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Register a new userFair Disaster Containment via Graph-Cut Problems
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Added by
Leonidas Tsepenekas
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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Graph cut problems form a fundamental problem type in combinatorial optimization, and are a central object of study in both theory and practice. In addition, the study of fairness in Algorithmic Design and Machine Learning has recently received significant attention, with many different notions proposed and analyzed in a variety of contexts. In this paper we initiate the study of fairness for graph cut problems by giving the first fair definitions for them, and subsequently we demonstrate appropriate algorithmic techniques that yield a rigorous theoretical analysis. Specifically, we incorporate two different definitions of fairness, namely demographic and probabilistic individual fairness, in a particular cut problem modeling disaster containment scenarios. Our results include a variety of approximation algorithms with provable theoretical guarantees.
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Let $G$ be an $n$-vertex graph with $m$ edges. When asked a subset $S$ of vertices, a cut query on $G$ returns the number of edges of $G$ that have exactly one endpoint in $S$. We show that there is a bounded-error quantum algorithm that determines all connected components of $G$ after making $O(log(n)^6)$ many cut queries. In contrast, it follows from results in communication complexity that any randomized algorithm even just to decide whether the graph is connected or not must make at least $Omega(n/log(n))$ many cut queries. We further show that with $O(log(n)^8)$ many cut queries a quantum algorithm can with high probability output a spanning forest for $G$. En route to proving these results, we design quantum algorithms for learning a graph using cut queries. We show that a quantum algorithm can learn a graph with maximum degree $d$ after $O(d log(n)^2)$ many cut queries, and can learn a general graph with $O(sqrt{m} log(n)^{3/2})$ many cut queries. These two upper bounds are tight up to the poly-logarithmic factors, and compare to $Omega(dn)$ and $Omega(m/log(n))$ lower bounds on the number of cut queries needed by a randomized algorithm for the same problems, respectively. The key ingredients in our results are the Bernstein-Vazirani algorithm, approximate counting with OR queries, and learning sparse vectors from inner products as in compressed sensing.
In this paper, we consider the problem of designing cut sparsifiers and sketches for directed graphs. To bypass known lower bounds, we allow the sparsifier/sketch to depend on the balance of the input graph, which smoothly interpolates between undirected and directed graphs. We give nearly matching upper and lower bounds for both for-all (cf. Benczur and Karger, STOC 1996) and for-each (Andoni et al., ITCS 2016) cut sparsifiers/sketches as a function of cut balance, defined the maximum ratio of the cut value in the two directions of a directed graph (Ene et al., STOC 2016). We also show an interesting application of digraph sparsification via cut balance by using it to give a very short proof of a celebrated maximum flow result of Karger and Levine (STOC 2002).
As important decisions about the distribution of societys resources become increasingly automated, it is essential to consider the measurement and enforcement of fairness in these decisions. In this work we build on the results of Dwork and Ilvento ITCS19, which laid the foundations for the study of fair algorithms under composition. In particular, we study the cohort selection problem, where we wish to use a fair classifier to select $k$ candidates from an arbitrarily ordered set of size $n>k$, while preserving individual fairness and maximizing utility. We define a linear utility function to measure performance relative to the behavior of the original classifier. We develop a fair, utility-optimal $O(n)$-time cohort selection algorithm for the offline setting, and our primary result, a solution to the problem in the streaming setting that keeps no more than $O(k)$ pending candidates at all time.
We present an $(e^{O(p)} frac{log ell}{loglogell})$-approximation algorithm for socially fair clustering with the $ell_p$-objective. In this problem, we are given a set of points in a metric space. Each point belongs to one (or several) of $ell$ groups. The goal is to find a $k$-medians, $k$-means, or, more generally, $ell_p$-clustering that is simultaneously good for all of the groups. More precisely, we need to find a set of $k$ centers $C$ so as to minimize the maximum over all groups $j$ of $sum_{u text{ in group }j} d(u,C)^p$. The socially fair clustering problem was independently proposed by Ghadiri, Samadi, and Vempala [2021] and Abbasi, Bhaskara, and Venkatasubramanian [2021]. Our algorithm improves and generalizes their $O(ell)$-approximation algorithms for the problem. The natural LP relaxation for the problem has an integrality gap of $Omega(ell)$. In order to obtain our result, we introduce a strengthened LP relaxation and show that it has an integrality gap of $Theta(frac{log ell}{loglogell})$ for a fixed $p$. Additionally, we present a bicriteria approximation algorithm, which generalizes the bicriteria approximation of Abbasi et al. [2021].
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