Fair Correlation Clustering in Forests

The study of algorithmic fairness received growing attention recently. This stems from the awareness that bias in the input data for machine learning systems may result in discriminatory outputs. For clustering tasks, one of the most central notions of fairness is the formalization by Chierichetti, Kumar, Lattanzi, and Vassilvitskii [NeurIPS 2017]. A clustering is said to be fair, if each cluster has the same distribution of manifestations of a sensitive attribute as the whole input set. This is motivated by various applications where the objects to be clustered have sensitive attributes that should not be over- or underrepresented. Most research on this version of fair clustering has focused on centriod-based objectives. In contrast, we discuss the applicability of this fairness notion to Correlation Clustering. The existing literature on the resulting Fair Correlation Clustering problem either presents approximation algorithms with poor approximation guarantees or severely limits the possible distributions of the sensitive attribute (often only two manifestations with a 1:1 ratio are considered). Our goal is to understand if there is hope for better results in between these two extremes. To this end, we consider restricted graph classes which allow us to characterize the distributions of sensitive attributes for which this form of fairness is tractable from a complexity point of view. While existing work on Fair Correlation Clustering gives approximation algorithms, we focus on exact solutions and investigate whether there are efficiently solvable instances. The unfair version of Correlation Clustering is trivial on forests, but adding fairness creates a surprisingly rich picture of complexities. We give an overview of the distributions and types of forests where Fair Correlation Clustering turns from tractable to intractable. As the most surprising insight, we consider the fact that the cause of the hardness of Fair Correlation Clustering is not the strictness of the fairness condition. We lift most of our results to also hold for the relaxed version of the fairness condition. Instead, the source of hardness seems to be the distribution of the sensitive attribute. On the positive side, we identify some reasonable distributions that are indeed tractable. While this tractability is only shown for forests, it may open an avenue to design reasonable approximations for larger graph classes

The First Proven Performance Guarantees for the Non-Dominated Sorting Genetic Algorithm II (NSGA-II) on a Combinatorial Optimization Problem

The Non-dominated Sorting Genetic Algorithm-II (NSGA-II) is one of the most prominent algorithms to solve multi-objective optimization problems. Recently, the first mathematical runtime guarantees have been obtained for this algorithm, however only for synthetic benchmark problems. In this work, we give the first proven performance guarantees for a classic optimization problem, the NP-complete bi-objective minimum spanning tree problem. More specifically, we show that the NSGA-II with population size $N \ge 4((n-1) w_{\max} + 1)$ computes all extremal points of the Pareto front in an expected number of $O(m^2 n w_{\max} \log(n w_{\max}))$ iterations, where $n$ is the number of vertices, $m$ the number of edges, and $w_{\max}$ is the maximum edge weight in the problem instance. This result confirms, via mathematical means, the good performance of the NSGA-II observed empirically. It also shows that mathematical analyses of this algorithm are not only possible for synthetic benchmark problems, but also for more complex combinatorial optimization problems. As a side result, we also obtain a new analysis of the performance of the global SEMO algorithm on the bi-objective minimum spanning tree problem, which improves the previous best result by a factor of $|F|$, the number of extremal points of the Pareto front, a set that can be as large as $n w_{\max}$. The main reason for this improvement is our observation that both multi-objective evolutionary algorithms find the different extremal points in parallel rather than sequentially, as assumed in the previous proofs.Comment: Author-generated version of a paper appearing in the proceedings of IJCAI 202

Fixed-Parameter Sensitivity Oracles

We combine ideas from distance sensitivity oracles (DSOs) and fixed-parameter tractability (FPT) to design sensitivity oracles for FPT graph problems. An oracle with sensitivity $f$ for an FPT problem $\Pi$ on a graph $G$ with parameter $k$ preprocesses $G$ in time $O(g(f,k) \cdot \textsf{poly}(n))$. When queried with a set $F$ of at most $f$ edges of $G$, the oracle reports the answer to the $\Pi$-with the same parameter $k$-on the graph $G-F$, i.e., $G$ deprived of $F$. The oracle should answer queries in a time that is significantly faster than merely running the best-known FPT algorithm on $G-F$ from scratch. We mainly design sensitivity oracles for the $k$-Path and the $k$-Vertex Cover problem. Following our line of research connecting fault-tolerant FPT and shortest paths problems, we also introduce parameterization to the computation of distance preservers. We study the problem, given a directed unweighted graph with a fixed source $s$ and parameters $f$ and $k$, to construct a polynomial-sized oracle that efficiently reports, for any target vertex $v$ and set $F$ of at most $f$ edges, whether the distance from $s$ to $v$ increases at most by an additive term of $k$ in $G-F$.Comment: 19 pages, 1 figure, abstract shortened to meet ArXiv requirements; accepted at ITCS'2

A Strategic Routing Framework and Algorithms for Computing Alternative Paths

Traditional navigation services find the fastest route for a single driver. Though always using the fastest route seems desirable for every individual, selfish behavior can have undesirable effects such as higher energy consumption and avoidable congestion, even leading to higher overall and individual travel times. In contrast, strategic routing aims at optimizing the traffic for all agents regarding a global optimization goal. We introduce a framework to formalize real-world strategic routing scenarios as algorithmic problems and study one of them, which we call Single Alternative Path (SAP), in detail. There, we are given an original route between a single origin-destination pair. The goal is to suggest an alternative route to all agents that optimizes the overall travel time under the assumption that the agents distribute among both routes according to a psychological model, for which we introduce the concept of Pareto-conformity. We show that the SAP problem is NP-complete, even for such models. Nonetheless, assuming Pareto-conformity, we give multiple algorithms for different variants of SAP, using multi-criteria shortest path algorithms as subroutines. Moreover, we prove that several natural models are in fact Pareto-conform. The implementation and evaluation of our algorithms serve as a proof of concept, showing that SAP can be solved in reasonable time even though the algorithms have exponential running time in the worst case

A Mathematical Runtime Analysis of the Non-dominated Sorting Genetic Algorithm III (NSGA-III)

The Non-dominated Sorting Genetic Algorithm II (NSGA-II) is the most prominent multi-objective evolutionary algorithm for real-world applications. While it performs evidently well on bi-objective optimization problems, empirical studies suggest that it is less effective when applied to problems with more than two objectives. A recent mathematical runtime analysis confirmed this observation by proving the NGSA-II for an exponential number of iterations misses a constant factor of the Pareto front of the simple 3-objective ONEMINMAX problem. In this work, we provide the first mathematical runtime analysis of the NSGA-III, a refinement of the NSGA-II aimed at better handling more than two objectives. We prove that the NSGA-III with sufficiently many reference points-a small constant factor more than the size of the Pareto front, as suggested for this algorithm-computes the complete Pareto front of the 3-objective ONEMINMAX benchmark in an expected number of O(n log n) iterations. This result holds for all population sizes (that are at least the size of the Pareto front). It shows a drastic advantage of the NSGA-III over the NSGA-II on this benchmark. The mathematical arguments used here and in the previous work on the NSGA-II suggest that similar findings are likely for other benchmarks with three or more objectives

Sensitivity to oxidative stress is not a definite predictor of thermal sensitivity in symbiotic dinoflagellates

Coral bleaching, the loss of symbiotic dinoflagellate algae (genus Symbiodinium) and/or photosynthetic algal pigments from corals, is thought to be primarily triggered by the thermal dysfunction of photosynthesis and a consequent build-up of reactive oxygen species. However, different corals exhibit differential bleaching susceptibilities, perhaps resulting from dissimilar abilities to deal with oxidative stress. We therefore tested whether thermal sensitivity in Symbiodinium is correlated with the capacity to deal with oxidative stress, by comparing the effects of increased temperature and hydrogen peroxide (HO) concentration on the photosynthetic performance of four cultured Symbiodinium ITS2 types (A1, B2, E, and F1) and two freshly isolated Symbiodinium ITS2 types (a temperate A-type and B1). Generally, the maximum quantum yield of photosystem II (F/F) declined in Symbiodinium with both increasing thermal and oxidative stress. However, not all types followed this pattern. Cultured Symbiodinium A1 showed a strong negative response to elevated temperature but little response to the addition of HO, while cultured F1 showed an increase in F/F at elevated temperature but a decline in this parameter to almost zero at high HO concentrations. Furthermore, both freshly isolated Symbiodinium types appeared to be relatively stress tolerant, with the temperate A-type showing an especially high resistance to both stressors. In conclusion, a range of Symbiodinium types were shown to differ in their susceptibilities to both thermal and oxidative stress, though in contrast to our original hypothesis, sensitivity to oxidative stress did not necessarily predict thermal sensitivity (or vice versa)

The First Proven Performance Guarantees for the Non-Dominated Sorting Genetic Algorithm II (NSGA-II) on a Combinatorial Optimization Problem

International audienceThe Non-dominated Sorting Genetic Algorithm-II (NSGA-II) is one of the most prominent algorithms to solve multi-objective optimization problems. Recently, the first mathematical runtime guarantees have been obtained for this algorithm, however only for synthetic benchmark problems. In this work, we give the first proven performance guarantees for a classic optimization problem, the NP-complete bi-objective minimum spanning tree problem. More specifically, we show that the NSGA-II with population size N ≥ 4((n − 1)w max + 1) computes all extremal points of the Pareto front in an expected number of O(m 2 nw max log(nw max)) iterations, where n is the number of vertices, m the number of edges, and w max is the maximum edge weight in the problem instance. This result confirms, via mathematical means, the good performance of the NSGA-II observed empirically. It also shows that mathematical analyses of this algorithm are not only possible for synthetic benchmark problems, but also for more complex combinatorial optimization problems. As a side result, we also obtain a new analysis of the performance of the global SEMO algorithm on the bi-objective minimum spanning tree problem, which improves the previous best result by a factor of |F |, the number of extremal points of the Pareto front, a set that can be as large as nw max. The main reason for this improvement is our observation that both multiobjective evolutionary algorithms find the different extremal points in parallel rather than sequentially, as assumed in the previous proofs