Clique Finder

The Maximum Clique Problem (MCP) is a classical NP-hard problem that has gained considerable attention due to its numerous real-world applications and theoretical complexity. It is inherently computationally complex, and so exact methods may require prohibitive computing time. Nature-inspired meta-heuristics have proven their utility in solving many NP-hard problems. In this research, we propose a simulated annealing-based algorithm that we call Clique Finder algorithm to solve the MCP. Our algorithm uses a logarithmic cooling schedule and two moves that are selected in an adaptive manner. The objective (error) function is the total number of missing links in the clique, which is to be minimized. The proposed algorithm was evaluated using benchmark graphs from the open-source library DIMACS, and results show that the proposed algorithm had a high success rate.

Clique Finder

The Maximum Clique Problem (MCP) is a classical NP-hard problem that has gained considerable attention due to its numerous real-world applications and theoretical complexity. It is inherently computationally complex, and so exact methods may require prohibitive computing time. Nature-inspired meta-heuristics have proven their utility in solving many NP-hard problems. In this research, we propose a simulated annealing-based algorithm that we call Clique Finder algorithm to solve the MCP. Our algorithm uses a logarithmic cooling schedule and two moves that are selected in an adaptive manner. The objective (error) function is the total number of missing links in the clique, which is to be minimized. The proposed algorithm was evaluated using benchmark graphs from the open-source library DIMACS, and results show that the proposed algorithm had a high success rate.

A modified simplex partition algorithm to test copositivity

A real symmetric matrix A is copositive if $$x^\top Ax\ge 0$$ x ⊤ A x ≥ 0 for all $$x\ge 0$$ x ≥ 0 . As A is copositive if and only if it is copositive on the standard simplex, algorithms to determine copositivity, such as those in Sponsel et al. (J Glob Optim 52:537–551, 2012) and Tanaka and Yoshise (Pac J Optim 11:101–120, 2015), are based upon the creation of increasingly fine simplicial partitions of simplices, testing for copositivity on each. We present a variant that decomposes a simplex $$\bigtriangleup $$ △ , say with n vertices, into a simplex $$\bigtriangleup _1$$ △ 1 and a polyhedron $$\varOmega _1$$ Ω 1 ; and then partitions $$\varOmega _1$$ Ω 1 into a set of at most $$(n-1)$$ ( n - 1 ) simplices. We show that if A is copositive on $$\varOmega _1$$ Ω 1 then A is copositive on $$\bigtriangleup _1$$ △ 1 , allowing us to remove $$\bigtriangleup _1$$ △ 1 from further consideration. Numerical results from examples that arise from the maximum clique problem show a significant reduction in the time needed to establish copositivity of matrices.

Numerical experiments with LP formulations of the maximum clique problem

The maximum clique problems calls for determining the size of the largest clique in a given graph. This graph problem affords a number of zero-one linear programming formulations. In this case study we deal with some of these formulations. We consider ways for tightening the formulations. We carry out numerical experiments to see the improvements the tightened formulations provide.

Using Machine Learning for Quantum Annealing Accuracy Prediction

Quantum annealers, such as the device built by D-Wave Systems, Inc., offer a way to compute solutions of NP-hard problems that can be expressed in Ising or quadratic unconstrained binary optimization (QUBO) form. Although such solutions are typically of very high quality, problem instances are usually not solved to optimality due to imperfections of the current generations quantum annealers. In this contribution, we aim to understand some of the factors contributing to the hardness of a problem instance, and to use machine learning models to predict the accuracy of the D-Wave 2000Q annealer for solving specific problems. We focus on the maximum clique problem, a classic NP-hard problem with important applications in network analysis, bioinformatics, and computational chemistry. By training a machine learning classification model on basic problem characteristics such as the number of edges in the graph, or annealing parameters, such as the D-Wave’s chain strength, we are able to rank certain features in the order of their contribution to the solution hardness, and present a simple decision tree which allows to predict whether a problem will be solvable to optimality with the D-Wave 2000Q. We extend these results by training a machine learning regression model that predicts the clique size found by D-Wave.

Tiếp cận Heuristic giải bài toán Clique lớn nhất trên đồ thị đơn vô hướng không trọng số

Bài toán Clique lớn nhất (Maximum Clique Problem) là bài toán tìm tập con lớn nhất của tập đỉnh trong đơnđồ thị vô hướng, sao cho hai đỉnh phân biệt trong nó luôn kề nhau. Đây là bài toán nổi tiếng thuộc lớp NP-complete, đượcứng dụng nhiều trong các lĩnh vực khai thác dữ liệu, phân tích mạng, truy xuất thông tin, y học, giáo dục và nhiều lĩnhvực khác liên quan đến mạng lưới toàn cầu. Có nhiều cách tiếp cận giải bài toán Clique lớn nhất như quy hoạch động,nhánh-cận, heuristic hay meta-heuristic – cho lời giải chính xác hay xấp xỉ. Trong bài báo này, nhóm tác giả phân tíchhai thuật giải tiếp cận heuristic gần đây và đề xuất các heuristic tăng độ chính xác của lời giải cho bài toán Clique lớnnhất. Phần thực nghiệm, nhóm tác giả so sánh chất lượng lời giải của thuật giải đề xuất trên 10 bộ dữ liệu từ DIMACS.

A Novel Rescheduling Algorithm for the Airline Recovery with Flight Priorities and Airport Capacity Constraints

Many flights experience delays at the airport due to bad weather, temporary closures of airports, unscheduled maintenance, etc., which emphasizes the urgent need for disruption management. It is widely accepted for Chinese airline companies to determine the flight timetable according to the lexicographic preference of flight priorities. Flight schedulers usually deal with the preceding flights as important as the latter flight of a higher priority. In this paper, we propose a build-in flight feasibility verification algorithm to improve the rescheduling algorithm. A novel model of the feasibility verification problem is given, which is equivalent to the model of a maximum clique problem for networks. Examples and tests show the advantage of our algorithm, and the algorithm runs fairly quickly and can be plugged in other scheduling algorithms easily.