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Proportional apportionment is the problem of assigning seats to parties according to their relative share of votes. Divisor methods are the de-facto standard solution, used in many countries. In recent literature, there are two algorithms…

Data Structures and Algorithms · Computer Science 2017-12-06 Raphael Reitzig , Sebastian Wild

Apportionment is the task of assigning resources to entities with different entitlements in a fair manner, and specifically a manner that is as proportional as possible. The best-known application is the assignment of parliamentary seats to…

Computer Science and Game Theory · Computer Science 2025-10-14 Julian Chingoma , Ulle Endriss , Ronald de Haan , Adrian Haret , Jan Maly

Apportionment refers to the well-studied problem of allocating legislative seats among parties or groups with different entitlements. We present a multi-level generalization of apportionment where the groups form a hierarchical structure,…

Theoretical Economics · Economics 2025-11-14 Ulrike Schmidt-Kraepelin , Warut Suksompong , Steven Wijaya

Traditionally, the problem of apportioning the seats of a legislative body has been viewed as a one-shot process with no dynamic considerations. While this approach is reasonable for some settings, dynamic aspects play an important role in…

Computer Science and Game Theory · Computer Science 2025-10-17 Javier Cembrano , Jose Correa , Svenja M. Griesbach , Victor Verdugo

In parliamentary elections, parties compete for a limited, typically fixed number of seats. Most parliaments are assembled using apportionment methods that distribute the seats based on the parties' vote counts. Common apportionment methods…

In the classic apportionment problem the goal is to decide how many seats of a parliament should be allocated to each party as a result of an election. The divisor methods provide a way of solving this problem by defining a notion of…

Optimization and Control · Mathematics 2021-08-10 Claire Mathieu , Victor Verdugo

Apportionment is the problem of distributing $h$ indivisible seats across states in proportion to the states' populations. In the context of the US House of Representatives, this problem has a rich history and is a prime example of…

Computer Science and Game Theory · Computer Science 2024-06-21 Paul Gölz , Dominik Peters , Ariel D. Procaccia

The apportionment problem constitutes a fundamental problem in democratic societies: How to distribute a fixed number of seats among a set of states in proportion to the states' populations? This--seemingly simple--task has led to a rich…

Computer Science and Game Theory · Computer Science 2024-11-01 Javier Cembrano , José Correa , Ulrike Schmidt-Kraepelin , Alexandros Tsigonias-Dimitriadis , Victor Verdugo

In the apportionment problem, a fixed number of seats must be distributed among parties in proportion to the number of voters supporting each party. We study a generalization of this setting, in which voters can support multiple parties by…

Computer Science and Game Theory · Computer Science 2022-03-31 Markus Brill , Paul Gölz , Dominik Peters , Ulrike Schmidt-Kraepelin , Kai Wilker

The problem of how to allocate to states the seats in the US House of Representatives is the most studied instance of what is termed the `apportionment problem'. We propose a new method of apportionment which is stochastic, which meets the…

Statistics Theory · Mathematics 2007-06-13 Geoffrey Grimmett

How to fairly apportion congressional seats to states has been debated for centuries. We present an alternative perspective on apportionment, centered not on states but "families" of state, sets of states with "divisor-method" quotas with…

General Economics · Economics 2022-10-20 Ross Hyman , Nicolaus Tideman

We consider three algorithms for allocating parliamentary seats by proportional representation. The usual approach to describing such algorithms is to compute a quota of votes that each party uses to "acquire'' representatives. This kind of…

Data Structures and Algorithms · Computer Science 2023-11-07 Raul Rojas

Many signal processing problems can be solved by maximizing the fitness of a segmented model over all possible partitions of the data interval. This letter describes a simple but powerful algorithm that searches the exponentially large…

We consider the distributed version of the Multiple Knapsack Problem (MKP), where $m$ items are to be distributed amongst $n$ processors, each with a knapsack. We propose different distributed approximation algorithms with a tradeoff…

Data Structures and Algorithms · Computer Science 2017-02-06 Ananth Murthy , Chandan Yeshwanth , Shrisha Rao

We consider distributed iterative algorithms for the averaging problem over time-varying topologies. Our focus is on the convergence time of such algorithms when complete (unquantized) information is available, and on the degradation of…

Optimization and Control · Mathematics 2009-01-14 Angelia Nedić , Alex Olshevsky , Asuman Ozdaglar , John Tsitsiklis

Given a set $P$ of $n$ points in $\mathbf{R}^d$, and a positive integer $k \leq n$, the $k$-dispersion problem is that of selecting $k$ of the given points so that the minimum inter-point distance among them is maximized (under Euclidean…

Computational Geometry · Computer Science 2025-11-04 Ke Chen , Adrian Dumitrescu

Divisor methods are well known to satisfy house monotonicity, which allows representative seats to be allocated sequentially. We focus on stationary divisor methods defined by a rounding cutpoint $c \in [0,1]$. For such methods with…

General Mathematics · Mathematics 2026-03-02 Michael A. Jones , Brittany Ohlinger , Jennifer Wilson

We consider the following problem: given an unsorted array of $n$ elements, and a sequence of intervals in the array, compute the median in each of the subarrays defined by the intervals. We describe a simple algorithm which uses O(n) space…

Data Structures and Algorithms · Computer Science 2009-01-14 Beat Gfeller , Peter Sanders

We present a sorting algorithm for the case of recurrent random comparison errors. The algorithm essentially achieves simultaneously good properties of previous algorithms for sorting $n$ distinct elements in this model. In particular, it…

Data Structures and Algorithms · Computer Science 2017-09-22 Barbara Geissmann , Stefano Leucci , Chih-Hung Liu , Paolo Penna

The process of drawing electoral district boundaries is known as political redistricting. Within this context, gerrymandering is the practice of drawing these boundaries such that they unfairly favor a particular political party, often…

Data Structures and Algorithms · Computer Science 2024-02-22 Jin-Yi Cai , Jacob Kruse , Kenneth Mayer , Daniel P. Szabo
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