Given a simple graph G = (V,E) with edge cost c ∈ℝ^|E| , a positive integer h, source s ∈ V and terminal t ∈ V , the hop-constrained cheapest path problem ( HCCPP ) seeks to find an s-t path of length at most h hops with the cheapest cost. This paper proposes a cut-based mixed integer programming (MIP) formulation, which has only one set of constraints that capture connectivity and hop constraints simultaneously, to solve the HCCPP when edge costs are nonnegative. As the only set of constraints of the formulation has exponentially many constraints, we show that their corresponding fractional separation problem is easy for h ∈{2,3} and hard for h ≥ 4 . We also propose a polytime algorithm for solving the integer separation problem. Furthermore, we show that our cut-based model is at least as strong as the jump-based model of Dahl (Operations Research Letters, 1999), which results in perfectness of our proposed model for h ∈{2,3} . We finally conduct a brief set of computational experiments to compare the performance of the cut formulation against the jump model.

Given a simple graph with vertex set and edge set , the minimum biclique cover problem seeks to cover all edges of the graph with a minimum number of bicliques (i.e., complete bipartite subgraphs). This paper proposes two compact mixed integer programming (MIP) formulations for solving the minimum biclique cover problem on general graphs: (i) A natural formulation in the edge space and (ii) an extended formulation in the edge and vertex spaces. While the natural MIP formulation of Cornaz and Fonlupt (Discrete Mathematics, 2006) has exponentially many constraints, our natural formulation enjoys only a polynomial number of their exponential "no-good" cuts, along with another set of polynomial valid inequalities. We also employ bounding and variable fixing procedures that help solve most of our social network instances, which are not solvable to optimality in a one-hour time limit without the bounding and fixing procedures. The instances that are not solved in the one-hour time limit are submitted to the 2024 Mixed Integer Programming Library (MIPLIB 2024).

This paper studies the problems of partitioning the vertices of a graph G = (V, E) into (or covering with) a minimum number of low-diameter clusters from the lenses of approximation algorithms and integer programming. Here, the low-diameter criterion is formalized by an s-club, which is a subset of vertices whose induced subgraph has diameter at most s. For these problems, we give O(n(1/2))-approximation algorithms for any even integer s, generalizing a previous algorithm for the case s = 2. (The O(& centerdot;) notation suppresses logarithmic factors and n := | V|.) Complementing this, we show that for any epsilon > 0 the problem is NP-hard to approximate within n(1/2 - epsilon) and n(1 - epsilon), for each fixed even and odd integer s, respectively, suggesting a contrast in approximability for even and odd values of s. Second, we develop new MIP-based heuristics (inspired by the approximation algorithms) that perform well in practice, solving more than half of previous benchmark instances in less than one second. To handle the remaining instances, we propose a MIP formulation with an exponentially large class of cut-like inequalities that we solve with a branch-and-cut algorithm. With it, we tackle more benchmark instances than previous approaches and in less time.

Motivated by a recent FIFPRO (the International Federation of Professional Footballers’ Associations) report on the adverse effects of international travel load and time-zone crossings on player performance and physical well-being, this paper develops an optimization framework to reduce the total distance traveled by national teams in the FIFA World Cup 2026. We propose a mixed integer programming model that minimizes the total internal distance traveled by the 48 teams in the group stage, subject to the structural constraints implied by FIFA’s official schedule. Computational results yield a feasible schedule that reduces total internal travel distance by 50 https://igorlucindo.github.io/fifa-world-cup-2026-scheduler-APP/ .

Decision trees are powerful supervised machine learning tools for classification and regression that attract many academic researchers and industry professionals. In particular, decision trees provide interpretability, which is often preferred over other higheraccuracy methods that are relatively uninterpretable. An optimal binary classification tree has two types of vertices and is obtainable by solving a bi-objective optimization problem that seeks to (i) maximize the number of correctly classified datapoints and (ii) minimize the number of branching vertices. In this paper, we propose two mixed integer linear optimization (MILO) formulations for designing optimal binary classification trees: a maximum flow-based formulation and a minimum cut-based formulation, both of which share a common base model. The formulations enforce connected feasible paths for training datapoints through flow-based or cut-based connectivity constraints while maintaining computational tractability without the need for decomposition techniques commonly used to combat scaling concerns. We show theoretical improvements on the strongest flow-based MILO formulation currently in the literature and conduct experiments on publicly available data sets to demonstrate our models' ability to scale, strength against traditional branch-and-bound approaches, and robustness in out-of-sample test performance. Our code and data are available on GitHub.

The software and data in this repository are a snapshot of the software and data that were used in the research reported on in the paper Partitioning a Graph into Low-Diameter Clusters by Jack Zhang, Lucas Silveira, Hamidreza Validi, Logan Smith, Austin Buchanan, and Illya V. Hicks.

Given graph G=(V,E) with vertex set V and edge set E, the max k-cut problem seeks to partition the vertex set V into at most k subsets that maximize the weight (number) of edges with endpoints in different parts. This paper proposes a graph folding procedure (i.e., a procedure that reduces the number of the vertices and edges of graph G) for the weighted max k-cut problem that may help reduce the problem’s dimensionality. While our theoretical results hold for any k ≥ 2 , our computational results show the effectiveness of the proposed preprocess only for k=2 and on two sets of instances. Furthermore, we observe that the preprocess improves the performance of a MIP solver on a set of large-scale instances of the max cut problem.

Quantum computing offers significant potential for solving NP-hard combinatorial (optimization) problems that are beyond the reach of classical computers. One way to tap into this potential is by reformulating combinatorial problems as a quadratic unconstrained binary optimization (QUBO) problem. The solution of the QUBO reformulation can then be addressed using adiabatic quantum computing devices or appropriate quantum computing algorithms on gate-based quantum computing devices. In general, QUBO reformulations of combinatorial problems can be readily obtained by properly penalizing the violation of the problem's constraints in the original problem's objective. However, characterizing tight (i.e., minimal but sufficient) penalty coefficients for this purpose is important and non-trivial for enabling the solution of the resulting QUBO in current and near-term quantum computing devices. Along these lines, we present closed-form characterizations of tight penalty coefficients for two distinct QUBO reformulations of the max k-cut problem whose values depend on the (weighted) degree of the vertices of the graph defining the problem. These findings contribute to the ongoing effort to make quantum computing a viable tool for solving combinatorial problems at scale. We support our theoretical results with illustrative examples and simple numerical results.

This paper proposes pandemic mitigation vaccination policies for Newfoundland and Labrador (NL) based on two compact mixed integer programming (MIP) models of the distance-based critical node detection problem (DCNDP). Our main focus is on two variants of the DCNDP that seek to minimize the number of connections with lengths of at most one (1-DCNDP) and two (2-DCNDP). A polyhedral study for the 1-DCNDP is conducted, and new aggregated inequalities are provided for the 2-DCNDP. The computational experiments show that the 2-DCNDP with aggregated inequalities outperforms the one with disaggregated inequalities for graphs with a density of at least 0.5%. We also study the strategic vaccine allocation problem as a real-world application of the DCNDP and conduct a set of computational experiments on a simulated contact network of NL. Our computational results demonstrate that the DCNDP-based strategies can have a better performance in comparison with the real-world strategies implemented during COVID-19.

Motivated by the importance of user engagement as a crucial element in cascading leaving of users from a social network, we study identifying a largest relaxed variant of a degree-based cohesive subgraph: the maximum anchored k-core problem. Given graph [Formula: see text] and integers k and b, the maximum anchored k-core problem seeks to find a largest subset of vertices [Formula: see text] that induces a subgraph with at least [Formula: see text] vertices of degree at least k. We introduce a new integer programming (IP) formulation for the maximum anchored k-core problem and conduct a polyhedral study on the polytope of the problem. We show the linear programming relaxation of the proposed IP model is at least as strong as that of a naïve formulation. We also identify facet-defining inequalities of the IP formulation. Furthermore, we develop inequalities and fixing procedures to improve the computational performance of our IP model. We use benchmark instances to compare the computational performance of the IP model with (i) the naïve IP formulation and (ii) two existing heuristic algorithms. Our proposed IP model can optimally solve half of the benchmark instances that cannot be solved to optimality either by the naïve model or the existing heuristic approaches. Funding: This work is funded by the National Science Foundation (NSF) [Grant DMS-2318790] titled AMPS: Novel Combinatorial Optimization Techniques for Smartgrids and Power Networks. Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoo.2022.0024 .

Liver transplantation has been a critical issue in the U.S. healthcare system for decades, and the region redesign aims to ameliorate this issue. This paper revisits two mixed integer programming (MIP) formulations of the liver region redesign problem proposed by Akshat et al. [1]. We study their first formulation considering two different modeling approaches: one compact formulation and one with exponentially many constraints. We also propose a set of variable fixing procedures and conduct a polyhedral study on their second formulation. Our computational results show that multiple unsolved instances are solved to optimality.

Motivated by applications in political districting, we consider the task of partitioning the n vertices of a planar graph into k connected components. We propose an extended formulation for this task that has two desirable properties: (i) it uses just O ( n ) variables, constraints, and nonzeros, and (ii) it is perfect . To explore its ability to solve real-world problems, we apply it to a political districting problem in which contiguity and population balance are imposed as hard constraints and compactness is optimized. Computational experiments show that, despite the model’s small size and integrality for connected partitioning, the population balance constraints are more troublesome to effectively impose. Nevertheless, we share our findings in hopes that others may find better ways to impose them.

The maximum independent set (MIS) seeks to find a subset of vertices with the maximum size such that no pair of its vertices are adjacent. This paper develops a recursive fixing procedure that generalizes the existing polytime algorithm to solve the maximum independent set problem on chordal graphs, which admit simplicial orderings. We prove that the generalized fixing procedure is safe; i.e., it does not remove all optimal solutions of the MIS problem from the solution space. Our computational results show that the proposed recursive fixing algorithm, along with the basic mixed integer programming (MIP) of the MIS, outperforms the pure MIP formulation of the problem. Our codes, data, and results are available on GitHub.

A tight continuous relaxation is a crucial factor in solving mixed integer formulations of many NP-hard combinatorial optimization problems. The (weighted) max k-cut problem is a fundamental combinatorial optimization problem with multiple notorious mixed integer optimization formulations. In this paper, we explore four existing mixed integer optimization formulations of the max k-cut problem. Specifically, we show that the continuous relaxation of a binary quadratic optimization formulation of the problem is: (i) stronger than the continuous relaxation of two mixed integer linear optimization formulations and (ii) at least as strong as the continuous relaxation of a mixed integer semidefinite optimization formulation. We also conduct a set of experiments on multiple sets of instances of the max k-cut problem using state-of-the-art solvers that empirically confirm the theoretical results in item (i). Furthermore, these numerical results illustrate the advances in the efficiency of global non-convex quadratic optimization solvers and more general mixed integer nonlinear optimization solvers. As a result, these solvers provide a promising option to solve combinatorial optimization problems. Our codes and data are available on GitHub.

The set partitioning problem and its decision variant (i.e., the exact cover problem) are combinatorial optimization problems that were historically crucial in the quantum optimization community. This problem is also employed in the main problem of the branch-and-price approach in many real-world optimization problems, including, but not limited to, redistricting and scheduling. Motivated by recent claims on the capability of quantum computers in “solving” hard combinatorial optimization problems, we propose a quadratic unconstrained binary optimization ( QUBO ) formulation for the set partitioning problem with penalty coefficients that are at least as tight as the existing naive ones. We also employ five reduction techniques of Garfinkel and Nemhauser (Operations Research 17(5):848–856, 1969) to reduce the size of an existing set of benchmark instances. We finally use variational quantum eigensolver ( VQE ) as a heuristic to find feasible solutions for the problem. Our computational experiments show the efficacy of employing the proposed penalty coefficients and the existing classical reduction techniques in the quantum context. Our codes and data are available on GitHub.

When constructing political districting plans, prominent criteria include population balance, contiguity, and compactness. The compactness of a districting plan, which is often judged by the “eyeball test”, has been quantified in many ways, e.g., Length-Width, Polsby-Popper, and Moment-of-Inertia. This paper considers the number of cut edges, which has recently gained traction in the redistricting literature as a measure of compactness because it is simple and reasonably agrees with the eyeball test. We study the stylized problem of minimizing the number of cut edges, subject to constraints on population balance and contiguity. With the integer programming techniques proposed in this paper, all county-level instances in the USA (and some tract-level instances) can be solved to optimality. Our techniques extend to minimize weighted cut edges (e.g., to minimize district perimeter length) or to impose compactness constraints. All data, code, and results are on GitHub.

Distributed networked systems form an essential resource for computation and applications ranging from commercial, military, scientific, and research communities. Allocation of resources on a given infrastructure is realized through various mapping systems that are tailored towards specific use cases of the requesting applications. While HPC system requests demand compute resources heavy on processor and memory, cloud applications may demand distributed web services that are composed of networked processing and some memory. All resource requests allocate on the infrastructure with some form of network connectivity. However, during mapping of resources, the features and topology constraints of network components are typically handled indirectly through abstractions of user requests. This paper is on a novel graph representation that enables precise mapping methods for distributed networked systems. The proposed graph representations are demonstrated to allocate specific network components and adjacency requirements of a requested graph on a given infrastructure. Furthermore, we report on application of business policy requirements that resulted in increased utilization and a gradual decrease in idle node count as requests are mapped using our proposed methods.

Decision trees are powerful tools for classification and regression that attract many researchers working in the burgeoning area of machine learning. One advantage of decision trees over other methods is their interpretability, which is often preferred over other higher accuracy methods that are relatively uninterpretable. A binary classification tree has two types of vertices: (i) branching vertices which have exactly two children and where datapoints are assessed on a set of discrete features; and (ii) leaf vertices at which datapoints are given a discrete prediction. An optimal binary classification tree can be obtained by solving a biobjective optimization problem that seeks to (i) maximize the number of correctly classified datapoints and (ii) minimize the number of branching vertices. In this paper, we propose four mixed integer linear optimization (MILO) formulations for designing optimal binary classification trees: two flow-based formulations and two-cut based formulations. We provide theoretical comparisons between our proposed formulations and the strongest flow-based MILO formulation of Aghaei et al. (2021). We conduct experiments on 13 publicly available datasets to show the models' ability to scale and the strength of a biobjective approach using Pareto frontiers. Our code and data are available on GitHub.

For nearly 60 years, operations research techniques have assisted in the creation of political districting plans, beginning with an integer programming model. This model, which seeks compactness as its objective, tends to generate districts that are contiguous, or nearly so, but provides no guarantee of contiguity. In the paper “Imposing contiguity constraints in political districting models” by Hamidreza Validi, Austin Buchanan, and Eugene Lykhovyd, the authors consider and analyze four different contiguity models (two old and two new). Their computer implementation can handle redistricting instances as large as Indiana (1,511 census tracts). Their fastest approach uses a branch-and-cut algorithm, where contiguity constraints are added in a callback. Critically, many variables can be fixed to zero a priori by Lagrangian arguments. All test instances and source code are publicly available.