequential manner. The performance of the proposed algorithms is validated using an experimental underwater acoustic sensor network.

Clustered maximum weight clique problem: Algorithms and empirical analysis

We introduce the Clustered Maximum Weight Clique Problem (CCP), a generalization of the Maximum Weight Clique Problem, that models an image acquisition scheduling problem for a satellite constellation. The solution of CCP represents satellite schedules that satisfy customer requests for satellite imagery. Each request has a priority, an area of interest, and a time window. Often, the area of interest is too large to be imaged by one satellite pass and it has to be divided into several smaller images. Each image has one or more opportunities for an acquisition by a satellite.The problem is modeled by a clustered weighted graph. A graph node represents one opportunity for an image acquisition by one satellite. A graph edge indicates that either two opportunities are not in conflict – can both be in a schedule, or two opportunities are not acquiring the same image. Each graph node has a weight that represents the area size of the image. The graph nodes are partitioned into clusters each of which encompasses all the opportunities of one customer request. The priority of the request is captured by the cluster weight. The time window of the request restricts the number of opportunities.The CCP deals with finding a clique of a maximum weight where the weight combines the node weights and the cluster weights. More precisely, the cluster weight is multiplied by the contribution of the sum of the weights of the clique nodes. The contribution is either a linear function or a piece-wise linear function, where the latter is meant to favour finalizing an already partially served customer request.The paper presents several mathematical programming formulations of the CCP and proposes matheuristic solution approaches. The computational study is performed on the clustered adaptations of the DIMACS and BHOSLIB benchmark instances for the Maximum Weight Clique Problem. The achieved results are encouraging.

A hybrid iterated local search heuristic for the maximum weight independent set problem

This paper presents a hybrid iterated local search (ILS) algorithm for the maximum weight independent set (MWIS) problem, a generalization of the classical maximum independent set problem. Two efficient neighborhood structures are proposed and they are explored using the variable neighborhood descent procedure. Moreover, we devise a perturbation mechanism that dynamically adjusts the balance between intensification and diversification during the search. The proposed algorithm was tested on two well-known benchmarks (DIMACS-W and BHOSLIB-W) and the results obtained were compared with those found by state-of-the-art heuristics and exact methods. Our heuristic outperforms the best-known heuristic for the MWIS as well as the best heuristics for the maximum weight clique problem. The results also show that the hybrid ILS was capable of finding all known optimal solutions in milliseconds.

Fast maximum weight clique extraction algorithm: Optimal tables for branch-and-bound

A new branch-and-bound algorithm for the maximum weight clique problem is proposed. The proposed algorithm consists of two phases, a precomputation phase and a branch-and-bound phase. In the precomputation phase, the weights of maximum weight cliques in many small subgraphs are calculated and stored in optimal tables. In the branch-and-bound phase, each problem is divided into smaller subproblems, and unnecessary subproblems are pruned using the optimal tables. We performed experiments with the proposed algorithm and five existing algorithms for several types of graphs. The results indicate that only the proposed algorithm can obtain exact solutions for all graphs and that it performs much faster than other algorithms for nearly all graphs.

Molecular beacon computing model for maximum weight clique problem

Given an undirected graph with weights on the vertices, the maximum weight clique problem requires finding the clique of the graph which has the maximum weight. The problem is a general form of the maximum clique problem. In this paper, we encode weight of vertex into a unique fixed length oligonucleotide segment and employ sticker model to solve the problem. The proposed method has two distinct characteristics. On one hand, we skip generating initial data pool that contains every possible solution to the problem of interest, the key point of which is constructing the solution instead of searching solution in the vast initial data pool according to logic constraints. On the other hand, oligonucleotide segments are treated like variables which store weights on vertices, no matter what kind of number the weights are, integer or real. Therefore, the proposed method can solve the problem with arbitrary weight values and be applied to solve the other weight-related problem. In addition, molecular beacon is also employed in order to overcome shortcomings of sticker model. Besides, we have analyzed the proposed algorithm’s feasibility.

An Exact Algorithm for the Maximum Weight Clique Problem in Large Graphs

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.

A Maximum Weight Clique Algorithm For Dense Circle Graphs With Many Shared Endpoints

Circle graphs are derived from the intersections of a set of chords. They have applications in VLSI design, bioinformatics, and chemistry. Some intractable problems on general graphs can be solved in polynomial time on circle graphs. As such, the study of circle graph algorithms has received attention. State-of-the-art algorithms for finding the maximum weight clique of a circle graph are very efficient when the graph is sparse. However, these algorithms require $\Theta(n^2)$ time when the graph is dense. We present an algorithm that is a factor of $\sqrt{n}$ faster for dense graphs in which many chord endpoints are shared. We also argue that these assumptions are practical.

Rate Aware Instantly Decodable Network Codes

This paper addresses the problem of reducing the delivery time of data messages to cellular users using instantly decodable network coding (IDNC) with physical-layer rate awareness. While most of the existing literature on IDNC does not consider any physical layer complications and abstracts the model as equally slotted time for all users, this paper proposes a cross-layer scheme that incorporates the different channel rates of the various users in the decision process of both the transmitted message combinations and the rates with which they are transmitted. The consideration of asymmetric rates for receivers reflects more practical application scenarios and introduces a new tradeoff between the choice of coding combinations for various receivers and the broadcasting rates. The completion time minimization problem in such a scenario is first shown to be intractable. The problem is thus approximated by reducing, at each transmission, the increase of an anticipated version of the completion time. This paper solves the problem by formulating it as a maximum weight clique problem over a newly designed rate-aware IDNC graph. The highest weight clique in the created graph being potentially not unique, this paper further suggests a multi-layer version of the proposed solution to improve the obtained results from the employed completion time approximation. Simulation results indicate that the cross-layer design largely outperforms the uncoded transmissions strategies and the classical IDNC scheme.

New Lower Bounds for Binary Constant-Dimension Subspace Codes

ABSTRACTLet denote the maximum cardinality of a set of k-dimensional subspaces of an n-dimensional vector space over the finite field of order q, , such that any two different subspaces have a distance of at least d. Lower bounds on can be obtained by explicitly constructing corresponding sets . When searching for such sets with a prescribed group of automorphisms, the search problem leads to instances of the maximum weight clique problem. The main focus is here on subgroups with small index in the normalizer of a Singer subgroup of . With a stochastic maximum weight clique algorithm and a systematic consideration of groups of the above mentioned type, new lower bounds on and for 8 ⩽ n ⩽ 11 are obtained.

Data dissemination in wireless sensor networks with instantly decodable network coding

Wireless sensor networks (WSNs) are widely applied in monitoring and control of environment parameters. It is sometimes necessary to disseminate data through wireless links after they are deployed in order to adjust configuration parameters of sensors or distribute management commands and queries to sensors. Several approaches have been proposed recently for data dissemination in WSNs. However, none of these approaches achieves both high efficiency and low complexity simultaneously. To address this problem, cluster-tree based network architecture, which divides a WSN into hierarchies and clusters is proposed. Upon this architecture, data is delivered from base station to all sensors in clusters hierarchy by hierarchy. In each cluster, father broadcasts data to all his children with instantly decodable network coding (IDNC), and a novel scheme targeting to maximize total transmission gain (MTTG) is proposed. This scheme employs a new packet scheduling algorithm to select IDNC packets, which uses weight status feedback matrix (WSFM) directly. Analysis and simulation results indicate that the transmission efficiency approximate to the best existing approach maximum weight clique, but with much lower computational overhead. Hence, the energy efficiency achieves both in data transmission and processing.

Hybrid Radio/Free-Space Optical Design for Next Generation Backhaul Systems

The deluge of date rate in today’s networks imposes a cost burden on the backhaul network design. Developing cost-efficient backhaul solutions becomes an exciting, yet challenging, problem. Traditional technologies for backhaul networks, including either radio-frequency (RF) backhauls or optical fibers (OF). While RF is a cost-effective solution as compared with OF, it supports the lower data rate requirements. Another promising backhaul solution is the free-space optics (FSO) as it offers both a high data rate and a relatively low cost. The FSO, however, is sensitive to nature conditions, e.g., rain, fog, and line-of-sight. This paper combines both the RF and FSO advantages and proposes a hybrid RF/FSO backhaul solution. It considers the problem of minimizing the cost of the backhaul network by choosing either OF or hybrid RF/FSO backhaul links between the base stations, so as to satisfy data rate, connectivity, and reliability constraints. It shows that under a specified realistic assumption about the cost of OF and hybrid RF/FSO links, the problem is equivalent to a maximum weight clique problem, which can be solved with moderate complexity. Simulation results show that the proposed solution shows a close-to-optimal performance, especially for reasonable prices of the hybrid RF/FSO links. They further reveal that the hybrid RF/FSO is a cost-efficient solution and a good candidate for upgrading the existing backhaul networks.

A Generalization of Join and an Algorithmic Recognition Problem

We consider a new graph operation c-join which generalizes join and co-join. We show that odd hole-free graphs ( odd antihole-free graphs) are closed under c-join and describe a polynomial time algorithm to recognize graphs that admit a c-join. The time complexity of the (a) recognition problem, (b) maximum weight independent set (MWIS) problem, and (c) minimum coloring (MC) problem for odd hole-free graphs are still unknown. Let H be an odd hole-free graph that contains an odd antihole as an induced subgraph and GH be the class of all graphs generated from the induced subgraphs of H by using c-join recursively. Then GH is odd hole-free, contains all P4-free graphs, complement of all bipartite graphs, and some imperfect graphs. We show that the MWIS problem, maximum weight clique (MWC) problem, MC problem, and minimum clique cover (MCC) problem can be solved efficiently for GH .

A Space-Time Graph Optimization Approach Based on Maximum Cliques for Action Detection

We present an efficient action detection method that takes a space-time (ST) graph optimization approach for real-world videos. Given an ST graph representing the entire action video, our method identifies a maximum-weight connected subgraph (MWCS) indicating an action region by applying an optimization approach based on clique information. We define an energy function based on maximum weight cliques for subregions of the graph and formulate it using an optimization problem that can be represented as a linear system. Our energy function includes the maximum and connectivity properties for finding the MWCS, and its optimization solution indicates the probability of belonging to the maximum subgraph for each node. Our graph optimization method efficiently solves the detection problem by applying the clique-based approach and simple linear system solver. We demonstrate that our detection method results in a more accurate localization compared with conventional methods through our experimental results with real-world data sets, such as the Hollywood and MSR action data sets. We also show that our method outperforms the state-of-the-art methods of action detection.