Computer Science

The problem of scheduling with testing in the framework of explorable uncertainty models environments where some preliminary action can influence the duration of a task. In the model, each job has an unknown processing time that can be revealed by running a test. Alternatively, jobs may be run untested for the duration of a given upper limit. Recently, Dürr et al. [5] have studied the setting where all testing times are of unit size and have given lower and upper bounds for the objectives of minimizing the sum of completion times and the makespan on a single machine. In this paper, we extend the problem to non-uniform testing times and present the first competitive algorithms. The general setting is motivated for example by online user surveys for market prediction or querying centralized databases in distributed computing. Introducing general testing times gives the problem a new flavor and requires updated methods with new techniques in the analysis. We present constant competitive ratios for the objective of minimizing the sum of completion times in the deterministic case, both in the non-preemptive and preemptive setting. For the preemptive setting, we additionally give a first lower bound. We also present a randomized algorithm with improved competitive ratio. Furthermore, we give tight competitive ratios for the objective of minimizing the makespan, both in the deterministic and the randomized setting.

For intractable problems on graphs of bounded treewidth, two graph parameters treedepth and vertex cover number have been used to obtain fine-grained complexity results. Although the studies in this direction are successful, we still need a systematic way for further investigations because the graphs of bounded vertex cover number form a rather small subclass of the graphs of bounded treedepth. To fill this gap, we use vertex integrity, which is placed between the two parameters mentioned above. For several graph problems, we generalize fixed-parameter tractability results parameterized by vertex cover number to the ones parameterized by vertex integrity. We also show some finer complexity contrasts by showing hardness with respect to vertex integrity or treedepth.

The Metric k -median problem over a metric space (X,d) is defined as follows: given a set L⊆X of facility locations and a set C⊆X of clients, open a set F⊆L of k facilities such that the total service cost, defined as Φ(F,C)≡ ∑ x∈C min f∈F d(x,f) , is minimised. The metric k -means problem is defined similarly using squared distances. In many applications there are additional constraints that any solution needs to satisfy. This gives rise to different constrained versions of the problem such as r -gather, fault-tolerant, outlier k -means/ k -median problem. Surprisingly, for many of these constrained problems, no constant-approximation algorithm is known. We give FPT algorithms with constant approximation guarantee for a range of constrained k -median/means problems. For some of the constrained problems, ours is the first constant factor approximation algorithm whereas for others, we improve or match the approximation guarantee of previous works. We work within the unified framework of Ding and Xu that allows us to simultaneously obtain algorithms for a range of constrained problems. In particular, we obtain a (3+ε) -approximation and (9+ε) -approximation for the constrained versions of the k -median and k -means problem respectively in FPT time. In many practical settings of the k -median/means problem, one is allowed to open a facility at any client location, i.e., C⊆L . For this special case, our algorithm gives a (2+ε) -approximation and (4+ε) -approximation for the constrained versions of k -median and k -means problem respectively in FPT time. Since our algorithm is based on simple sampling technique, it can also be converted to a constant-pass log-space streaming algorithm.

The three-in-a-tree problem asks for an induced tree of the input graph containing three mandatory vertices. In 2006, Chudnovsky and Seymour [Combinatorica, 2010] presented the first polynomial time algorithm for this problem, which has become a critical subroutine in many algorithms for detecting induced subgraphs, such as beetles, pyramids, thetas, and even and odd-holes. In 2007, Derhy and Picouleau [Discrete Applied Mathematics, 2009] considered the natural generalization to k mandatory vertices, proving that, when k is part of the input, the problem is NP -complete, and ask what is the complexity of four-in-a-tree. Motivated by this question and the relevance of the original problem, we study the parameterized complexity of k -in-a-tree. We begin by showing that the problem is W[1] -hard when jointly parameterized by the size of the solution and minimum clique cover and, under the Exponential Time Hypothesis, does not admit an n o(k) time algorithm. Afterwards, we use Courcelle's Theorem to prove fixed-parameter tractability under cliquewidth, which prompts our investigation into which parameterizations admit single exponential algorithms; we show that such algorithms exist for the unrelated parameterizations treewidth, distance to cluster, and distance to co-cluster. In terms of kernelization, we present a linear kernel under feedback edge set, and show that no polynomial kernel exists under vertex cover nor distance to clique unless NP⊆coNP/poly . Along with other remarks and previous work, our tractability and kernelization results cover many of the most commonly employed parameters in the graph parameter hierarchy.

Let n be the size of a parametrized problem and k the parameter. We present a full kernel for Path Contraction and Cluster Editing/Deletion as well as a kernel for Feedback Vertex Set whose sizes are all polynomial in k , that are computable in polynomial time, and use O(poly(k)logn) bits. By first executing the new kernelizations and subsequently the best known polynomial-time kernelizations for the problem under consideration, we obtain the best known kernels in polynomial time with O(poly(k)logn) bits.

An instance of colorful k-center consists of points in a metric space that are colored red or blue, along with an integer k and a coverage requirement for each color. The goal is to find the smallest radius \r{ho} such that there exist balls of radius \r{ho} around k of the points that meet the coverage requirements. The motivation behind this problem is twofold. First, from fairness considerations: each color/group should receive a similar service guarantee, and second, from the algorithmic challenges it poses: this problem combines the difficulties of clustering along with the subset-sum problem. In particular, we show that this combination results in strong integrality gap lower bounds for several natural linear programming relaxations. Our main result is an efficient approximation algorithm that overcomes these difficulties to achieve an approximation guarantee of 3, nearly matching the tight approximation guarantee of 2 for the classical k-center problem which this problem generalizes.

The rise of algorithmic decision-making has created an explosion of research around the fairness of those algorithms. While there are many compelling notions of individual fairness, beginning with the work of Dwork et al., these notions typically do not satisfy desirable composition properties. To this end, Dwork and Ilvento introduced the fair cohort selection problem, which captures a specific application where a single fair classifier is composed with itself to pick a group of candidates of size exactly k . In this work we introduce a specific instance of cohort selection where the goal is to choose a cohort maximizing a linear utility function. We give approximately optimal polynomial-time algorithms for this problem in both an offline setting where the entire fair classifier is given at once, or an online setting where candidates arrive one at a time and are classified as they arrive.

As important decisions about the distribution of society's 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 ITCS'19, 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.

Constrained submodular maximization problems encompass a wide variety of applications, including personalized recommendation, team formation, and revenue maximization via viral marketing. The massive instances occurring in modern day applications can render existing algorithms prohibitively slow, while frequently, those instances are also inherently stochastic. Focusing on these challenges, we revisit the classic problem of maximizing a (possibly non-monotone) submodular function subject to a knapsack constraint. We present a simple randomized greedy algorithm that achieves a 5.83 approximation and runs in O(nlogn) time, i.e., at least a factor n faster than other state-of-the-art algorithms. The robustness of our approach allows us to further transfer it to a stochastic version of the problem. There, we obtain a 9 -approximation to the best adaptive policy, which is the first constant approximation for non-monotone objectives. Experimental evaluation of our algorithms showcases their improved performance on real and synthetic data.

We consider online algorithms for the k -server problem on trees. Chrobak and Larmore proposed a k -competitive algorithm for this problem that has the optimal competitive ratio. However, a naive implementation of their algorithm has O(n) time complexity for processing each query, where n is the number of nodes in the tree. We propose a new time-efficient implementation of this algorithm that has O(nlogn) time complexity for preprocessing and O( k 2 +k⋅logn) time for processing a query. We also propose a quantum algorithm for the case where the nodes of the tree are presented using string paths. In this case, no preprocessing is needed, and the time complexity for each query is O( k 2 n − − √ logn) . When the number of queries is o( n √ k 2 logn ) , we obtain a quantum speed-up on the total runtime compared to our classical algorithm. We also give a simple quantum algorithm to find the first marked element in a collection of m objects, that works even in the presence of two-sided bounded errors on the input oracle. It has worst-case complexity O( m − − √ ) . In the particular case of one-sided errors on the input, it has expected time complexity O( x − − √ ) where x is the position of the first marked element. Compare with previous work, our algorithm can handle errors in the input oracle.