Linear Programming Formulation Research Articles

On Interdicting Dense Clusters in a Network

Given a vertex-weighted undirected graph with blocking costs of its vertices and edges, we seek a minimum cost subset of vertices and edges to block such that the weight of any γ-quasi-clique in the interdicted graph is at most some predefined threshold parameter. The value of [Formula: see text] specifies the edge density of cohesive vertex groups of interest in the network. The considered weighted γ-quasi-clique interdiction problem can be viewed as a natural generalization of several variations of the clique blocker problem previously studied in the literature. From the application perspective, this setting is primarily motivated by the problem of disrupting adversarial (“dark”) networks (e.g., social or communication networks), where γ-quasi-cliques represent “tightly knit” groups of adversaries that we aim to dismantle. We first address the theoretical computational complexity of the problem. We then exploit some basic characterization of its feasible solutions to derive a linear integer programming (IP) formulation. This linear IP model can be solved using a lazy-fashioned branch-and-cut scheme. We also propose a combinatorial branch-and-bound algorithm for solving this problem. The computational performance of the developed exact solution schemes is studied using a test bed of randomly generated and real-life networks. Finally, some interesting insights and observations are also provided using a well-known example of a terrorist network. History: Accepted by Russel Bent, Area Editor for Network Optimization: Algorithms & Applications. Funding: The work of S. Butenko was partially supported by the Air Force Office of Scientific Research under Award FA9550-23-1-0300. The work of O. A. Prokopyev was partially supported by the Office of Naval Research under Award ONR N00014-22-1-2678. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2023.0027 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2023.0027 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ . The online appendix is available at https://doi.org/10.1287/ijoc.2023.0027 .

Re-Supplying Autonomous Mobile Parcel Lockers in Last-Mile Distribution

This paper investigates a practical last-mile delivery scenario where a fleet of trucks replenishes autonomous mobile parcel lockers (AMPLs) in an urban setting. The lockers move along specified paths within restricted zones to reach customers’ locations. Ensuring seamless coordination between trucks and AMPLs requires the identification of suitable locations to exchange empty or loaded modular lockers. We first introduce a mixed-integer linear programming (MILP) formulation for the investigated problem. The proposed formulation establishes the basis for optimizing meeting point selection and routing decisions. Additionally, the study introduces a cluster-based simulated annealing (CSA) algorithm tailored for addressing larger-scale instances of the studied problem. The CSA algorithm incorporates the K-means clustering method with specialized operators rooted in an extensive neighborhood search, aiming to improve the effectiveness of solution discovery. We generated a new set of benchmark instances to assess the MILP formulation’s efficiency and the proposed metaheuristic algorithm and conducted comprehensive numerical experiments.

The Exact Subset MultiCover Problem

In this paper, we study the Exact Subset MultiCover problem (or ESM), which can be seen as an extension of the well-known Set Cover problem. Let (U,f) be a multiset built from set U={e1,e2,…,em} and function f:U→N⁎. ESM is defined as follows: given (U,f) and a collection S={S1,S2,…,Sn} of n subsets of U, is it possible to find a multiset (S′,g) with S′={S1′,S2′,…,Sn′} and g:S′→N, such that (i) Si′⊆Si for every 1≤i≤n, and (ii) each element of U appears as many times in (U,f) as in (S′,g)? We study this problem under an algorithmic viewpoint and provide diverse complexity results such as polynomial cases, NP-hardness proofs and FPT algorithms. We also study two variants of ESM: (i) Exclusive Exact Subset MultiCover (EESM), which asks that each element of U appears in exactly one subset Si′ of S′; (ii) Maximum Exclusive Exact Subset MultiCover (Max-EESM), an optimisation version of EESM, which asks that a maximum number of elements of U appear in exactly one subset Si′ of S′. For both variants, we provide several complexity results; in particular we present a 2-approximation algorithm for Max-EESM, that we prove to be tight. For these three problems, we also provide an Integer Linear Programming (ILP) formulation.

Tscluster: A python package for the optimal temporal clustering framework

Temporal clustering extends the conventional task of data clustering by grouping time series data according to shared temporal trends across sociospatial units, with diverse applications in the social sciences, especially urban science. The two dominant methods are as follows: Time Series Clustering (TSC), with dynamic cluster centres but static labels for each entity, and Sequence Label Analysis (SLA), with static cluster centres but dynamic labels. To implement the universe of models spanning the design space between TSC and SLA, we present tscluster, an open-source Python framework. tscluster offers: (1) several innovative techniques, such as Bounded Dynamic Clustering (BDC), that are not available in existing libraries, allowing users to set an upper bound on the number of label changes and identify the most dynamically evolving time series; (2) a user-friendly interface for applying and comparing these methods; (3) globally optimal solutions for the clustering objective by employing a mixed-integer linear programming formulation, enhancing the reproducibility and robustness of the results in contrast to existing methods based on initialization-sensitive local optimization; and (4) a suite of visualization tools for interpretability and comparison of clustering results. We present our framework using a case study of neighbourhood change in Toronto, comparing two methods available in tscluster. Supplemental materials provide an additional case study of local business development in Chicago and a detailed mathematical exposition of our framework. tscluster can be installed via PyPI (pypi.org/project/tscluster), and the source code is accessible on Github (github.com/tscluster-project/tscluster). Documentation is available online at the tscluster website (tscluster.readthedocs.io).

Communication Complexity of Entanglement-Assisted Multi-Party Computation

We consider a quantum and a classical version of a multi-party function computation problem with n players, where players 2,…,n need to communicate appropriate information to player 1 so that a “generalized” inner product function with an appropriate promise can be calculated. In the quantum version of the protocol, the players have access to entangled qudits but the communication is still classical. The communication complexity of a given protocol is the total number of classical bits that need to be communicated. When n is prime, and for our chosen function, we exhibit a quantum protocol (with complexity (n−1)(logn) bits) and a classical protocol (with complexity ((n−1)2(logn2) bits)). Furthermore, we present an integer linear programming formulation for determining a lower bound on the classical communication complexity. This demonstrates that our quantum protocol is strictly better than classical protocols.

A hybrid local search algorithm for the continuous energy-constrained scheduling problem

AbstractWe consider the continuous energy-constrained scheduling problem (CECSP). A set of jobs has to be processed on a continuous, shared resource. A schedule for a job consists of a start time, completion time, and a resource consumption profile. The goal is to find a schedule such that each job does not start before its release time, is completed before its deadline, satisfies its full resource requirement, and respects its lower and upper bounds on resource consumption during processing. The objective is to minimize the total weighted completion time. We present a hybrid local search approach, using simulated annealing and linear programming, and compare it to a mixed-integer linear programming (MILP) formulation. We show that the hybrid local search approach matches the MILP formulation in solution quality for small instances and is able to find a feasible solution for larger instances in reasonable time.

Minimizing the earliness–tardiness for the customer order scheduling problem in a dedicated machine environment

AbstractThe customer order scheduling problem has garnered considerable attention in the recent scheduling literature. It is assumed that each of several customer orders consists of several jobs, and each customer order is completed only if each job of the order is completed. In this paper, we consider the customer order scheduling problem in a machine environment where each customer places exactly one job on each machine. The objective is to minimize the earliness–tardiness, where tardiness is defined as the time an order is finished past its due date, and earliness is the time a job is finished before its due date or the completion time of the corresponding order, whichever is later. Even though the earliness–tardiness criterion is an important objective for just-in-time production, this problem has not been studied in the context of the customer order scheduling problem. We provide a mixed-integer linear programming (MILP) formulation for this problem and derive multiple problem properties. Furthermore, we develop six different heuristics for this problem configuration. They follow the structure of the iterated greedy algorithm and additionally use a refinement function in which they differ. In a computational experiment, the algorithms were compared with each other and outperformed a solver solution of the MILP, which proves their ability to efficiently solve the problem configuration.

Proactive scheduling for mmWave Wireless LANs

To cope with growing wireless bandwidth demand, millimeter wave (mmWave) communication has been identified as a promising technology to deliver Gbps throughput. However, due to the susceptibility of mmWave signals to blockage, applications can experience significant performance variability as users move around due to rapid and significant variation in channel conditions. In this context, proactive schedulers that make use of future data rate prediction have potential to bring a significant performance improvement as compared to traditional schedulers. In this work, we explore the possibility of proactive scheduling that uses mobility prediction and some knowledge of the environment to predict future channel conditions. We present both an optimal proactive scheduler, which is based on an integer linear programming formulation and provides an upper bound on proactive scheduling performance, and a greedy heuristic proactive scheduler that is suitable for practical implementation. Extensive simulation results show that proactive scheduling has the potential to increase average user data rate by up to 35% over the classic proportional fair scheduler without any loss of fairness and incurring only a small increase in jitter. The results also show that the efficient proactive heuristic scheduler achieves from 60% to 75% of the performance gains of the optimal proactive scheduler. Finally, the results show that proactive scheduling performance is sensitive to the quality of mobility prediction and, thus, use of state-of-the-art mobility prediction techniques will be necessary to realize its full potential.

Routing and charging scheduling for the electric carsharing system with mobile charging vehicles

Electric carsharing systems are expected to be an optional alternative to private vehicles for decreasing the urban traffic congestion and emissions. However, the temporal and spatial imbalance of the charging demand of shared electric vehicles adds to the managerial complexity of electric carsharing systems. This paper integrates mobile charging vehicles into the electric carsharing system to address this imbalance. Mobile charging vehicles can dwell at stations to provide elastic charging capacity, and thereby decrease both the waiting time of shared electric vehicles at busy stations and the investments in fixed charging piles at suburban stations. In this paper, a mixed integer linear programming formulation is proposed based on a time-space network, in which the routes of shared electric vehicles, charging schedules of shared electric vehicles, and routes of mobile charging vehicles are optimized simultaneously. Then, an algorithm based on Lagrangian relaxation is proposed. Specifically, the proposed formulation is decomposed into three independent subproblems. We propose three exact algorithms for these subproblems, and a tailored multistep repair algorithm is designed to generate feasible solutions. A case study in Hefei, China demonstrates the performance of the proposed algorithm and the effects of the number of SEVs, the number of MCVs, the number of fixed charging piles, trip component, battery capacity, and revenue on the operation of the electric carsharing system.

Last-train timetable synchronization for service compatibility maximization in urban rail transit networks with arrival uncertainties

The last train services of urban rail transit offer all passengers the final chance to reach their destinations. In the context of multimodal transport, considering the random delayed arrivals of late-night vehicles of other transport modes at urban hub stations and their adverse effects on multimodal passengers transferring to urban rail transit, a probabilistic scenario set is established to determine the arrival uncertainties of multimodal passengers. In each scenario, the timetable synchronization problem is formulated as a mixed-integer nonlinear programming model that requires a performance trade-off between destination reachability and the remaining path distance of both non-multimodal and multimodal passengers, integrally termed last train service compatibility. Through linearization techniques, the model is transformed into an equivalent mixed-integer linear programming form, which can be efficiently solved using the Gurobi solver to obtain the optimal last train timetable for each scenario and form an alternative scheme set. An improved probabilistic scenario set-based regret value theory is then developed, in which a novel regret value calculation method is proposed. The scheme with the minimum total weighted opportunity loss in all scenarios is selected as the optimal robust. Real case experiments based on the Chengdu–Chongqing high-speed railway line and Chengdu metro network are conducted to test the performance of our model. The results show that compared with the original timetable, the optimized timetable reduces the number of unreachable passengers by 34.29% and the sum of the average remaining path distance of non-multimodal and multimodal passengers by 57.43% with the help of path planning for all passengers. The proposed approach is proven not only to balance the demands of both reachable and unreachable passengers, but also to significantly improve the robustness of the last train timetable to arrival uncertainties of multimodal passengers and reduce their service inequity.

Proof Complexity and Beyond

Proof complexity is a multi-disciplinary research area that addresses questions of the general form “how difficult is it to prove certain mathematical facts?” The current workshop focussed on recent advances in our understanding that the analysis of an appropriately tailored concept of “proof” underlies many of the arguments in algorithms, geometry or combinatorics research that make the core of modern theoretical computer science. These include the analysis of practical Boolean satisfiability (SAT) solving algorithms, the size of linear or semidefinite programming formulations of combinatorial optimization problems, the complexity of solving total NP search problems by local methods, and the complexity of describing winning strategies in two-player round-based games, to name just a few important examples.

Lossless Approximate Pattern Matching: Automated Design of Efficient Search Schemes.

This study introduces a pioneering approach to automate the creation of search schemes for lossless approximate pattern matching. Search schemes are combinatorial structures that define a series of searches over a partitioned pattern. Each search specifies the processing order of these parts and the cumulative lower and upper bounds on the number of errors in each part of the pattern. Together, these searches ensure the identification of all approximate occurrences of a search pattern within a predefined limit of k errors. While existing literature offers designed schemes for up to k = 4 errors, designing search schemes for larger k values incurs escalating computational costs. Our method integrates a greedy algorithm and a novel Integer Linear Programming (ILP) formulation to design efficient search schemes for up to k = 7 errors. Comparative analyses demonstrate the superiority of our ILP-optimal schemes over alternative strategies in both theoretical and practical contexts. Additionally, we propose a dynamic scheme selection technique tailored to specific search patterns, further enhancing efficiency. Combined, this yields runtime reductions of up to 53% for higher k values. To facilitate search scheme generation, we present Hato, an open-source software tool (AGPL-3.0 license) employing the greedy algorithm and utilizing CPLEX for ILP solving. Furthermore, we introduce Columba 1.2, an open-source lossless read-mapper (AGPL-3.0 license) implemented in C++. Columba surpasses existing state-of-the-art tools by identifying all approximate occurrences of 100,000 Illumina reads (150 bp) in the human reference genome within 24 seconds (maximum edit distance of 4) and 75 seconds (maximum edit distance of 6) using a single CPU core. Notably, our study showcases Columba's capability to align 100,000 reads of length 50, with high error rates and up to an edit distance of 7, in a mere 2 hours and 15 minutes. This achievement is unmatched by other lossless aligners, which require over 3 hours for edit distance 5 alignments. Moreover, Columba exhibits a mapping rate four times higher than that of a lossy tool for this dataset.

A new mathematical model and meta-heuristic algorithm for order batching, depot selection, and assignment problem with multiple depots and pickers

The ‘order picking problem’ involves the efficient and organized retrieval of items from shelves to fulfill customer orders. In this study, we consider a new configuration of warehouse system with multiple pickers and depots (m-pickers & n-depots) for manually operated ‘picker-to-parts’ warehouses. The efficiency of the process is measured based on two metrics: i- total order picking distance of each picker ii- total number of pickers assigned to each of depots. The handled problem is called as ‘Order Batching, Depot Selection and Assignment Problem with Multiple Depots and Multiple Pickers (OBDSAPMDMP)11OBDSAPMDMP = Order Batching, Depot Selection, and Assignment Problem with Multiple Depots and Multiple Pickers.’. To solve this complex problem, a new bi-objective Mixed-Integer Linear Programming (MILP) formulation for small-sized problems and a meta-heuristic called ‘Dependent Harmony Search (DHS)22DHS = Dependent Harmony Search.’ for large-sized problems are proposed. The performance of DHS algorithm is evaluated by comparing the optimal results attained by MILP model. For the problem size of 10 orders, the average gap (%) in distance between the solution of DHS and MILP is 4.22%, although in some experiments DHS can find the optimal solution in a very short time. Also, in related analysis, it is seen that constructing multiple depots instead of one left-most located depot decreases total order picking distance by 7.11% on average.

Traveling Salesman Problem with Backend Information Processing

This paper introduces the Traveling Salesman Problem (TSP) with Backend Information Processing (bTSP), which integrates backend processing times into path planning to minimize the makespan. The study develops a mixed integer linear programming formulation and conducts a theoretical analysis to understand the relationship between the TSP and the bTSP. The results show that an optimal solution for the bTSP can be efficiently derived by leveraging the connection to the standard TSP.

A generic stochastic network-based formulation for production optimization of district heating systems

AbstractDistrict heating (DH) systems are an important component in the EU strategy to reach the emission goals, since they allow an efficient supply of heat while using the advantages of sector coupling between different energy carriers such as power, heat, gas and biomass. Most DH systems use several different types of units to produce heat for hundreds or thousands of households (e.g. natural gas-fired boilers, electric boilers, biomass-fired units, waste heat from industry, solar thermal units). Furthermore, combined heat and power units units are often included to use the synergy effects of excess heat from electricity production. To address the challenge of providing optimization tools for a vast variety of different system configurations, we propose a generic mixed-integer linear programming formulation for the operational production optimization in DH systems. The model is based on a network structure that can represent different system setups while the underlying model formulation stays the same. Therefore, the model can be used for most DH systems although they might use different combinations of technologies in different system layouts. The mathematical formulation is based on stochastic programming to account for the uncertainty of energy prices and production from non-dispatchable units such as waste heat and solar heat. Furthermore, the model is easily adaptable to different application cases in DH systems such as operational planning, bidding to electricity markets and long-term evaluation. We present results from three real cases in Denmark with different requirements.

Hazard-consistent scenario selection for long-term storm surge risk assessment over extended coastal regions

The recent destructive hurricane seasons and concerns related to the future influence of climate change have increased the relevance of coastal storm hazards and, in particular, of storm surge hazard estimation when discussing the resilience of coastal communities. This hazard is generally represented as surge inundation probabilities over a large number of individual locations in the geographic domain of interest, and is typically assessed utilizing an ensemble of storm scenarios (i.e., storm events) that are representative of the regional climatology. This paper investigates the storm ensemble selection within this setting, with the objective of identifying a small number of storm scenarios that are consistent with some chosen hazard descriptions over a large geographic region. Beyond the storm events, the occurrence rates (i.e., weights) are also updated. Following past works, a two-stage optimization is adopted for the storm selection. The inner-loop identifies the occurrence rates for a given storm subset, formulating the problem as a linear programming optimization for the sum of absolute deviation for the predicted hazard. The outer-loop searches for the best subset with the desired number of storms, adopting a genetic algorithm integer optimization for minimizing the aforementioned deviation. This work extends this implementation to extended coastal regions, with many locations of interest. In this case, it is computationally intractable to consider all the locations in the domain within the linear programming formulation, and for this reason, a subset of representative locations is chosen through cluster analysis. The hazard description for only these locations is used in the storm ensemble selection. For clustering, different strategies using correlation between locations based on geospatial information, surge response, or a combination of both are examined. Additionally, the correlation in the hazard description is better integrated into the storm selection by establishing a modification of the objective function adopted for the outer optimization loop. Applications to different North Atlantic coastal domains are presented as case studies.

Accurate Flow Decomposition via Robust Integer Linear Programming.

Minimum flow decomposition (MFD) is a common problem across various fields of Computer Science, where a flow is decomposed into a minimum set of weighted paths. However, in Bioinformatics applications, such as RNA transcript or quasi-species assembly, the flow is erroneous since it is obtained from noisy read coverages. Typical generalizations of the MFD problem to handle errors are based on least-squares formulations or modelling the erroneous flow values as ranges. All of these are thus focused on error handling at the level of individual edges. In this paper, we interpret the flow decomposition problem as a robust optimization problem and lift error-handling from individual edges to solution paths. As such, we introduce a new minimum path-error flow decomposition problem, for which we give an Integer Linear Programming formulation. Our experimental results reveal that our formulation can account for errors significantly better, by lowering the inaccuracy rate by 30-50% compared to previous error-handling formulations, with computational requirements that remain practical.

Service Selection under Uncertainty

Cloud computing has emerged as a popular computing paradigm, providing scalable and pay-per-use services to execute a variety of tasks. However, selecting the most suitable cloud services for a client can be challenging, as it involves taking into account the service characteristics and a set of client restrictions. In this paper, we formalize the problem of Service Selection under Uncertainty (SSuU) as an optimization problem. Our goal is to allocate tasks to appropriate cloud services while considering the probability of Service Level Agreement (SLA) violations and a set of client restrictions. We introduce an efficient dynamic programming approach to calculate the probability of SLA violations incrementally over time. We prove that the SSuU problem is strongly NP-complete, and propose an Integer Linear Programming (ILP) formulation and an Iterated Local Search-based algorithm for tackling it. To facilitate evaluation, we also introduce a set of benchmark instances for the SSuU problem. We extensively evaluate our proposed solutions on 94 input instances and compare them to the exact method (i.e., which always produces the optimal solution). Our results demonstrate that the metaheuristic approach is significantly faster and leads to exact solutions in the great majority of the evaluation scenarios.

Learning the Graph Structure of Regular Vine-Copulas from Dependence Lists

Regular vine copulas (R-vines) provide a comprehensive framework for modeling high-dimensional dependencies using a hierarchy of trees and conditional pair-copulas. While the graphical structure of R-vines is traditionally derived from data, this work introduces a novel approach by utilizing a (conditional) pairwise dependence list. Our primary goal is to construct R-vine graphs that include the maximum possible number of dependence relationships specified in such lists. To tackle this optimization challenge, characterized by exponential growth in the search space and the structural constraints of R-vines, we propose two distinct methodologies: A 0-1 linear programming formulation and a Genetic Algorithm (GA). Additionally, the Randomized Constructive Technique (RCT) is employed to generate the initial population of the GA, serving as a baseline for our comparison. Experimental results reveal the superior performance of the GA over the RCT in terms of success rate, incorporating more relationships than RCT into the constructed R-vine graphs and achieving near-optimal or optimal graph structures.

A tight formulation for the dial-a-ride problem

Ridepooling services play an increasingly important role in modern transportation systems. With soaring demand and growing fleet sizes, the underlying route planning problems become increasingly challenging. In this context, we consider the dial-a-ride problem (DARP): Given a set of transportation requests with pick-up and delivery locations, passenger numbers, time windows, and maximum ride times, an optimal routing for a fleet of vehicles, including an optimized passenger assignment, needs to be determined. We present tight mixed-integer linear programming (MILP) formulations for the DARP by combining two state-of-the-art models into novel location-augmented-event-based formulations. Strong valid inequalities and lower and upper bounding techniques are derived to further improve the formulations. We then demonstrate the theoretical and computational superiority of the new models: First, the linear programming relaxations of the new formulations are stronger than existing location-based approaches. Second, extensive numerical experiments on benchmark instances show that computational times are on average reduced by 53.9% compared to state-of-the-art event-based approaches.