Multistage Committee Election

Abstract

Electing a single committee of a small size is a classical and well-understood voting situation. Being interested in a sequence of committees, we introduce and study two time-dependent multistage models based on simple Plurality voting. Therein, we are given a sequence of voting profiles (stages) over the same set of agents and candidates, and our task is to find a small committee for each stage of high score. In the conservative model we additionally require that any two consecutive committees have a small symmetric difference. Analogously, in the revolutionary model we require large symmetric differences. We prove both models to be NP-hard even for a constant number of agents, and, based on this, initiate a parameterized complexity analysis for the most natural parameters and combinations thereof. Among other results, we prove both models to be in XP yet W[1]-hard regarding the number of stages, and that being revolutionary seems to be "easier" than being conservative: If the (upper-resp. lower-) bound on the size of symmetric differences is constant, the conservative model remains NP-hard while the revolutionary model becomes polynomial-time solvable.

A General Approach to Approximate Multistage Subgraph Problems

“…P in each stage. Most common choices for transition qualities are symmetric difference cost (q (S i , S i+1 ) := |S i S i+1 | with Ψ = min; see, e.g., [1,4,5,[9][10][11]15]) and intersection profit (q ∩ (S i , S i+1 ) := |S i ∩ S i+1 | with Ψ = max; see, e.g., [2,3,6,11]).…”

“…In many cases, generalizing a polynomial-time solvable problem to a multistage setting renders it NP-hard (e.g., Multistage Shortest s-t-Path [11] or Multistage Perfect Matching [15]). There is some work on identifying parameters that allow for fixed-parameter tractability of NP-hard multistage problems [5,[9][10][11]16]. Another popular approach to tackle such problems are approximation algorithms [1][2][3][4]8].…”

Parameterized Algorithms for Diverse Multistage Problems

“…We present a general framework which allows us to prove fixed-parameter tractability of Diverse Multistage Π parameterized by the diversity ℓ for several problems Π. This includes finding diverse matchings, but also diverse commitees (answering an open question by Bredereck et al [11]), diverse s-t paths, and diverse independent sets in matroids such as spanning forests. Finally, we show that similar results cannot be expected for finding diverse vertex covers.…”

“…Here, given a sequence of instances of some decision problem, the task is to find a sequence of solutions such that the diversity, i.e., the size of the symmetric difference of any two consecutive solutions is at least ℓ. This problem has already received some attention in the literature: Fluschnik et al [22] studied the problem of finding diverse s-t paths and Bredereck et al [11] considered series of committee elections. In a similar setting, but aiming for large symmetric difference between every two (i.e., not just consecutive) solutions, Baste et al [7] provide a framework for parameterization by treewidth, while Fomin et al [24,25] focus on the case that all problems are defined on the same graph and study matching, independent set, and matroids.…”

Bipartite Temporal Graphs and the Parameterized Complexity of Multistage 2-Coloring

“…Related work. The multistage framework is still young, but several problems have been investigated in it, mostly in the last couple of years, including Matching [3,9,21], Knapsack [4], s-t Path [20], Vertex Cover [19], Committee Election [7], and others [2]. The framework has also been extended to goals other than minimizing the number of changes in the solution between layers [22,25].…”