Implicit Enumeration Algorithm Research Articles

Algorithms for the global domination problem

A dominating set D in a graph G is a subset of its vertices such that every its vertex that does not belong to set D is adjacent to at least one vertex from set D. A set of vertices of graph G is a global dominating set if it is a dominating set for both, graph G and its complement. The objective is to find a global dominating set with the minimum cardinality. Neither exact nor approximation algorithm existed for the problem known to be NP-hard. We show that it remains NP-hard for restricted types of graphs. At the same time, we specify some families of graphs for which the three heuristics, that we propose here, are optimal. Given the complexity status of the problem, our aim was the development of powerful heuristic algorithms that work well in practice for large-scaled instances. To measure the efficiency of our heuristics, we formulated the problem as an integer linear program (ILP) and also we developed an alternative implicit enumeration (IE) algorithm obtaining guaranteed optimal solutions for the existing benchmark instances with up to 8000 vertices. Remarkably, for 56.75% of these instances, at least one of our heuristics also created an optimal solution, where an average absolute error for the remaining instances was a single vertex. The average approximation ratio was 1.005, whereas for the largest benchmark instances with up to 25000 vertices our heuristics delivered solutions in less than 2 min.

Mobility as a Service: An exploration of exact and heuristic algorithms for a new multi-modal multi-objective journey planning problem

Mobility as a Service (MaaS) is a term coined for the development and implementation of multi-modal trip planner recommendation systems. Multi-modal journeys can include public transport, private transport and hire-able e-scooters which can beneficially augment public transport journeys. This study proposes and tackles a new and more general multi-modal multi-objective journey planning problem than considered previously. Our aim is to generate Pareto sets of multi-modal journeys which minimise the objectives of cost, travel time, inconvenience, CO2 emissions, and calorie expenditure, as well as find journeys that optimally balance the trade-offs between them. Commuters can then choose between green, cheap, fast, car-free or convenient journeys. This work proposes implicit enumeration and heuristic algorithms which are analysed in terms of their theoretical time complexity, empirically in terms of solution time and optimality gap using a series of Manhattan and random structure transport networks. We show that our algorithms can generate solutions of equal quality to RAPTOR—a prominent existing journey planning algorithm. For this problem we reveal that the number of network nodes is a huge computational bottleneck, leading to the suggestion that future research can benefit from limiting the number of network nodes that are considered as transfer points. While our heuristic can generate good quality Pareto sets quicker than the enumeration procedure for the largest instances considered, we show that the enumeration procedure with pruning rules can still be the most effective strategy for this particular problem.

Competitive network restructuring with spatially loyal customers. A bilevel facility delocation problem

In this paper we address the problem faced by a company that needs to decide the number and location of facilities to close in order to reduce the size of their retail distribution network, whilst minimising the market share that is lost to their competitor, who is also engaged in a network reduction process. The problem is modelled as a bilevel competitive delocation model and is formulated as a variant of the well-known (r|p)-centroid problem. The formulation takes into account the existence of a certain level of customer loyalty, which implies that customers may decide to visit a more distant facility of their regular provider rather than rely on a closer competitor’s facility. A depth-first search implicit enumeration algorithm is proposed for solving a diverse set of instances of the problem. The analysis of an instance based on a real-life setting, and intensive numerical experimentation, provide evidence of the effectiveness of the algorithm for the solution of real-size problems, and confirm the practical validity of our modelling approach. Our results also highlight the importance of considering spatial loyalty in network restructuring scenarios.

Exact and heuristic algorithms for the domination problem

In a simple connected graph G=(V,E), a subset of vertices S⊆V is a dominating set if any vertex v∈V∖S is adjacent to some vertex x from this subset. A number of real-life problems can be modeled using this problem which is known to be among the difficult NP-hard problems in its class. We formulate the problem as an integer liner program (ILP) and compare the performance with the two earlier existing exact state-of-the-art algorithms and exact implicit enumeration and heuristic algorithms that we propose here. Our exact algorithm was able to find optimal solutions much faster than ILP and the above two exact algorithms for middle-dense instances. For graphs with a considerable size, our heuristic algorithm was much faster than both, ILP and our exact algorithm. It found an optimal solution for more than half of the tested instances, whereas it improved the earlier known state-of-the-art solutions for almost all the tested benchmark instances. Among the instances where the optimum was not found, it gave an average approximation error of 1.18.

A tri‐level optimization strategy incorporating wind power against coordinated cyber‐physical attacks

AbstractWith the fast development of cyber‐physical power systems, coordinated cyber‐physical attacks have become one of the main threats to the security of modern power grid. Considering that it is difficult for attackers to obtain full information of the entire power grid, the coordinated cyber‐physical attack problem for partitioned power grid is studied in this paper, and a tri‐level defender‐attacker‐dispatcher optimization model with consideration of wind power uncertainty in a specific coordinated cyber‐physical attack mode is proposed. The Karush‐Kuhn‐Tucker (KKT) condition and an improved implicit enumeration algorithm are applied to obtain the solution of the complicated optimization problem effectively. In addition, two risk assessment indices pertinent to the physical lines and load measurements are proposed according to the obtained total amount of optimized load shedding and coordinated attack‐defense strategy under valid attack‐defense scenarios. Taking a partitioned IEEE 39‐bus test system as example, the risks of physical transmission lines and load measurements in different areas under attacks are elaborately studied, and the impact of wind power uncertainty on attack‐defense strategies is discussed as well. Numerical simulation results illustrate the effectiveness of the proposed optimization model and risk assessment indices, and some significant conclusions are drawn.

Combining a multidimensional risk evaluation with an implicit enumeration algorithm to tackle the portfolio selection problem of a natural gas pipeline

In pipelines, dealing with the dilemma of investing in a combination of the most critical sections without exceeding resources is a relevant issue. Some studies recommend using additive models with mathematical programming to solve portfolio problems, considering interval scale transformation. However, such a joint approach is inadequate when probabilistic consequences are used for evaluating a model based on Multi-Attribute Utility Theory (MAUT). Thus, it is not possible to ensure a better solution because the axiomatic structure of Utility Theory consequently prevents scale transformation. Therefore, this paper proposes a model to overcome bias in additive models for portfolio selection when an interval scale is the only form of measurement. Thus, a multidimensional risk evaluation is conducted and an implicit enumeration algorithm is used. A numerical application is applied to validate the proposed model, and a sensitivity analysis is conducted to verify its robustness. Results show that this model ensures a combination of sections with the highest values of aggregate risk, given the restrictions. Moreover, compared to the joint approach (without scale transformation), the proposed model also finds better solutions in over 99% of simulated cases. Furthermore, it can help managers effectively to allocate complementary resources to where they are most needed.

Provably optimal sparse solutions to overdetermined linear systems with non-negativity constraints in a least-squares sense by implicit enumeration

Computing sparse solutions to overdetermined linear systems is a ubiquitous problem in several fields such as regression analysis, signal and image processing, information theory and machine learning. Additional non-negativity constraints in the solution are useful for interpretability. Most of the previous research efforts aimed at approximating the sparsity constrained linear least squares problem, and/or finding local solutions by means of descent algorithms. The objective of the present paper is to report on an efficient and modular implicit enumeration algorithm to find provably optimal solutions to the NP-hard problem of sparsity-constrained non-negative least squares. We focus on the problem where the system is assumed to be over-determined where the matrix has full column rank. Numerical results with real test data as well as comparisons of competing methods and an application to hyperspectral imaging are reported. Finally, we present a Python library implementation of our algorithm.

A Programmed Algorithm for Resource Scheduling in Construction.(Dept.C)

If the critical path determined completion time of the construction project is constrained by the lack of resources, which are needed to carry out some of the activities, then the analysis required for scheduling the activities is much more involved. There are two major categories of procedures for solving this problem. The first involves the use of some heuristic rules in determining priorities among activities competing for available resources. The second involves some form of mathematical programming to produce the optimal schedule. Heuristic scheduling rules. computer programmed to give good feasible schedules have been the basis for all practical systems to date. In the same time, the optimal solution of the problem has not been paid enough attention in the construction management literature. The increasing availability of more powerful computer systems should see more emphasis on the development of procedures to give the optimal solution for large complex projects which occur in construction practice. In this paper a Computer program is developed for implementing an implicit enumeration algorithm to give the optimal solution of the problem for construction projects. The algorithm has been chosen to satisfy large projects and its efficiency has been established by specifying a lower bound on the length of the schedule and by sorting project activities and resources in a way such that the computation time is greatly reduced. Experience in applying the algorithm to an actual project is reported.

The impact of congestion on protection decisions in supply networks under disruptions

We analyze the impact of congestion on the protection decisions of supply networks for mitigating major disruption. We present a tri-level mixed integer programming model to identify critical facilities to secure and backup plans with appropriate capacity levels and response speeds during disruption. The congestion arising at the facilities as a result of flow reallocation from the disrupted facilities is modelled as a convex cost function. We present an implicit enumeration algorithm to solve the tri-level model. The computational results demonstrate the efficiency of the solution method. Our analysis depicts that a decentralized protection strategy is recommended under congestion.

Orientational variable-length strip covering problem: A branch-and-price-based algorithm

In this article, we study a challenging optimization problem, namely the orientational variable-length strip covering problem. Given a large rectangular region and multiple orientational strips with variable lengths, the strips should be positioned and their lengths be determined such that their union can cover the large region with the minimal total area. This problem is motivated by the real-world problem in which multiple Earth observation satellites are used to image a large region in a cooperative pattern. It is a challenging nonlinear combinatorial optimization problem in continuous space. We propose a set-covering-problem-like integer programming formulation for the problem based on grid discretization and prove that the optimal solutions can be achieved when the strips take discrete values. As there is a large number of columns, we use a column generation method to solve the linear relaxation problem and an enumeration algorithm to solve the subproblem. To accelerate the solution, we propose a provable dominance rule to greatly reduce the solution space of the subproblem, enabling the application of an implicit enumeration (IE) algorithm. Then, we propose a cell-gathered approximate pricing method based on the definition of nested father–child grids. As a result, an efficient branch-and-price-based algorithm (BPBA) is developed. The numerical tests on random instances show that the dominance rule can reduce the subproblem’s solutions by almost 74% and the BPBA can run five times faster than Gurobi (commercial solving software) to find comparable solutions on average.

Toward an Efficient Resolution for a Single-machine Bi-objective Scheduling Problem with Rejection

Abstract We consider a single-machine bi-objective scheduling problem with rejection. In this problem, it is possible to reject some jobs. Four algorithms are provided to solve this scheduling problem. The two objectives are the total weighted completion time and the total rejection cost. The aim is to determine the set of efficient solutions. Four heuristics are described; they are implicit enumeration algorithms forming a branching tree, each one having two versions according to the root of the tree corresponding either to acceptance or rejection of all the jobs. The algorithms are first illustrated by a didactic example. Then they are compared on a large set of instances of various dimension and their respective performances are analysed.

Minimax-Regret Robust Defensive Strategy Against False Data Injection Attacks

This paper develops a multi-level game-theoretic framework for determining a cost-effective defensive strategy for protecting power systems from false data injection attacks like load redistribution attacks. First, a multi-level optimization problem considering interactions among defenders, attackers and operators is modeled based on the minimax-regret decision rule, which is then reformulated as an equivalent bi-level mixed-integer linear programming problem. Next, an implicit enumeration algorithm is developed to find a globally optimal solution to this complex bi-level problem. Several acceleration techniques are introduced to improve the computation efficiency of the proposed method for large-scale power system applications. Last, the proposed defensive strategy is validated by case studies based on a six-bus test system and a modified two-area RTS-96 system.

Optimizing fortification plan of capacitated facilities with maximum distance limits

Facilities are the backbone of many supply systems and are always vulnerable to potential attacks. Considering that when disruptions occur, limits on transportation distances are essential for many important supply systems like emergency items and perishable items, this paper presents fortification plan of capacitated facilities with maximum distance limits. The problem is formulated as a tri-level mixed integer problem, which considers capacitated facilities, interdiction, and fortification budget limit simultaneously. Then it is simplified to a bi-level problem and solved by implicit enumeration algorithm performed on a binary tree. Computational results obtained from 64 randomly generated test instances are reported along with the benefit of maximum distance limits.

The p-center flow-refueling facility location problem

The p-center flow-refueling facility location problem locates p refueling facilities to minimize the maximum percentage deviation of all drivers. It is a social equity resource allocation model as opposed to a social efficiency p-maximum coverage flow-refueling model. We propose a nonlinear integer program based on link formulation and analyze its relationship with p-maximum coverage and set covering location flow-refueling problems. We develop a link-based implicit enumeration algorithm with an embedded vehicle range-constrained shortest path subproblem to optimally solve the problem. The computational results show that multiple optimal solutions may exist but that they are associated with different total trip distances. In addition, the maximum deviation may not decrease as the vehicle range or the number of refueling facilities marginally increases.

Empirical Analysis of Traveling Backwards and Passenger Flows Reassignment on a Metro Network with Automatic Fare Collection (AFC) Data and Train Diagram

Metro (subway) systems are becoming overcrowded in some of China’s mega-cities, such as Beijing, Shanghai, and Guangzhou. As many passengers are unable to board trains on an overcrowded metro network during peak periods, some of them are willing to spend more time and energy by traveling backwards in order to secure a seat or even room for standing. Traditional studies on travel behavior analysis and transit assignment models seldom deal with this situation. We propose a methodology including the affinity propagation cluster method with between-within-proportion (BWP) index and an adaptive “0-1” model named the traveling backwards model (TBM) to identify the phenomenon of “traveling backwards” and to reassign passenger flows on a metro network using automatic fare collection (AFC) data and actual train diagrams. As a numerical example, this integrated approach is applied to the Beijing metro system. Our research shows that the affinity propagation cluster method with BWP index and implicit enumeration algorithm for TBM work well. TBM is a good replacement for the existing assignment model and travel behavior analysis, especially for those mega-cities’ networks in peak hours.

Scheduling a Single Machine with Primary and Secondary Objectives

We study a scheduling problem in which jobs with release times and due dates are to be processed on a single machine. With the primary objective to minimize the maximum job lateness, the problem is strongly NP-hard. We describe a general algorithmic scheme to minimize the maximum job lateness, with the secondary objective to minimize the maximum job completion time. The problem of finding the Pareto-optimal set of feasible solutions with these two objective criteria is strongly NP-hard. We give the dominance properties and conditions when the Pareto-optimal set can be formed in polynomial time. These properties, together with our general framework, provide the theoretical background, so that the basic framework can be expanded to (exponential-time) implicit enumeration algorithms and polynomial-time approximation algorithms (generating the Pareto sub-optimal frontier with a fair balance between the two objectives). Some available in the literature experimental results confirm the practical efficiency of the proposed framework.

An implicit enumeration algorithm for the hub interdiction median problem with fortification

Hubs are intermediate facilities that play a pivotal role in efficient functioning of transportation and telecommunication systems. Like any other service infrastructure, hub facilities can be subject to natural or man-made disruptions after installation. In this paper, we address the problem of optimally allocating protective resources among a set of p existing hub facilities in such a manner that the damage inflicted by an intentional strike against the service system is minimized. Casting the problem as a Stackelberg game, the leader (i.e., the network protector or defender) fortifies q of the p operating hubs in order to minimize the impact of the upcoming strike, whereas the follower (i.e., the attacker) tries to identify and interdict r of the p−q unprotected hubs that their loss would diminish the network performance the most. A bilevel programming formulation is presented to model the problem and using a min-max approach the model is reduced to a single level mixed integer programming (MIP) model. Furthermore, an efficient exact solution algorithm based on implicit enumeration is proposed for solving the problem. Extensive computational experiments show the capability of the proposed algorithm to obtain the optimal solutions in short computational times. Some managerial insights are also derived based on the obtained numerical results.

An Efficient Tri-Level Optimization Model for Electric Grid Defense Planning

The work reported in this paper aims at developing a practical and efficient tool for utility transmission planners to protect critical transmission assets from potential physical attacks. A trilevel min–max–min optimization model is proposed to represent actions of a planner, an attacker, and an operator. These three agents share a common objective function, which is the system load shedding. The strong duality theorem is used to merge the middle- and lower-level problems into a single-level one. The resulting mixed-integer bilevel model is solved by a Benders decomposition technique using primal cuts. Two case studies are presented as applications of this novel technique. Additionally, the proposed technique is compared with an implicit enumeration algorithm.

A variable neighborhood search for the network design problem with relays

Given a set of commodities to be routed over a network, the network design problem with relays involves selecting a route for each commodity and determining the location of relays where the commodities must be reprocessed at certain distance intervals. We propose a hybrid approach based on variable neighborhood search. The variable neighborhood algorithm searches for the route for each commodity and the optimal relay locations for a given set of routes are determined by an implicit enumeration algorithm. We show that dynamic programming can be used to determine the optimal relay locations for a single commodity. Dynamic programming is embedded into the implicit enumeration algorithm to solve the relay location problem optimally for multiple commodities. The special structure of the problem is leveraged for computational efficiency. In the variable neighborhood search algorithm, the routes of the current solution are perturbed and reconstructed to generate neighbor solutions using random and greedy construction heuristics. Computational experiments on three sets of problems (80 instances) show that the variable neighborhood search algorithm with optimal relay allocations outperforms all existing algorithms in the literature.

Algorithms for job scheduling problems with distinct time windows and general earliness/tardiness penalties

This paper addresses the single machine scheduling problem with distinct time windows and sequence-dependent setup times. The objective is to minimize the total weighted earliness and tardiness. The problem involves determining the job execution sequence and the starting time for each job in the sequence. An implicit enumeration algorithm denoted IE and a general variable neighborhood search algorithm denoted GVNS are proposed to determine the job scheduling. IE is an exact algorithm, whereas GVNS is a heuristic algorithm. In order to define the starting times, an O(n2) idle time insertion algorithm (ITIA) is proposed. IE and GVNS use the ITIA algorithm to determine the starting time for each job. However, the IE algorithm is only valid for instances with sequence-independent setup times, and takes advantage of theoretical results generated for this problem. Computational experiments show that the ITIA algorithm is more efficient than the only other equivalent algorithm found in the literature. The IE algorithm allows the optimal solutions of all instances with up to 15 jobs to be determined within a feasible computational time. For larger instances, GVNS produces better-quality solutions requiring less computational time compared with the other algorithm from the literature.