Proportional Fairness in Clustering: A Social Choice Perspective

We study the proportional clustering problem of Chen et al. [ICML'19] and relate it to the area of multiwinner voting in computational social choice. We show that any clustering satisfying a weak proportionality notion of Brill and Peters [EC'23] simultaneously obtains the best known approximations to the proportional fairness notion of Chen et al. [ICML'19], but also to individual fairness [Jung et al., FORC'20] and the "core" [Li et al. ICML'21]. In fact, we show that any approximation to proportional fairness is also an approximation to individual fairness and vice versa. Finally, we also study stronger notions of proportional representation, in which deviations do not only happen to single, but multiple candidate centers, and show that stronger proportionality notions of Brill and Peters [EC'23] imply approximations to these stronger guarantees

Parameterized Complexity of Geodetic Set

A vertex set S of a graph G is geodetic if every vertex of G lies on a shortest path between two vertices in S. Given a graph G and k ? ?, the NP-hard Geodetic Set problem asks whether there is a geodetic set of size at most k. Complementing various works on Geodetic Set restricted to special graph classes, we initiate a parameterized complexity study of Geodetic Set and show, on the negative side, that Geodetic Set is W[1]-hard when parameterized by feedback vertex number, path-width, and solution size, combined. On the positive side, we develop fixed-parameter algorithms with respect to the feedback edge number, the tree-depth, and the modular-width of the input graph

The PACE 2021 Parameterized Algorithms and Computational Experiments Challenge: Cluster Editing

The Parameterized Algorithms and Computational Experiments challenge (PACE) 2021 was devoted to engineer algorithms solving the NP-hard Cluster Editing problem, also known as Correlation Clustering: Given an undirected graph the task is to compute a minimum number of edges to insert or remove in a way that the resulting graph is a cluster graph, that is, a graph in which each connected component is a clique. Altogether 67 participants from 21 teams, 11 countries, and 3 continents submitted their implementations to the competition. In this report, we describe the setup of the challenge, the selection of benchmark instances, and the ranking of the participating teams. We also briefly discuss the approaches used in the submitted solvers

Parameterized Algorithms for Diverse Multistage Problems

The world is rarely static - many problems need not only be solved once but repeatedly, under changing conditions. This setting is addressed by the multistage view on computational problems. We study the diverse multistage variant, where consecutive solutions of large variety are preferable to similar ones, e.g. for reasons of fairness or wear minimization. While some aspects of this model have been tackled before, we introduce a framework allowing us to prove that a number of diverse multistage problems are fixed-parameter tractable by diversity, namely Perfect Matching, s-t Path, Matroid Independent Set, and Plurality Voting. This is achieved by first solving special, colored variants of these problems, which might also be of independent interest

Vertex Cover and Feedback Vertex Set Above and Below Structural Guarantees

Vertex Cover parameterized by the solution size k is the quintessential fixed-parameter tractable problem. FPT algorithms are most interesting when the parameter is small. Several lower bounds on k are well-known, such as the maximum size of a matching. This has led to a line of research on parameterizations of Vertex Cover by the difference of the solution size k and a lower bound. The most prominent cases for such lower bounds for which the problem is FPT are the matching number or the optimal fractional LP solution. We investigate parameterizations by the difference between k and other graph parameters including the feedback vertex number, the degeneracy, cluster deletion number, and treewidth with the goal of finding the border of fixed-parameter tractability for said difference parameterizations. We also consider similar parameterizations of the Feedback Vertex Set problem

An Adaptive Version of Brandes\u27 Algorithm for Betweenness Centrality

Betweenness centrality - measuring how many shortest paths pass through a vertex - is one of the most important network analysis concepts for assessing the relative importance of a vertex. The well-known algorithm of Brandes [2001] computes, on an n-vertex and m-edge graph, the betweenness centrality of all vertices in O(nm) worst-case time. In follow-up work, significant empirical speedups were achieved by preprocessing degree-one vertices and by graph partitioning based on cut vertices. We further contribute an algorithmic treatment of degree-two vertices, which turns out to be much richer in mathematical structure than the case of degree-one vertices. Based on these three algorithmic ingredients, we provide a strengthened worst-case running time analysis for betweenness centrality algorithms. More specifically, we prove an adaptive running time bound O(kn), where k < m is the size of a minimum feedback edge set of the input graph

Finding Balance-Fair Short Paths in Graphs

The computation of short paths in graphs with edge lengths is a pillar of graph algorithmics and network science. In a more diverse world, however, not every short path is equally valuable. We contribute to a broader view on path finding by injecting a natural fairness aspect. Our fairness notion relates to vertex-colored graphs. Herein, we seek to find short paths where all colors should appear with roughly the same frequency. Among other results, we prove the introduced problems to be computationally hard (NP-hard and parameterized hard with respect to the number of colors), while also presenting an encouraging algorithmic result ("fixed-parameter tractability") related to the length of the sought solution path