Graph partitioning is a fundamental family of problems in mathematics and computer science. Many applications, including power grid islanding, geographical districting, and network partitioning, require balanced part sizes and also that each part induces a connected subgraph. In this work, we study the structure of balanced connected graph partitions produced by connected recursive bisection (CRB). CRB repeatedly splits each part in half while maintaining part connectivity and terminates when it reaches a specified number of parts. Given a final partition produced by CRB, recovering the sequence of bisections is equivalent to finding a sequence of perfect matchings in the quotient graph, in which each vertex represents a part and edges correspond to part adjacencies. We define this sequence of matchings as a perfect hierarchical matching (PHM). The union of the matchings induces a spanning tree in the quotient graph that we denote as a PHM tree. We first characterize PHM trees by an efficient edge labeling scheme and count the number of PHM trees of the complete graph up to isomorphism. We prove tight max-degree and diameter bounds for PHM trees, as well as a matroid-like edge exchange property. We give an efficient algorithm for recognizing series-parallel graphs with PHMs and generalize it to a slicewise polynomial (XP) algorithm on graphs with constant treewidth. We conclude with open problems.

Gerrymandering—the deliberate manipulation of electoral district boundaries for political advantage—is a persistent challenge in U.S. elections. In this work, we introduce and analyze Votemandering, a strategic blend of gerrymandering and targeted political campaigning devised to gain more seats by circumventing fairness measures. Votemandering leverages accurate demographic and socio-political data, bolstered by advancements in technology and data analytics, to influence voter decisions in pursuit of subtle gerrymandering strategies. We formulate votemandering as a Mixed Integer Program (MIP) that performs fairness-constrained gerrymandering over multiple election rounds. We analyze the influence of various redistricting constraints and parameters on votemandering efficacy. We explore the interconnectedness of gerrymandering, substantial campaign budgets, and strategic campaigning, illustrating their collective potential to generate biased electoral maps. A case study of Wisconsin State Senate redistricting reveals significant votemandering potential. Our findings underscore the need for reforms in the redistricting process beyond enforcing thresholds for specific fairness metrics.

Following the decennial census, each state in the U.S. redraws its congressional and state legislative district boundaries, which must satisfy various legal criteria. For example, Arizona’s Constitution describes six legal criteria, including contiguity, population balance, competitiveness, compactness, and the preservation of communities of interest, political subdivisions and majority–minority districts, each of which is to be enforced “to the extent practicable.” Optimization algorithms are well suited to draw district maps, although existing models and methods have limitations that inhibit their ability to draw legally-valid maps. Adapting existing optimization methods presents two major challenges: the complexity of modeling to achieve multiple and subjective criteria, and the computational intractability when dealing with large redistricting input graphs. In this paper, we present a multi-stage optimization framework tailored to redistricting in Arizona. This framework combines key features from existing methods, such as a multilevel algorithm that reduces graph input sizes and a larger local search neighborhood that encourages faster exploration of the solution space. This framework heuristically optimizes geographical compactness and political competitiveness while ensuring that other criteria in Arizona’s Constitution are satisfied relative to existing norms. Compared to Arizona’s enacted map (CD118) to be used until 2032, the most compact map produced by the algorithm is 41% more compact, and the most competitive map has five more competitive districts. To enable accessibility and to promote future research, we have created Optimap, a publicly accessible tool to interact with a part of this framework. Beyond the creation of these maps, this case study demonstrates the positive impact of adapting optimization-based methodologies for political redistricting in practice.

Every 10 years, U.S. states redraw their congressional and state legislative district plans. This process decides the political landscape for the subsequent 10 years. Prior to the 2021 redistricting cycle, Missouri enacted new criteria for state legislative districts. The Missouri League of Women Voters (LWV-MO) contacted the authors to analyze the potential impact of these new criteria on the map-drawing process. We apply recombination (a spanning tree method) within a local search optimization framework to analyze the interplay between political geography, constitutional requirements, and political fairness in Missouri. We use this framework to produce district plans that satisfy the new criteria and prioritize different aspects of fairness. The results, quantified by several measures of fairness, reveal an inherent Republican advantage in Missouri because of the state’s political geography and constitutional requirements. We conclude that Missouri’s political geography and constitutional requirements prevent the optimization framework from substantially improving political fairness in state legislative plans. In contrast, the framework can substantially improve political fairness in Missouri congressional plans, which are not subject to the new requirements. The LWV-MO used this work to advocate for fairness and transparency in their testimonies for the Missouri redistricting commission’s public hearings. History: This paper was refereed. Funding: This material is based upon work supported by the National Science Foundation Graduate Research Fellowship Program [Grant DGE-1746047]. S. H. Jacobson was supported by the Air Force Office of Scientific Research [Grant FA9550-19-1-0106]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/inte.2022.0037 .

States in the United States redraw their electoral district boundaries every 10 years. This redistricting process can be contentious and has long-lasting consequences for political representation. To reduce bias in the redistricting process, some states require bipartisan commissions; however, bipartisan commissions can still involve partisan tension if the political parties cannot compromise. We propose an optimization framework to facilitate compromise between two redistricting stakeholders. This framework seeks a midpoint between two stakeholder plans with respect to a distance metric. A midpoint can help the stakeholders visualize a potential compromise that incorporates district structure common to both of their proposed plans. First, we consider multiple distance metrics and evaluate whether midpoints with respect to these metrics are achievable and align with redistricting requirements. Then we formulate a mixed-integer linear program to find a midpoint (or any fractional point) between two given plans with respect to the transfer distance. This formulation incorporates district structure from both given plans by fixing variables; consequently, it is possible to solve some realistically sized instances exactly in reasonable amounts of time. We present experiments on grid instances and Missouri’s congressional redistricting instance to demonstrate how this method can quickly generate compromise options that align with redistricting requirements. Funding: This material is based on work supported by the National Science Foundation Graduate Research Fellowship Program [Grant DGE-1746047]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoo.2023.0029 .

The Spherical Zoning Problem (SZP) seeks to partition a finite 3D volume comprised of convex polytope cells, or units, into k zones such that each zone’s surface is spherical (i.e., homeomorphic to a sphere). A special case of the classical Connected Graph Partition Problem, SZP is motivated by applications in 3D geospatial partitioning such as airspace sectorization and oceanographic clustering. For general graph partitioning problems, local search optimization often considers the flip operation, which transfers one cell from its current zone, or part, to a neighboring zone in each iteration. Applying local search optimization to SZP requires new computational methods to assess flip feasibility; methods tailored to planar graph partitioning (e.g., for geographic districting) break down when faced with the spherical zones constraint due to inherent complexities when extending from 2D to 3D environments. This paper introduces the 3D geo-graph, a model for enforcing zone sphericity during a local search flip for solving SZP. The time complexity of SZP flip verification with the 3D geo-graph is, in the worst case, linear in the number of cells sharing some boundary with the flipped cell and linearithmic in the number of faces, edges, and vertices on the cell’s surface. Computational experiments on space-filling honeycombs demonstrate that the 3D geo-graph enables efficient flip verification for large-scale SZP instances.

The authors conceived of the bisection protocol during a research meeting discussing recent political redistricting literature, in particular, the I-cut-you-freeze protocol preprint. After establishing the theoretical results for the continuous nongeometric setting, they discussed ways to implement both protocols on real-world data, culminating in the Iowa case study and computational experiments with 17 other states.

Gerrymandering, the deliberate manipulation of electoral district boundaries for political advantage, is a persistent issue in U.S. redistricting cycles. This paper introduces and analyzes a new phenomenon, 'votemandering'- a strategic blend of gerrymandering and targeted political campaigning, devised to gain more seats by circumventing fairness measures. It leverages accurate demographic and socio-political data to influence voter decisions, bolstered by advancements in technology and data analytics, and executes better-informed redistricting. Votemandering is established as a Mixed Integer Program (MIP) that performs fairness-constrained gerrymandering over multiple election rounds, via unit-specific variables for campaigns. To combat votemandering, we present a computationally efficient heuristic for creating and testing district maps that more robustly preserve voter preferences. We analyze the influence of various redistricting constraints and parameters on votemandering efficacy. We explore the interconnectedness of gerrymandering, substantial campaign budgets, and strategic campaigning, illustrating their collective potential to generate biased electoral maps. A Wisconsin State Senate redistricting case study substantiates our findings on real data, demonstrating how major parties can secure additional seats through votemandering. Our findings underscore the practical implications of these manipulations, stressing the need for informed policy and regulation to safeguard democratic processes.

The COVID-19 pandemic hastened hundreds of thousands of deaths in the United States. Many of these excess deaths are directly attributed to COVID-19, but others stem from the pandemic’s social, economic, and health care system disruptions. This study compares provisional mortality data for age and sex subgroups across different time windows, with and without COVID-19 deaths, and assesses whether mortality risks are returning to pre-pandemic levels. Using provisional mortality reports from the CDC, we compute mortality risks for 22 age and sex subgroups in 2021 and compare against 2015–2019 using odds ratios. We repeat this comparison for the first twelve full months of the COVID-19 pandemic in the United States (April 2020–March 2021) against the next twelve full months (April 2021–March 2022). Mortality risks for most subgroups were significantly higher in 2021 than in 2015–2019, both with and without deaths involving COVID-19. For ages 25–54, Year 2 (April 2021–March 2022) was more fatal than Year 1 (April 2020–March 2021), whereas total mortality risks for the 65 + age groups declined. Given so many displaced deaths in the first two years of the COVID-19 pandemic, mortality risks in the next few years may fall below pre-pandemic levels. Provisional mortality data suggest this is already happening for the 75 + age groups when excluding COVID-19 deaths.

Transmission of SARS-CoV-2 between passengers on airplanes is a significant concern and reducing the transmission of SARS-CoV-2 or other viruses aboard aircraft could save lives. Solving the Airplane Seating Assignment Problem (ASAP) produces seating arrangements that minimize transmission risks between passengers aboard an aircraft, but the chosen risk model affects the optimal seating arrangement. We analyze previous risk models and introduce two new risk models, masked and unmasked, based on previous experiments performed aboard real aircraft to test aerosol dispersion of SARS-CoV-2 sized particles. We make recommendations on when each risk model is applicable and the types of seating arrangements that are optimal for each risk model.

SARS-CoV-2, the virus that causes COVID-19, began infecting humans in late 2019 and has since spread to over 57 million people and caused over 1.75 million deaths, as of December 27, 2020. In response to reduced demand and travel restrictions as a result of COVID-19, airlines experienced a 94% reduction in passenger capacity worldwide in April and an estimated 60% reduction in passengers transported for all of 2020. SARS-CoV-2 has been shown to spread on airplanes by infected passengers, so minimizing the risk of secondary infections aboard aircraft may save lives. We present the airplane seating assignment problem (ASAP) to minimize transmission risks on airplanes, and we provide two models to solve ASAP. We show that both models can be effectively solved using a standard commercial solver and that seating assignments provided by these models have lower aggregate risk than the strategy of blocking the middle seats, given the same number of passengers. The available risk models for aircraft are based on influenza data, and hence risk models based on SARS-CoV-2 should be developed to maximize the benefits of our research.

AbstractEach year, the NCAA Division I Men’s Basketball Tournament attracts popular attention, including bracket challenges where fans seek to pick the winners of the tournament’s games. However, the quantity and unpredictable nature of games suggest a single bracket will likely select some winning teams incorrectly even if created with insightful and sophisticated methods. Hence, rather than focusing on creating a single bracket to perform well, a challenge participant may wish to create a pool of brackets that likely contains at least one high-scoring bracket. This paper proposes a power model to estimate tournament outcome probabilities based on past tournament data. Bracket pools are generated for the 2013–2019 tournaments using six generators, five using the power model and one using the Bradley-Terry model. The generated brackets are assessed by the ESPN scoring system and compared to those produced by a traditional pick favorite approach as well as the highest scoring brackets in the ESPN Tournament Challenge for each year.