A Lagrangian relaxation approach to combinatorial exchange in freight logistics

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.

An alternative framework to Lagrangian relaxation approach for job shop scheduling

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.

A new upper bound for the maximum weight clique problem

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.

A Lagrangian relaxation (LR) approach was presented which is with machine capacity relaxation and operation precedence relaxation for solving a flexible job shop (FJS) scheduling problem from the steelmaking-refining-continuous casting process. Unlike the full optimization of LR problems in traditional LR approaches, the machine capacity relaxation is optimized asymptotically, while the precedence relaxation is optimized approximately due to the NP-hard nature of its LR problem. Because the standard subgradient algorithm (SSA) cannot solve the Lagrangian dual (LD) problem within the partial optimization of LR problem, an effective deflected-conditional approximate subgradient level algorithm (DCASLA) was developed, named as Lagrangian relaxation level approach. The efficiency of the DCASLA is enhanced by a deflected-conditional epsilon-subgradient to weaken the possible zigzagging phenomena. Computational results and comparisons show that the proposed methods improve significantly the efficiency of the LR approach and the DCASLA adopting capacity relaxation strategy performs best among eight methods in terms of solution quality and running time.

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 prize collecting traveling salesman problem (PCTSP) is a generalization of the traveling salesman problem (TSP) in which a tour starting at a root node must visit a subset of nodes to collect a prescribed amount of total prize minimizing the summation of travel costs and penalty costs associated with nonvisited nodes. We propose new formulations and compare them with adaptations to the PCTSP of strong formulations from the literature for the asymmetric TSP. One of the formulations proposed in this paper provided the tighter linear relaxation bounds on computational experiments based on benchmark instances for the asymmetric TSP. We then develop a Lagrangian relaxation approach embedding heuristic schemes to obtain both lower and upper bounds. The proposed heuristic has shown to be successful in obtaining optimal or near‐optimal solutions with information from the Lagrangian subproblems. Our Lagrangian approach can thereupon provide small optimality gaps, being an efficient alternative to bound optimal values of large instances.

Computation of approximate [formula omitted]-points for large scale single machine scheduling problem

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.

Job shop scheduling problems with weighted quadratic tardiness costs are formulated as a class of transition firing problems of timed Petri nets, and a sequential Lagrangian relaxation approach is proposed to solve the scheduling problems. It is shown that the approach can not only avoid the solution oscillation in the Lagrangian relaxation approach proposed by Hoitomt and Luh, but also generate good near-optimal schedules. >

The authors present a new method for determining the unit-commitment schedule of a power system which operates under fuel utilisation constraints. The proposed method employs the Lagrangian relaxation approach to determine a feasible suboptimal schedule. Then, linear programming is applied to improve the feasible solution, as well as seeking the optimal economic dispatch of the committed generators. The algorithm is implemented on a VAX11/780 machine to solve the unit-commitment problem of a system which consists of 26 thermal units and 6 fuel-constrained units. Numerical results presented in this paper describe the usefulness and practicality of the proposed method.

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.