Parameterized Matroid-Constrained Maximum Coverage

In this paper, we introduce the concept of Density-Balanced Subset in a matroid, in which independent sets can be sampled so as to guarantee that (i) each element has the same probability to be sampled, and (ii) those events are negatively correlated. These Density-Balanced Subsets are subsets in the ground set of a matroid in which the traditional notion of uniform random sampling can be extended. We then provide an application of this concept to the Matroid-Constrained Maximum Coverage problem. In this problem, given a matroid ? = (V, ?) of rank k on a ground set V and a coverage function f on V, the goal is to find an independent set S ? ? maximizing f(S). This problem is an important special case of the much-studied submodular function maximization problem subject to a matroid constraint; this is also a generalization of the maximum k-cover problem in a graph. In this paper, assuming that the coverage function has a bounded frequency ? (i.e., any element of the underlying universe of the coverage function appears in at most ? sets), we design a procedure, parameterized by some integer ?, to extract in polynomial time an approximate kernel of size ? ? k that is guaranteed to contain a 1 - (? - 1)/? approximation of the optimal solution. This procedure can then be used to get a Fixed-Parameter Tractable Approximation Scheme (FPT-AS) providing a 1 - ? approximation in time (?/?)^O(k) ? |V|^O(1). This generalizes and improves the results of [Manurangsi, 2019] and [Huang and Sellier, 2022], providing the first FPT-AS working on an arbitrary matroid. Moreover, as the kernel has a very simple characterization, it can be constructed in the streaming setting

Parameterized Matroid-Constrained Maximum Coverage

In this paper, we introduce the concept of Density-Balanced Subset in a matroid, in which independent sets can be sampled so as to guarantee that (i) each element has the same probability to be sampled, and (ii) those events are negatively correlated. These Density-Balanced Subsets are subsets in the ground set of a matroid in which the traditional notion of uniform random sampling can be extended. We then provide an application of this concept to the Matroid-Constrained Maximum Coverage problem. In this problem, given a matroid $\mathcal{M} = (V, \mathcal{I})$ of rank $k$ on a ground set $V$ and a coverage function $f$ on $V$, the goal is to find an independent set $S \in \mathcal{I}$ maximizing $f(S)$. This problem is an important special case of the much-studied submodular function maximization problem subject to a matroid constraint; this is also a generalization of the maximum $k$-cover problem in a graph. In this paper, assuming that the coverage function has a bounded frequency $\mu$ (i.e., any element of the underlying universe of the coverage function appears in at most $\mu$ sets), we design a procedure, parameterized by some integer $\rho$, to extract in polynomial time an approximate kernel of size $\rho \cdot k$ that is guaranteed to contain a $1 - (\mu - 1)/\rho$ approximation of the optimal solution. This procedure can then be used to get a Fixed-Parameter Tractable Approximation Scheme (FPT-AS) providing a $1 - \varepsilon$ approximation in time $(\mu/\varepsilon)^{O(k)} \cdot |V|^{O(1)}$. This generalizes and improves the results of [Manurangsi, 2019] and [Huang and Sellier, 2022], providing the first FPT-AS working on an arbitrary matroid. Moreover, because of its simplicity, the kernel construction can be performed in the streaming setting

Parameterized Complexity of Multi-winner Determination: More Effort Towards Fixed-Parameter Tractability

We study the parameterized complexity of Winners Determination for three prevalent $k$-committee selection rules, namely the minimax approval voting (MAV), the proportional approval voting (PAV), and the Chamberlin-Courant's approval voting (CCAV). It is known that Winners Determination for these rules is NP-hard. Moreover, these problems have been studied from the parameterized complexity point of view with respect to some natural parameters recently. However, many results turned out to be W[1]-hard or W[2]-hard. Aiming at deriving more fixed-parameter algorithms, we revisit these problems by considering more natural and important single parameters, combined parameters, and structural parameters.Comment: 31 pages, 2 figures, AAMAS 201

FPT-Algorithms for the l-Matchoid Problem with Linear and Submodular Objectives

We design a fixed-parameter deterministic algorithm for computing a maximum weight feasible set under a $\ell$-matchoid of rank $k$, parameterized by $\ell$ and $k$. Unlike previous work that presumes linear representativity of matroids, we consider the general oracle model. Our result, combined with the lower bounds of Lovasz, and Jensen and Korte, demonstrates a separation between the $\ell$-matchoid and the matroid $\ell$-parity problems in the setting of fixed-parameter tractability. Our algorithms are obtained by means of kernelization: we construct a small representative set which contains an optimal solution. Such a set gives us much flexibility in adapting to other settings, allowing us to optimize not only a linear function, but also several important submodular functions. It also helps to transform our algorithms into streaming algorithms. In the streaming setting, we show that we can find a feasible solution of value $z$ and the number of elements to be stored in memory depends only on $z$ and $\ell$ but totally independent of $n$. This shows that it is possible to circumvent the recent space lower bound of Feldman et al., by parameterizing the solution value. This result, combined with existing lower bounds, also provides a new separation between the space and time complexity of maximizing an arbitrary submodular function and a coverage function in the value oracle model

Multi-Winner Voting with Approval Preferences

Approval-based committee (ABC) rules are voting rules that output a fixed-size subset of candidates, a so-called committee. ABC rules select committees based on dichotomous preferences, i.e., a voter either approves or disapproves a candidate. This simple type of preferences makes ABC rules widely suitable for practical use. In this book, we summarize the current understanding of ABC rules from the viewpoint of computational social choice. The main focus is on axiomatic analysis, algorithmic results, and relevant applications.Comment: This is a draft of the upcoming book "Multi-Winner Voting with Approval Preferences

Multi-Winner Voting with Approval Preferences

From fundamental concepts and results to recent advances in computational social choice, this open access book provides a thorough and in-depth look at multi-winner voting based on approval preferences. The main focus is on axiomatic analysis, algorithmic results and several applications that are relevant in artificial intelligence, computer science and elections of any kind. What is the best way to select a set of candidates for a shortlist, for an executive committee, or for product recommendations? Multi-winner voting is the process of selecting a fixed-size set of candidates based on the preferences expressed by the voters. A wide variety of decision processes in settings ranging from politics (parliamentary elections) to the design of modern computer applications (collaborative filtering, dynamic Q&A platforms, diversity in search results, etc.) share the problem of identifying a representative subset of alternatives. The study of multi-winner voting provides the principled analysis of this task. Approval-based committee voting rules (in short: ABC rules) are multi-winner voting rules particularly suitable for practical use. Their usability is founded on the straightforward form in which the voters can express preferences: voters simply have to differentiate between approved and disapproved candidates. Proposals for ABC rules are numerous, some dating back to the late 19th century while others have been introduced only very recently. This book explains and discusses these rules, highlighting their individual strengths and weaknesses. With the help of this book, the reader will be able to choose a suitable ABC voting rule in a principled fashion, participate in, and be up to date with the ongoing research on this topic

LIPIcs, Volume 274, ESA 2023, Complete Volume

LIPIcs, Volume 274, ESA 2023, Complete Volum