Efficient Algorithms for Electing Successive Committees

TL;DR

The paper addresses the problem of electing a sequence of $ au$ consecutive, size-$k$ committees under a per-candidate frequency bound $f$, using approval or ordinal preferences to maximize a chosen quality $oldsymbol{ angle extalpha}$($oldsymbol{ angle extbeta}$). It develops parameterized algorithms that are fixed-parameter tractable with respect to the number of candidates $m$ (and related parameters), including exact DP and color-coding techniques, plus randomized and derandomized variants. Key results include $oxed{O^ star(2^m)}$ time for $f=1$ across all studied $eta$, reductions to handle $f eq 1$, and $oxed{O^ star(4^{fm})}$-type algorithms for general $f$ with constant $k$, as well as $oxed{O^ star(m!(k+1)^m)}$ for broader settings. These contributions extend the practical applicability of successive committee elections to realistic scenarios with limited candidate pools or horizons, and lay groundwork for empirical validation and further parameterized analyses.

Abstract

In a recently introduced model of successive committee elections (Bredereck et al., AAAI-20) for a given set of ordinal or approval preferences one aims to find a sequence of a given length of "best" same-size committees such that each candidate is a member of a limited number of consecutive committees. However, the practical usability of this model remains limited, as the described task turns out to be NP-hard for most selection criteria already for seeking committees of size three. Non-trivial or somewhat efficient algorithms for these cases are lacking too. Motivated by a desire to unlock the full potential of the described temporal model of committee elections, we devise (parameterized) algorithms that effectively solve the mentioned hard cases in realistic scenarios of a moderate number of candidates or of a limited time horizon.