scholarly journals A Semi-exact Algorithm for Quickly Computing A Maximum Weight Clique in Large Sparse Graphs

2021 ◽  
Vol 72 ◽  
pp. 39-67
Author(s):  
Shaowei Cai ◽  
Jinkun Lin ◽  
Yiyuan Wang ◽  
Darren Strash
Keyword(s):  
Large Scale ◽  
Heuristic Algorithms ◽  
State Of The Art ◽  
Exact Algorithm ◽  
Exact Algorithms ◽  
Maximum Weight ◽  
Large Graphs ◽  
Sparse Graphs ◽  
Short Time ◽  
Maximum Weight Clique

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.

scholarly journals An Exact Algorithm Based on MaxSAT Reasoning for the Maximum Weight Clique Problem

2016 ◽  
Vol 55 ◽  
pp. 799-833 ◽  
Author(s):  
Zhiwen Fang ◽  
Chu-Min Li ◽  
Ke Xu
Keyword(s):  
Upper Bound ◽  
Optimal Solution ◽  
Maximum Clique ◽  
Exact Algorithm ◽  
Search Space ◽  
Exact Algorithms ◽  
Maximum Weight ◽  
Winner Determination ◽  
Sound Transformation ◽  
Maximum Weight Clique

Recently, MaxSAT reasoning is shown very effective in computing a tight upper bound for a Maximum Clique (MC) of a (unweighted) graph. In this paper, we apply MaxSAT reasoning to compute a tight upper bound for a Maximum Weight Clique (MWC) of a wighted graph. We first study three usual encodings of MWC into weighted partial MaxSAT dealing with hard clauses, which must be satisfied in all solutions, and soft clauses, which are weighted and can be falsified. The drawbacks of these encodings motivate us to propose an encoding of MWC into a special weighted partial MaxSAT formalism, called LW (Literal-Weighted) encoding and dedicated for upper bounding an MWC, in which both soft clauses and literals in soft clauses are weighted. An optimal solution of the LW MaxSAT instance gives an upper bound for an MWC, instead of an optimal solution for MWC. We then introduce two notions called the Top-k literal failed clause and the Top-k empty clause to extend classical MaxSAT reasoning techniques, as well as two sound transformation rules to transform an LW MaxSAT instance. Successive transformations of an LW MaxSAT instance driven by MaxSAT reasoning give a tight upper bound for the encoded MWC. The approach is implemented in a branch-and-bound algorithm called MWCLQ. Experimental evaluations on the broadly used DIMACS benchmark, BHOSLIB benchmark, random graphs and the benchmark from the winner determination problem show that our approach allows MWCLQ to reduce the search space significantly and to solve MWC instances effectively. Consequently, MWCLQ outperforms state-of-the-art exact algorithms on the vast majority of instances. Moreover, it is surprisingly effective in solving hard and dense instances.

scholarly journals Finding A Small Vertex Cover in Massive Sparse Graphs: Construct, Local Search, and Preprocess

2017 ◽  
Vol 59 ◽  
pp. 463-494 ◽  
Author(s):  
Shaowei Cai ◽  
Jinkun Lin ◽  
Chuan Luo
Keyword(s):  
Local Search ◽  
Real World ◽  
Large Scale ◽  
Heuristic Algorithms ◽  
Search Algorithm ◽  
Vertex Cover ◽  
Search Algorithms ◽  
Theory And Practice ◽  
Sparse Graphs ◽  
Massive Graphs

The problem of finding a minimum vertex cover (MinVC) in a graph is a well known NP-hard combinatorial optimization problem of great importance in theory and practice. Due to its NP-hardness, there has been much interest in developing heuristic algorithms for finding a small vertex cover in reasonable time. Previously, heuristic algorithms for MinVC have focused on solving graphs of relatively small size, and they are not suitable for solving massive graphs as they usually have high-complexity heuristics. This paper explores techniques for solving MinVC in very large scale real-world graphs, including a construction algorithm, a local search algorithm and a preprocessing algorithm. Both the construction and search algorithms are based on low-complexity heuristics, and we combine them to develop a heuristic algorithm for MinVC called FastVC. Experimental results on a broad range of real-world massive graphs show that, our algorithms are very fast and have better performance than previous heuristic algorithms for MinVC. We also develop a preprocessing algorithm to simplify graphs for MinVC algorithms. By applying the preprocessing algorithm to local search algorithms, we obtain two efficient MinVC solvers called NuMVC2+p and FastVC2+p, which show further improvement on the massive graphs.

Fast Filtering of Search Results Sorted by Attribute

2022 ◽  
Vol 40 (2) ◽  
pp. 1-24
Author(s):  
Franco Maria Nardini ◽  
Roberto Trani ◽  
Rossano Venturini
Keyword(s):  
Heuristic Algorithms ◽  
State Of The Art ◽  
Optimal Algorithm ◽  
Computational Cost ◽  
Approximation Error ◽  
Optimal Solution ◽  
Exact Algorithm ◽  
Performance Bounds ◽  
Search Results ◽  
Filtering Problem

Modern search services often provide multiple options to rank the search results, e.g., sort “by relevance”, “by price” or “by discount” in e-commerce. While the traditional rank by relevance effectively places the relevant results in the top positions of the results list, the rank by attribute could place many marginally relevant results in the head of the results list leading to poor user experience. In the past, this issue has been addressed by investigating the relevance-aware filtering problem, which asks to select the subset of results maximizing the relevance of the attribute-sorted list. Recently, an exact algorithm has been proposed to solve this problem optimally. However, the high computational cost of the algorithm makes it impractical for the Web search scenario, which is characterized by huge lists of results and strict time constraints. For this reason, the problem is often solved using efficient yet inaccurate heuristic algorithms. In this article, we first prove the performance bounds of the existing heuristics. We then propose two efficient and effective algorithms to solve the relevance-aware filtering problem. First, we propose OPT-Filtering, a novel exact algorithm that is faster than the existing state-of-the-art optimal algorithm. Second, we propose an approximate and even more efficient algorithm, ϵ-Filtering, which, given an allowed approximation error ϵ, finds a (1-ϵ)–optimal filtering, i.e., the relevance of its solution is at least (1-ϵ) times the optimum. We conduct a comprehensive evaluation of the two proposed algorithms against state-of-the-art competitors on two real-world public datasets. Experimental results show that OPT-Filtering achieves a significant speedup of up to two orders of magnitude with respect to the existing optimal solution, while ϵ-Filtering further improves this result by trading effectiveness for efficiency. In particular, experiments show that ϵ-Filtering can achieve quasi-optimal solutions while being faster than all state-of-the-art competitors in most of the tested configurations.

scholarly journals Cardinality Encodings for Graph Optimization Problems

2017 ◽  
Author(s):  
Alexey Ignatiev ◽  
Antonio Morgado ◽  
Joao Marques-Silva
Keyword(s):  
Optimization Problems ◽  
State Of The Art ◽  
Maximum Clique ◽  
Maximum Clique Problem ◽  
Concrete Case ◽  
Large Graphs ◽  
Sparse Graphs ◽  
Maximum Satisfiability ◽  
Graph Optimization ◽  
Sparse Networks

Different optimization problems defined on graphs find application in complex network analysis. Existing propositional encodings render impractical the use of propositional satisfiability (SAT) and maximum satisfiability (MaxSAT) solvers for solving a variety of these problems on large graphs. This paper has two main contributions. First, the paper identifies sources of inefficiency in existing encodings for different optimization problems in graphs. Second, for the concrete case of the maximum clique problem, the paper develops a novel encoding which is shown to be far more compact than existing encodings for large sparse graphs. More importantly, the experimental results show that the proposed encoding enables existing SAT solvers to compute a maximum clique for large sparse networks, often more efficiently than the state of the art.

scholarly journals Multi space RB preconditioners for large-scale parametrized PDEs parametrized PDEs

2018 ◽  
Author(s):  
Niccolò Dal Santo
Keyword(s):  
Large Scale ◽  
State Of The Art ◽  
Navier Stokes ◽  
Reduced Basis ◽  
Fine Grid ◽  
Reduction Techniques ◽  
Efficient Construction ◽  
Short Time ◽  
Proper Orthogonal

We introduce a new two-level preconditioner for the efficient solution of large scale linear systems arising from the finite element (FE) discretization of parametrized unsteady Navier-Stokes (NS) equations. The proposed preconditioner combines a reduced basis (RB) solver, which plays the role of coarse component, with a fine grid preconditioner, in our numerical experiments a SIMPLE preconditioner. The RB coarse component is iteration dependent and is built upon a new Multi Space Reduced Basis (MSRB) method, where a RB space is built through the proper orthogonal decomposition algorithm and is tailored to each step of the iterative method at hand. The resulting operator is used as preconditioner in the flexible GMRES method. The Krylov iterations employed to solve the resulting preconditioned system target small tolerances with a very small iteration count and in a very short time. We show in this poster how to address the well-posedness of the RB coarse components and the efficient construction of the resulting preconditioner by means of hyper-reduction techniques. Simulations are carried out to evaluate the performance of the proposed MSRB preconditioner in a large scale computational setting related to the NS equations in parametrized carotid bifurcations and compared to state of the art preconditioners.

New algorithms for pattern matching with wildcards and length constraints

2015 ◽  
Vol 07 (03) ◽  
pp. 1550032 ◽  
Author(s):  
Abdullah N. Arslan ◽  
Betsy George ◽  
Kirsten Stor
Keyword(s):  
Pattern Matching ◽  
Polynomial Time ◽  
Original Problem ◽  
Heuristic Algorithms ◽  
Optimal Solution ◽  
Exact Algorithm ◽  
Exact Algorithms ◽  
Worst Case ◽  
Special Cases ◽  
New Algorithms

The pattern matching with wildcards and length constraints problem is an interesting problem in the literature whose computational complexity is still open. There are polynomial time exact algorithms for its special cases. There are heuristic algorithms, and online algorithms that do not guarantee an optimal solution to the original problem. We consider two special cases of the problem for which we develop offline solutions. We give an algorithm for one case with provably better worst case time complexity compared to existing algorithms. We present the first exact algorithm for the second case. This algorithm uses integer linear programming (ILP) and it takes polynomial time under certain conditions.

A State-of-the-Art Review of Artificial Bee Colony in the Optimization of Single and Multiple Criteria

2013 ◽  
Vol 4 (4) ◽  
pp. 23-45 ◽  
Author(s):  
B. S. P. Mishra ◽  
S. Dehuri ◽  
G.-N. Wang
Keyword(s):  
Honey Bees ◽  
Artificial Bee Colony ◽  
Heuristic Algorithms ◽  
Optimization Problems ◽  
State Of The Art ◽  
Exact Algorithm ◽  
Multi Objective Optimization ◽  
Abc Algorithm ◽  
Bee Colony ◽  
Time And Space Complexity

Nowadays computers are used to solve a variety and multitude of complex problems facing in every sphere of peoples’ life. However, many of the problems are intractable in nature exact algorithm might need centuries to manage with formidable challenges. In such cases heuristic or in a broader sense meta-heuristic algorithms that find an approximate solution but have acceptable time and space complexity play indispensable role. In this article, the authors present a state-of-the-art review on meta-heuristic algorithm popularly known as artificial bee colony (ABC) inspired by honey bees. Moreover, the ABC algorithm for solving single and multi-objective optimization problems have been studied. A few potential application areas of ABC are highlighted as an end note of this article.

An Exact Algorithm for Minimum Vertex Cover Problem

Mathematics ◽  
2019 ◽  
Vol 7 (7) ◽  
pp. 603
Author(s):  
Luzhi Wang ◽  
Shuli Hu ◽  
Mingyang Li ◽  
Junping Zhou
Keyword(s):  
Lower Bound ◽  
Branch And Bound ◽  
Heuristic Algorithms ◽  
Vertex Cover ◽  
Exact Algorithm ◽  
Search Space ◽  
Exact Algorithms ◽  
The Other ◽  
Branch And Bound Algorithm ◽  
Cover Problem

In this paper, we propose a branch-and-bound algorithm to solve exactly the minimum vertex cover (MVC) problem. Since a tight lower bound for MVC has a significant influence on the efficiency of a branch-and-bound algorithm, we define two novel lower bounds to help prune the search space. One is based on the degree of vertices, and the other is based on MaxSAT reasoning. The experiment confirms that our algorithm is faster than previous exact algorithms and can find better results than heuristic algorithms.

scholarly journals The Airport Gate Assignment Problem: A Survey

2014 ◽  
Vol 2014 ◽  
pp. 1-27 ◽  
Author(s):  
Abdelghani Bouras ◽  
Mageed A. Ghaleb ◽  
Umar S. Suryahatmaja ◽  
Ahmed M. Salem
Keyword(s):  
Real Time ◽  
Assignment Problem ◽  
Heuristic Algorithms ◽  
State Of The Art ◽  
Operational Efficiency ◽  
Exact Algorithms ◽  
Metaheuristic Algorithms ◽  
Research Trend ◽  
Gate Assignment ◽  
Time Basis

The airport gate assignment problem (AGAP) is one of the most important problems operations managers face daily. Many researches have been done to solve this problem and tackle its complexity. The objective of the task is assigning each flight (aircraft) to an available gate while maximizing both conveniences to passengers and the operational efficiency of airport. This objective requires a solution that provides the ability to change and update the gate assignment data on a real time basis. In this paper, we survey the state of the art of these problems and the various methods to obtain the solution. Our survey covers both theoretical and real AGAP with the description of mathematical formulations and resolution methods such as exact algorithms, heuristic algorithms, and metaheuristic algorithms. We also provide a research trend that can inspire researchers about new problems in this area.

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