integer programming (IP), as the name suggests is an integer-variable-based approach commonly used to formulate real-world optimization problems with constraints. Currently, quantum algorithms reformulate the IP into ...
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integer programming (IP), as the name suggests is an integer-variable-based approach commonly used to formulate real-world optimization problems with constraints. Currently, quantum algorithms reformulate the IP into an unconstrained form through the use of binary variables, which is an indirect and resource-consuming way of solving it. We develop an algorithm that maps and solves an IP problem in its original form to any quantum system possessing a large number of accessible internal degrees of freedom that are controlled with sufficient accuracy. This work leverages the principle of superposition to solve the optimization problem. Using a single Rydberg atom as an example, we associate the integer values to electronic states belonging to different manifolds and implement a selective superposition of different states to solve the full IP problem. The optimal solution is found within a few microseconds for prototypical IP problems with up to eight variables and four constraints. This also includes non-linear IP problems, which are usually harder to solve with classical algorithms when compared to their linear counterparts. Our algorithm for solving IP is benchmarked by a well-known classical algorithm (branch and bound) in terms of the number of steps needed for convergence to the solution. This approach carries the potential to improve the solutions obtained for larger-size problems using hybrid quantum-classical algorithms.
Chudnovsky and Seymour proposed the Three-in-a-tree algorithm which solves the following problem in polynomial time: given three fixed vertices in a simple finite graph, check whether an induced tree containing these ...
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Chudnovsky and Seymour proposed the Three-in-a-tree algorithm which solves the following problem in polynomial time: given three fixed vertices in a simple finite graph, check whether an induced tree containing these vertices exists. In this paper, we deal with a generalization of this problem, referred to henceforth as k-in-a-tree. The k-in-a-tree checks whether a graph contains an induced tree spanning k given vertices. When k is part of the input, the problem is known to be NP-complete. If k = 4 is a fixed given number, its complexity is an open question, although there are efficient algorithms for restricted cases such as claw-free graphs, graphs with a girth of at least k and chordal graphs. We present mixed-integer programming formulations for this problem, and we show that instances with up to 25,000 vertices can be solved in reasonable computational time.
We study a discrete version of the classical classification problem in Euclidean space, to be called the geodesic classification problem. It is defined on a graph, where some vertices are initially assigned a class an...
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We study a discrete version of the classical classification problem in Euclidean space, to be called the geodesic classification problem. It is defined on a graph, where some vertices are initially assigned a class and the remaining ones must be classified. This vertex partition into classes is grounded on the concept of geodesic convexity on graphs, as a replacement for Euclidean convexity in the multidimensional space. We propose two new integer programming models along with branch-and-bound algorithms to solve them. We also carry out a polyhedral study of the associated polyhedra, which produced families of facetdefining inequalities and separation algorithms. Finally, we run computational experiments to evaluate the computational efficiency and the classification accuracy of the proposed approaches by comparing them with classic solution methods for the Euclidean convexity classification problem. (c) 2023 Elsevier B.V. All rights reserved.
Round robin tournaments are omnipresent in sport competitions and beyond. We investigate three in-teger programming formulations for scheduling a round robin tournament, one of which we call the matching formulation. ...
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Round robin tournaments are omnipresent in sport competitions and beyond. We investigate three in-teger programming formulations for scheduling a round robin tournament, one of which we call the matching formulation. We analytically compare their linear relaxations, and find that the relaxation of the matching formulation is stronger than the other relaxations, while still being solvable in polynomial time. In addition, we provide an exponentially sized class of valid inequalities for the matching formu-lation. Complementing our theoretical assessment of the strength of the different formulations, we also experimentally show that the matching formulation is superior on a broad set of instances. Finally, we describe a branch-and-price algorithm for finding round robin tournaments that is based on the matching formulation. (c) 2023 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license ( http://***/licenses/by/4.0/ )
Given an integer dimension K and a simple, undirected graph G with positive edge weights, the Distance Geometry Problem (DGP) aims to find a realization function mapping each vertex to a coordinate in R-K such that th...
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Given an integer dimension K and a simple, undirected graph G with positive edge weights, the Distance Geometry Problem (DGP) aims to find a realization function mapping each vertex to a coordinate in R-K such that the distance between pairs of vertex coordinates is equal to the corresponding edge weights in G. The so-called discretization assumptions reduce the search space of the realization to a finite discrete one, which can be explored via the branch-and-prune (BP) algorithm. Given a discretization vertex order in G, the BP algorithm constructs a binary tree where the nodes at a layer provide all possible coordinates of the vertex corresponding to that layer. The focus of this paper is on finding optimal BP trees for a class of discretizable DGPs. More specifically, we aim to find a discretization vertex order in G that yields a BP tree with the least number of branches. We propose an integer programming formulation and three constraint programming formulations that all significantly outperform the state-of-the-art cutting-plane algorithm for this problem. Moreover, motivated by the difficulty in solving instances with a large and low-density input graph, we develop two hybrid decomposition algorithms, strengthened by a set of valid inequalities, which further improve the solvability of the problem. Summary of Contribution: We present a new model to solve a combinatorial optimization problem on graphs, MIN DOUBLE, which comes from the highly active area of distance geometry and has applications in a wide variety of fields. We use integer programming (IP) and present the first constraint programming(CP) models and hybrid decomposition methods, implemented as a branch-and-cut procedure, for MIN DOUBLE. Through an extensive computational study, we show that our approaches advance the state of the art for MIN DOUBLE. We accomplish this by not only combining generic techniques from IP and CP but also exploring the structure of the problem in developing valid inequal
This paper studies a general class of interdiction problems in which the solution space of both the leader and follower are characterized by two discrete sets denoted the leader's strategy set and the follower'...
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This paper studies a general class of interdiction problems in which the solution space of both the leader and follower are characterized by two discrete sets denoted the leader's strategy set and the follower's structure set. In this setting, the interaction between any strategy-structure pair is assumed to be binary, in the sense that the strategy selected by the leader either interacts or not with the follower's choice of structure and, if it does, then the structure becomes unavailable for the follower. There are many interdiction games defined by this setup, including problems where the leader wishes to attack some type of network structures, such as shortest paths, minimum spanning trees, and minimum dominating sets, among others. We study a general set-covering type of formulation that can be used for solving this class of problems and analyze several properties of the convex hull of its solutions. We develop a wide class of valid inequalities that generalizes and extends several others that have appeared in the literature in recent years. We provide conditions for them to be facet-defining and conclude with a general discussion about their separation. Several examples of problems in the context of network interdiction are presented to help with the exposition. (C) 2022 Elsevier B.V. All rights reserved.
It is natural to formulate sequencing problems as integer programming models. However, there are a number of possible formulations the practical value of which can be significantly different. In this paper, we first p...
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It is natural to formulate sequencing problems as integer programming models. However, there are a number of possible formulations the practical value of which can be significantly different. In this paper, we first propose a novel classification of integer programming formulations for single-machine sequencing. Next, we present associated mixed-integer linear programming models for total tardiness minimization. Finally, we conduct an extensive computational study on randomly generated instances. For the unweighted case, the position-indexed formulation with linearly many constraints outperforms others, whereas for the weighted case, it is best to use the sparse reformulation of the time-indexed formulation. integer programming turns out to be a viable option for many practical problem sizes.
Fire-related emergency vehicle scheduling has always been an important part of autonomous vehicle emergency management. To maintain the fire safety of buildings and reduce the transportation loss of emergency resource...
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Fire-related emergency vehicle scheduling has always been an important part of autonomous vehicle emergency management. To maintain the fire safety of buildings and reduce the transportation loss of emergency resources, this paper constructs a multi-objective integer planning model for fire emergency resource scheduling, which minimizes the time spent in the transportation of fire materials and considers the transportation needs of potential fire vehicles. The fixed-point iterative algorithm is realized and optimized, and distributed computing is adopted to improve the solving speed. In this paper, the comparison and simulation experiments with Branch and Bound, improved particle swarm optimization and other algorithms are carried out. The results show that the model and the algorithm can effectively deal with the fire emergency vehicle scheduling problem under different simulated fires and multiple fires occurring at the same time, and have certain advantages in numerical stability and convergence speed.
Nowadays, reaching a high level of employee satisfaction in efficient schedules is an important and difficult task faced by companies. We tackle a new variant of the personnel scheduling problem under unknown demand b...
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Nowadays, reaching a high level of employee satisfaction in efficient schedules is an important and difficult task faced by companies. We tackle a new variant of the personnel scheduling problem under unknown demand by considering employee satisfaction via endogenous uncertainty depending on the combination of their preferred and received schedules. We address this problem in the context of reserve staff scheduling, an unstudied operational problem from the transit industry. To handle the challenges brought by the two uncertainty sources, regular employee and reserve employee absences, we formulate this problem as a two-stage stochastic integer program with mixed-integer recourse. The first-stage decisions consist in finding the days off of the reserve employees. After the unknown regular employee absences are revealed, the second-stage decisions are to schedule the reserve staff duties. We incorporate reserve employees' days-off preferences into the model to examine how employee satisfaction may affect their own absence rates.
In organizational and academic settings, the strategic formation of teams is paramount, necessitating an approach that transcends conventional methodologies. This study introduces a novel application of multicriteria ...
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In organizational and academic settings, the strategic formation of teams is paramount, necessitating an approach that transcends conventional methodologies. This study introduces a novel application of multicriteria integer programming (MCIP), which simultaneously accommodates multiple criteria, thereby innovatively addressing the complex task of team formation. Unlike traditional single-objective optimization methods, our research designs a comprehensive framework capable of modeling a wide array of factors, including skill levels, backgrounds, and personality traits. The objective function of this framework is optimized to maximize within-team diversity while minimizing both conflict levels and variance in diversity between teams. Central to our approach is a two-stage optimization process. Initially, it segments the population into subgroups using a weighted heterogeneous multivariate K-means algorithm, allowing for a targeted and nuanced team assembly. This is followed by the application of a surrogate optimization technique within these subgroups, efficiently navigating the complexities of MCIP for large-scale applications. Our approach is further enhanced by the inclusion of explicit constraints such as potential interpersonal conflicts, a factor often overlooked in previous studies. The results from our study demonstrate the optimality and robustness of our model across simulation scenarios with different data heterogeneity levels. The contributions of this study are manifold, addressing critical gaps in the existing literature with a theory-backed, empirically validated framework for advanced team formation. Beyond theoretical implications, our work provides a practical guide for implementing conflict-aware, sophisticated team formation strategies in real-world scenarios. This advancement paves the way for future research to explore and enhance this model, providing more sophisticated and efficient team formation strategies.
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