Exact algorithms for a scheduling problem with unrelated parallel machines and sequence and machine-dependent setup times

A scheduling problem with unrelated parallel machines, sequence and machine-dependent setup times, due dates and weighted jobs is considered in this work. A branch-and-bound algorithm (B&B) is developed and a solution provided by the metaheuristic GRASP is used as an upper bound. We also propose a set of instances for this type of problem. The results are compared to the solutions provided by two mixed integer programming models (MIP) with the solver CPLEX 9.0. We carry out computational experiments and the algorithm performs extremely well on instances with up to 30 jobs.     

极小化加权完工时间的多资源工序的资源分配问题

针对现实中广泛存在的多资源工序的资源分配问题,考虑基于资源使用的优先次序约束,以最小化加权完工时间为优化目标,构建一类新的资源分配混合整数线性规划模型.其次,提出Benders分解和禁忌搜索的混合算法,该混合算法以Benders分解为基本框架,将原问题分为提供资源分配方案的主问题和计算工序加权完工时间的子问题,并通过改进数学模型和添加禁忌搜索提高混合算法的收敛速度.最后,通过300个随机仿真算例测试结果表明,在相同时间下求解小规模问题时,所提的Benders分解混合算法能获得距离商业求解器CPLEX最优解平均差距为0.86%的满意解;在求解大规模问题时,所提出的算法的性能表现优于CPLEX、禁忌搜索算法、变邻域搜索算法和Benders分解嵌入遗传算法的混合方法,能给出更好的资源分配方案,与CPLEX相比,上界和下界分别改善了4.74%和9.62%.     

An efficient local search for partial vertex cover problem

In this paper, an efficient local search framework, namely GRASP-PVC, is proposed to solve the minimum partial vertex cover problem. In order to speed up the convergence, a novel least-cost vertex selecting strategy is applied into GRASP-PVC. As far as we know, no heuristic algorithms have ever been reported to solve this momentous problem and we compare GRASP-PVC with a commercial integer programming solver CPLEX as well as a 2-approximation algorithm on two standard benchmark libraries called DIMACS and BHOSLIB. Experimental results evince that GRASP-PVC finds much better partial vertex covers than CPLEX and the approximation algorithm on most instances. Additional experimental results also confirm the validity of the least-cost vertex selecting strategy.

     

The labeled maximum matching problem

Given a graph G where a label is associated with each edge, we address the problem of looking for a maximum matching of G using the minimum number of different labels, namely the labeled maximum matching problem. It is a relatively new problem whose application is related to the timetabling problem. We prove it is NP-complete and present four different mathematical formulations. Moreover, we propose an exact algorithm based on a branch-and-bound approach to solve it. We evaluate the performance of our algorithm on a wide set of instances and compare our computational times with the ones required by CPLEX to solve the proposed mathematical formulations. Test results show the effectiveness of our procedure, that hugely outperforms the solver.     

Heuristics for a bidding problem

In this paper, we study a bidding problem which can be modeled as a set packing problem. A simulated annealing heuristic with three local moves, including an embedded branch-and-bound move, is developed for the problem. We compared the heuristic with the CPLEX 8.0 solver and the current best non-exact method, Casanova, using the standard CATS benchmark and other realistic test sets. Results show that the heuristic outperforms CPLEX and Casanova.     

Approximation Algorithms and Hardness Results for Packing Element-Disjoint Steiner Trees in Planar Graphs

We study the problem of packing element-disjoint Steiner trees in graphs. We are given a graph and a designated subset of terminal nodes, and the goal is to find a maximum cardinality set of element-disjoint trees such that each tree contains every terminal node. An element means a non-terminal node or an edge. (Thus, each non-terminal node and each edge must be in at most one of the trees.) We show that the problem is APX-hard when there are only three terminal nodes, thus answering an open question. Our main focus is on the special case when the graph is planar. We show that the problem of finding two element-disjoint Steiner trees in a planar graph is NP-hard. Similarly, the problem of finding two edge-disjoint Steiner trees in a planar graph is NP-hard. We design an algorithm for planar graphs that achieves an approximation guarantee close to 2. In fact, given a planar graph that is k element-connected on the terminals (k is an upper bound on the number of element-disjoint Steiner trees), the algorithm returns $\lfloor\frac{k}{2} \rfloor-1$ element-disjoint Steiner trees. Using this algorithm, we get an approximation algorithm for the edge-disjoint version of the problem on planar graphs that improves on the previous approximation guarantees. We also show that the natural LP relaxation of the planar problem has an integrality ratio approaching?2.     

Efficient lower bounds and heuristics for the variable cost and size bin packing problem

We consider the variable cost and size bin packing problem, a generalization of the well-known bin packing problem, where a set of items must be packed into a set of heterogeneous bins characterized by possibly different volumes and fixed selection costs. The objective of the problem is to select bins to pack all items at minimum total bin-selection cost. The paper introduces lower bounds and heuristics for the problem, the latter integrating lower and upper bound techniques. Extensive numerical tests conducted on instances with up to 1000 items show the effectiveness of these methods in terms of computational effort and solution quality. We also provide a systematic numerical analysis of the impact on solution quality of the bin selection costs and the correlations between these and the bin volumes. The results show that these correlations matter and that solution methods that are un-biased toward particular correlation values perform better.     

A beam search approach to the container loading problem

The single container loading problem is a three-dimensional packing problem in which a container has to be filled with a set of boxes. The objective is to maximize the space utilization of the container. This problem has wide applications in the logistics industry. In this work, a new constructive approach to this problem is introduced. The approach uses a beam search strategy. This strategy can be viewed as a variant of the branch-and-bound search that only expands the most promising nodes at each level of the search tree. The approach is compared with state-of-the-art algorithms using 16 well-known sets of benchmark instances. Results show that the new approach outperforms all the others for each set of instances.     

A branch-and-bound procedure for a single-machine earliness scheduling problem with two agents

This paper addresses a two-agent scheduling problem on a single machine where the objective is to minimize the total weighted earliness cost of all jobs, while keeping the earliness cost of one agent below or at a fixed level Q. A mixed-integer programming (MIP) model is first formulated to find the optimal solution which is useful for small-size problem instances. To solve medium- to large-size problem instances, a branch-and-bound algorithm incorporating with several dominance properties and a lower bound is then provided to derive the optimal solution. A simulated annealing heuristic algorithm incorporating with a heuristic procedure is developed to derive the near-optimal solutions for the problem. A computational experiment is also conducted to evaluate the performance of the proposed algorithms.     

Frequency-driven tabu search for the maximum s-plex problem

The maximum s-plex problem is an important model for social network analysis and other studies. In this study, we present an effective frequency-driven multi-neighborhood tabu search algorithm (FD-TS) to solve the problem on very large networks. The proposed FD-TS algorithm relies on two transformation operators (Add and Swap) to locate high-quality solutions, and a frequency-driven perturbation operator (Press) to escape and search beyond the identified local optimum traps. We report computational results for 47 massive real-life (sparse) graphs from the SNAP Collection and the 10th DIMACS Challenge, as well as 52 (dense) graphs from the 2nd DIMACS Challenge (results for 48 more graphs are also provided in the Appendix). We demonstrate the effectiveness of our approach by presenting comparisons with the current best-performing algorithms.     

Branch-and-bound method for minimizing the weighted completion time scheduling problem on a single machine with release dates

In this paper, we consider a single-machine scheduling problem with release dates. The aim is to minimize the total weighted completion time. This problem is known to be strongly NP-hard. We propose two new lower bounds that can be, respectively, computed in O(n²) and in O(nlogn) time where n is the number of jobs. We prove a sufficient and necessary condition for local optimality, which can also be considered as a priority rule. We present an efficient heuristic using such a condition. We also propose some dominance properties. A branch-and-bound algorithm incorporating the heuristic, the lower bounds and the dominance properties is proposed and tested on a large set of instances.     

Heuristic and exact algorithms for the max–min optimization of the multi-scenario knapsack problem

We are concerned with a variation of the standard 0–1 knapsack problem, where the values of items differ under possible S scenarios. By applying the ‘pegging test’ the ordinary knapsack problem can be reduced, often significantly, in size; but this is not directly applicable to our problem. We introduce a kind of surrogate relaxation to derive upper and lower bounds quickly, and show that, with this preprocessing, the similar pegging test can be applied to our problem. The reduced problem can be solved to optimality by the branch-and-bound algorithm. Here, we make use of the surrogate variables to evaluate the upper bound at each branch-and-bound node very quickly by solving a continuous knapsack problem. Through numerical experiments we show that the developed method finds upper and lower bounds of very high accuracy in a few seconds, and solves larger instances to optimality faster than the previously published algorithms.     

A Best-first Branch-and-bound Algorithm for Orthogonal Rectangular Packing Problems

In this paper we discuss the problem of packing a set of small rectangles (pieces) in an enclosing final rectangle. We present first a best-first branch-and-bound exact algorithm and second a heuristic approach in order to solve exactly and approximately this problem. The performances of the proposed approaches are evaluated on several randomly generated problem instances. Computational results show that the proposed exact algorithm is able to solve small and medium problem instances within reasonable execution time. The derived heuristic performs very well in the sense that it produces high-quality solutions within small computational time.     

A branch-and-bound algorithm for single machine scheduling with quadratic earliness and tardiness penalties

This paper considers the problem of scheduling a single machine, in which the objective function is to minimize the weighted quadratic earliness and tardiness penalties and no machine idle time is allowed. We develop a branch and bound algorithm involving the implementation of lower and upper bounding procedures as well as some dominance rules. The lower bound is designed based on a lagrangian relaxation method and the upper bound includes two phases, one for constructing initial schedules and the other for improving them. Computational experiments on a set of randomly generated instances show that one of the proposed heuristics, used as an upper bound, has an average gap less than 1.3% for instances optimally solved. The results indicate that both the lower and upper bounds are very tight and the branch-and-bound algorithm is the first algorithm that is able to optimally solve problems with up to 30 jobs in a reasonable amount of time.     

On the hardness of finding near-optimal multicuts in directed acyclic graphs

Given an edge-weighted (di)graph and a list of source-sink pairs of vertices of this graph, the minimum multicut problem consists in selecting a minimum-weight set of edges (or arcs), whose removal leaves no path from each source to the corresponding sink. This is a well-known NP-hard problem, and improving several previous results, we show that it remains APX-hard in unweighted directed acyclic graphs (DAG), even with only two source-sink pairs. This is also true if we remove vertices instead of arcs.     

Single machine weighted earliness–tardiness penalty problem with a common due date

In this paper, we have considered a class of single machine job scheduling problems where the objective is to minimize the weighted sum of earliness–tardiness penalties of jobs. The weights are job-independent but they depend on whether a job is early or tardy. The restricted version of the problem where the common due date is smaller than a critical value, is known to be NP-complete. While dynamic programming formulation runs out of memory for large problem instances, depth-first branch-and-bound formulation runs slow for large problems since it uses a tree search space. In this paper, we have suggested an algorithm to optimally solve large instances of the restricted version of the problem. The algorithm uses a graph search space. Unlike dynamic programming, the algorithm can output optimal solutions even when available memory is limited. It has been found to run faster than dynamic programming and depth-first branch-and-bound formulations and can solve much larger instances of the problem in reasonable time. New upper and lower bounds have been proposed and used. Experimental findings are given in detail.Scope and purposeA class of single machine problems arising out of scheduling jobs in JIT environment has been considered in this paper. The objective is to minimize the total weighted earliness–tardiness penalties of jobs. In this paper, we have presented a new algorithm and conducted extensive empirical runs to show that the new algorithm performs much better than the existing approaches in solving large instances of the problem.     

Towards optimal kernel for edge-disjoint triangle packing

We study the kernelization of the Edge-Disjoint Triangle Packing (Etp) problem, in which we are asked to find k edge-disjoint triangles in an undirected graph. Etp is known to be fixed-parameter tractable with a 4k vertex kernel. The kernelization first finds a maximal triangle packing which contains at most 3k vertices, then the reduction rules are used to bound the size of the remaining graph. The constant in the kernel size is so small that a natural question arises: Could 4k be already the optimal vertex kernel size for this problem? In this paper, we answer the question negatively by deriving an improved 3.5k vertex kernel for the problem.     

Minimum cost flows with minimum quantities

We consider a variant of the well-known minimum cost flow problem where the flow on each arc in the network is restricted to be either zero or above a given lower bound. The problem was recently shown to be weakly NP-complete even on series-parallel graphs. We start by showing that the problem is strongly NP-complete and cannot be approximated in polynomial time (unless P=NP) up to any polynomially computable function even when the graph is bipartite and the given instance is guaranteed to admit a feasible solution. Moreover, we present a pseudo-polynomial-time exact algorithm and a fully polynomial-time approximation scheme (FPTAS) for the problem on series-parallel graphs.     

On the Power of Non-adaptive Learning Graphs

We introduce a notion of the quantum query complexity of a certificate structure. This is a formalization of a well-known observation that many quantum query algorithms only require the knowledge of the position of possible certificates in the input string, not the precise values therein. Next, we derive a dual formulation of the complexity of a non-adaptive learning graph and use it to show that non-adaptive learning graphs are tight for all certificate structures. By this, we mean that there exists a function possessing the certificate structure such that a learning graph gives an optimal quantum query algorithm for it. For a special case of certificate structures generated by certificates of bounded size, we construct a relatively general class of functions having this property. The construction is based on orthogonal arrays and generalizes the quantum query lower bound for the k-sum problem derived recently by Belovs and ?palek (Proceeding of 4th ACM ITCS, 323–328, 2013). Finally, we use these results to show that the learning graph for the triangle problem by Lee et al. (Proceeding of 24th ACM-SIAM SODA, 1486–1502, 2013) is almost optimal in the above settings. This also gives a quantum query lower bound for the triangle sum problem.