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Samir Khuller

Publications and source records attributed to Samir Khuller.

35 records · Page 2Linked to original sources
arXiv2018

On the cost of essentially fair clusterings

Clustering is a fundamental tool in data mining. It partitions points into groups (clusters) and may be used to make decisions for each point based on its group. However, this process may harm protected (minority) classes if the clustering algorithm does not adequately represent them in desirable clusters -- especially if the data is already biased. At NIPS 2017, Chierichetti et al. proposed a model for fair clustering requiring the representation in each cluster to (approximately) preserve the global fraction of each protected class. Restricting to two protected classes, they developed both a 4-approximation for the fair $k$-center problem and a $O(t)$-approximation for the fair $k$-median problem, where $t$ is a parameter for the fairness model. For multiple protected classes, the best known result is a 14-approximation for fair $k$-center. We extend and improve the known results. Firstly, we give a 5-approximation for the fair $k$-center problem with multiple protected classes. Secondly, we propose a relaxed fairness notion under which we can give bicriteria constant-factor approximations for all of the classical clustering objectives $k$-center, $k$-supplier, $k$-median, $k$-means and facility location. The latter approximations are achieved by a framework that takes an arbitrary existing unfair (integral) solution and a fair (fractional) LP solution and combines them into an essentially fair clustering with a weakly supervised rounding scheme. In this way, a fair clustering can be established belatedly, in a situation where the centers are already fixed.

cs.DS↗
arXiv2017

Select and Permute: An Improved Online Framework for Scheduling to Minimize Weighted Completion Time

In this paper, we introduce a new online scheduling framework for minimizing total weighted completion time in a general setting. The framework is inspired by the work of Hall et al. [Mathematics of Operations Research, Vol 22(3):513-544, 1997] and Garg et al. [Proc. of Foundations of Software Technology and Theoretical Computer Science, pp. 96-107, 2007], who show how to convert an offline approximation to an online scheme. Our framework uses two offline approximation algorithms (one for the simpler problem of scheduling without release times, and another for the minimum unscheduled weight problem) to create an online algorithm with provably good competitive ratios. We illustrate multiple applications of this method that yield improved competitive ratios. Our framework gives algorithms with the best or only known competitive ratios for the concurrent open shop, coflow, and concurrent cluster models. We also introduce a randomized variant of our framework based on the ideas of Chakrabarti et al. [Proc. of International Colloquium on Automata, Languages, and Programming, pp. 646-657, 1996] and use it to achieve improved competitive ratios for these same problems.

cs.DS↗
arXiv2016

Scheduling Distributed Clusters of Parallel Machines: Primal-Dual and LP-based Approximation Algorithms [Full Version]

The Map-Reduce computing framework rose to prominence with datasets of such size that dozens of machines on a single cluster were needed for individual jobs. As datasets approach the exabyte scale, a single job may need distributed processing not only on multiple machines, but on multiple clusters. We consider a scheduling problem to minimize weighted average completion time of N jobs on M distributed clusters of parallel machines. In keeping with the scale of the problems motivating this work, we assume that (1) each job is divided into M "subjobs" and (2) distinct subjobs of a given job may be processed concurrently. When each cluster is a single machine, this is the NP-Hard concurrent open shop problem. A clear limitation of such a model is that a serial processing assumption sidesteps the issue of how different tasks of a given subjob might be processed in parallel. Our algorithms explicitly model clusters as pools of resources and effectively overcome this issue. Under a variety of parameter settings, we develop two constant factor approximation algorithms for this problem. The first algorithm uses an LP relaxation tailored to this problem from prior work. This LP-based algorithm provides strong performance guarantees. Our second algorithm exploits a surprisingly simple mapping to the special case of one machine per cluster. This mapping-based algorithm is combinatorial and extremely fast. These are the first constant factor approximations for this problem.

cs.DS↗
arXiv2016

LP Rounding and Combinatorial Algorithms for Minimizing Active and Busy Time

We consider fundamental scheduling problems motivated by energy issues. In this framework, we are given a set of jobs, each with a release time, deadline and required processing length. The jobs need to be scheduled on a machine so that at most g jobs are active at any given time. The duration for which a machine is active (i.e., "on") is referred to as its active time. The goal is to find a feasible schedule for all jobs, minimizing the total active time. When preemption is allowed at integer time points, we show that a minimal feasible schedule already yields a 3-approximation (and this bound is tight) and we further improve this to a 2-approximation via LP rounding techniques. Our second contribution is for the non-preemptive version of this problem. However, since even asking if a feasible schedule on one machine exists is NP-hard, we allow for an unbounded number of virtual machines, each having capacity of g. This problem is known as the busy time problem in the literature and a 4-approximation is known for this problem. We develop a new combinatorial algorithm that gives a 3-approximation. Furthermore, we consider the preemptive busy time problem, giving a simple and exact greedy algorithm when unbounded parallelism is allowed, i.e., g is unbounded. For arbitrary g, this yields an algorithm that is 2-approximate.

cs.DS↗
arXiv2016

Minimizing Uncertainty through Sensor Placement with Angle Constraints

We study the problem of sensor placement in environments in which localization is a necessity, such as ad-hoc wireless sensor networks that allow the placement of a few anchors that know their location or sensor arrays that are tracking a target. In most of these situations, the quality of localization depends on the relative angle between the target and the pair of sensors observing it. In this paper, we consider placing a small number of sensors which ensure good angular $α$-coverage: given $α$ in $[0,π/2]$, for each target location $t$, there must be at least two sensors $s_1$ and $s_2$ such that the $\angle(s_1 t s_2)$ is in the interval $[α, π-α]$. One of the main difficulties encountered in such problems is that since the constraints depend on at least two sensors, building a solution must account for the inherent dependency between selected sensors, a feature that generic Set Cover techniques do not account for. We introduce a general framework that guarantees an angular coverage that is arbitrarily close to $α$ for any $α<= π/3$ and apply it to a variety of problems to get bi-criteria approximations. When the angular coverage is required to be at least a constant fraction of $α$, we obtain results that are strictly better than what standard geometric Set Cover methods give. When the angular coverage is required to be at least $(1-1/δ)\cdotα$, we obtain a $\mathcal{O}(\log δ)$- approximation for sensor placement with $α$-coverage on the plane. In the presence of additional distance or visibility constraints, the framework gives a $\mathcal{O}(\logδ\cdot\log k_{OPT})$-approximation, where $k_{OPT}$ is the size of the optimal solution. We also use our framework to give a $\mathcal{O}(\log δ)$-approximation that ensures $(1-1/δ)\cdot α$-coverage and covers every target within distance $3R$.

cs.CG↗
arXiv2015

On Correcting Inputs: Inverse Optimization for Online Structured Prediction

Algorithm designers typically assume that the input data is correct, and then proceed to find "optimal" or "sub-optimal" solutions using this input data. However this assumption of correct data does not always hold in practice, especially in the context of online learning systems where the objective is to learn appropriate feature weights given some training samples. Such scenarios necessitate the study of inverse optimization problems where one is given an input instance as well as a desired output and the task is to adjust the input data so that the given output is indeed optimal. Motivated by learning structured prediction models, in this paper we consider inverse optimization with a margin, i.e., we require the given output to be better than all other feasible outputs by a desired margin. We consider such inverse optimization problems for maximum weight matroid basis, matroid intersection, perfect matchings, minimum cost maximum flows, and shortest paths and derive the first known results for such problems with a non-zero margin. The effectiveness of these algorithmic approaches to online learning for structured prediction is also discussed.

cs.LG↗
arXiv2013

Analyzing the Optimal Neighborhood: Algorithms for Budgeted and Partial Connected Dominating Set Problems

We study partial and budgeted versions of the well studied connected dominating set problem. In the partial connected dominating set problem, we are given an undirected graph G = (V,E) and an integer n', and the goal is to find a minimum subset of vertices that induces a connected subgraph of G and dominates at least n' vertices. We obtain the first polynomial time algorithm with an O(\ln Δ) approximation factor for this problem, thereby significantly extending the results of Guha and Khuller (Algorithmica, Vol. 20(4), Pages 374-387, 1998) for the connected dominating set problem. We note that none of the methods developed earlier can be applied directly to solve this problem. In the budgeted connected dominating set problem, there is a budget on the number of vertices we can select, and the goal is to dominate as many vertices as possible. We obtain a (1/13)(1 - 1/e) approximation algorithm for this problem. Finally, we show that our techniques extend to a more general setting where the profit function associated with a subset of vertices is a monotone "special" submodular function. This generalization captures the connected dominating set problem with capacities and/or weighted profits as special cases. This implies a O(\ln q) approximation (where q denotes the quota) and an O(1) approximation algorithms for the partial and budgeted versions of these problems. While the algorithms are simple, the results make a surprising use of the greedy set cover framework in defining a useful profit function.

cs.DS↗
arXiv2013

Data Placement and Replica Selection for Improving Co-location in Distributed Environments

Increasing need for large-scale data analytics in a number of application domains has led to a dramatic rise in the number of distributed data management systems, both parallel relational databases, and systems that support alternative frameworks like MapReduce. There is thus an increasing contention on scarce data center resources like network bandwidth; further, the energy requirements for powering the computing equipment are also growing dramatically. As we show empirically, increasing the execution parallelism by spreading out data across a large number of machines may achieve the intended goal of decreasing query latencies, but in most cases, may increase the total resource and energy consumption significantly. For many analytical workloads, however, minimizing query latencies is often not critical; in such scenarios, we argue that we should instead focus on minimizing the average query span, i.e., the average number of machines that are involved in processing of a query, through colocation of data items that are frequently accessed together. In this work, we exploit the fact that most distributed environments need to use replication for fault tolerance, and we devise workload-driven replica selection and placement algorithms that attempt to minimize the average query span. We model a historical query workload trace as a hypergraph over a set of data items, and formulate and analyze the problem of replica placement by drawing connections to several well-studied graph theoretic concepts. We develop a series of algorithms to decide which data items to replicate, and where to place the replicas. We show effectiveness of our proposed approach by presenting results on a collection of synthetic and real workloads. Our experiments show that careful data placement and replication can dramatically reduce the average query spans resulting in significant reductions in the resource consumption.

cs.DB↗
arXiv2012

LP Rounding for k-Centers with Non-uniform Hard Capacities

In this paper we consider a generalization of the classical k-center problem with capacities. Our goal is to select k centers in a graph, and assign each node to a nearby center, so that we respect the capacity constraints on centers. The objective is to minimize the maximum distance a node has to travel to get to its assigned center. This problem is NP-hard, even when centers have no capacity restrictions and optimal factor 2 approximation algorithms are known. With capacities, when all centers have identical capacities, a 6 approximation is known with no better lower bounds than for the infinite capacity version. While many generalizations and variations of this problem have been studied extensively, no progress was made on the capacitated version for a general capacity function. We develop the first constant factor approximation algorithm for this problem. Our algorithm uses an LP rounding approach to solve this problem, and works for the case of non-uniform hard capacities, when multiple copies of a node may not be chosen and can be extended to the case when there is a hard bound on the number of copies of a node that may be selected. In addition we establish a lower bound on the integrality gap of 7(5) for non-uniform (uniform) hard capacities. In addition we prove that if there is a (3-eps)-factor approximation for this problem then P=NP. Finally, for non-uniform soft capacities we present a much simpler 11-approximation algorithm, which we find as one more evidence that hard capacities are much harder to deal with.

cs.DS↗
arXiv2012

A Model for Minimizing Active Processor Time

We introduce the following elementary scheduling problem. We are given a collection of n jobs, where each job has an integer length as well as a set Ti of time intervals in which it can be feasibly scheduled. Given a parameter B, the processor can schedule up to B jobs at a timeslot t so long as it is "active" at t. The goal is to schedule all the jobs in the fewest number of active timeslots. The machine consumes a fixed amount of energy per active timeslot, regardless of the number of jobs scheduled in that slot (as long as the number of jobs is non-zero). In other words, subject to all units of each job being scheduled in its feasible region and at each slot at most B jobs being scheduled, we are interested in minimizing the total time during which the machine is active. We present a linear time algorithm for the case where jobs are unit length and each Ti is a single interval. For general Ti, we show that the problem is NP-complete even for B = 3. However when B = 2, we show that it can be efficiently solved. In addition, we consider a version of the problem where jobs have arbitrary lengths and can be preempted at any point in time. For general B, the problem can be solved by linear programming. For B = 2, the problem amounts to finding a triangle-free 2-matching on a special graph. We extend the algorithm of Babenko et. al. to handle our variant, and also to handle non-unit length jobs. This yields an O(sqrt(L)m) time algorithm to solve the preemptive scheduling problem for B = 2, where L is the sum of the job lengths. We also show that for B = 2 and unit length jobs, the optimal non-preemptive schedule has at most 4/3 times the active time of the optimal preemptive schedule; this bound extends to several versions of the problem when jobs have arbitrary length.

cs.DS↗
arXiv2009

On Computing Compression Trees for Data Collection in Sensor Networks

We address the problem of efficiently gathering correlated data from a wired or a wireless sensor network, with the aim of designing algorithms with provable optimality guarantees, and understanding how close we can get to the known theoretical lower bounds. Our proposed approach is based on finding an optimal or a near-optimal {\em compression tree} for a given sensor network: a compression tree is a directed tree over the sensor network nodes such that the value of a node is compressed using the value of its parent. We consider this problem under different communication models, including the {\em broadcast communication} model that enables many new opportunities for energy-efficient data collection. We draw connections between the data collection problem and a previously studied graph concept, called {\em weakly connected dominating sets}, and we use this to develop novel approximation algorithms for the problem. We present comparative results on several synthetic and real-world datasets showing that our algorithms construct near-optimal compression trees that yield a significant reduction in the data collection cost.

cs.NI↗
arXiv2002

Designing Multi-Commodity Flow Trees

The traditional multi-commodity flow problem assumes a given flow network in which multiple commodities are to be maximally routed in response to given demands. This paper considers the multi-commodity flow network-design problem: given a set of multi-commodity flow demands, find a network subject to certain constraints such that the commodities can be maximally routed. This paper focuses on the case when the network is required to be a tree. The main result is an approximation algorithm for the case when the tree is required to be of constant degree. The algorithm reduces the problem to the minimum-weight balanced-separator problem; the performance guarantee of the algorithm is within a factor of 4 of the performance guarantee of the balanced-separator procedure. If Leighton and Rao's balanced-separator procedure is used, the performance guarantee is O(log n). This improves the O(log^2 n) approximation factor that is trivial to obtain by a direct application of the balanced-separator method.

cs.DS↗
arXiv2002

Balancing Minimum Spanning and Shortest Path Trees

This paper give a simple linear-time algorithm that, given a weighted digraph, finds a spanning tree that simultaneously approximates a shortest-path tree and a minimum spanning tree. The algorithm provides a continuous trade-off: given the two trees and epsilon > 0, the algorithm returns a spanning tree in which the distance between any vertex and the root of the shortest-path tree is at most 1+epsilon times the shortest-path distance, and yet the total weight of the tree is at most 1+2/epsilon times the weight of a minimum spanning tree. This is the best tradeoff possible. The paper also describes a fast parallel implementation.

cs.DS↗
arXiv2002

Approximating the Minimum Equivalent Digraph

The MEG (minimum equivalent graph) problem is, given a directed graph, to find a small subset of the edges that maintains all reachability relations between nodes. The problem is NP-hard. This paper gives an approximation algorithm with performance guarantee of pi^2/6 ~ 1.64. The algorithm and its analysis are based on the simple idea of contracting long cycles. (This result is strengthened slightly in ``On strongly connected digraphs with bounded cycle length'' (1996).) The analysis applies directly to 2-Exchange, a simple ``local improvement'' algorithm, showing that its performance guarantee is 1.75.

cs.DS↗
arXiv2002

Low-Degree Spanning Trees of Small Weight

The degree-d spanning tree problem asks for a minimum-weight spanning tree in which the degree of each vertex is at most d. When d=2 the problem is TSP, and in this case, the well-known Christofides algorithm provides a 1.5-approximation algorithm (assuming the edge weights satisfy the triangle inequality). In 1984, Christos Papadimitriou and Umesh Vazirani posed the challenge of finding an algorithm with performance guarantee less than 2 for Euclidean graphs (points in R^n) and d > 2. This paper gives the first answer to that challenge, presenting an algorithm to compute a degree-3 spanning tree of cost at most 5/3 times the MST. For points in the plane, the ratio improves to 3/2 and the algorithm can also find a degree-4 spanning tree of cost at most 5/4 times the MST.

cs.DS↗
arXiv2002

On Strongly Connected Digraphs with Bounded Cycle Length

The MEG (minimum equivalent graph) problem is, given a directed graph, to find a small subset of the edges that maintains all reachability relations between nodes. The problem is NP-hard. This paper gives a proof that, for graphs where each directed cycle has at most three edges, the MEG problem is equivalent to maximum bipartite matching, and therefore solvable in polynomial time. This leads to an improvement in the performance guarantee of the previously best approximation algorithm for the general problem in ``Approximating the Minimum Equivalent Digraph'' (1995).

cs.DS↗