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Dependent Randomized Rounding via Exchange Properties of Combinatorial Structures (Extended Abstract)
"... We consider the problem of randomly rounding a fractional solution x in an integer polytope P ⊆ [0, 1] n to a vertex X of P, so that E[X] = x. Our goal is to achieve concentration properties for linear and submodular functions of the rounded solution. Such dependent rounding techniques, with conce ..."
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Cited by 26 (3 self)
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We consider the problem of randomly rounding a fractional solution x in an integer polytope P ⊆ [0, 1] n to a vertex X of P, so that E[X] = x. Our goal is to achieve concentration properties for linear and submodular functions of the rounded solution. Such dependent rounding techniques, with concentration bounds for linear functions, have been developed in the past for two polytopes: the assignment polytope (that is, bipartite matchings and bmatchings) [32], [19], [23], and more recently for the spanning tree polytope [2]. These schemes have led to a number of new algorithmic results. In this paper we describe a new swap rounding technique which can be applied in a variety of settings including matroids and matroid intersection, while providing Chernofftype concentration bounds for linear and submodular functions of the rounded solution. In addition to existing techniques based on negative correlation, we use a martingale argument to obtain an exponential tail estimate for monotone submodular functions. The rounding scheme explicitly exploits exchange properties of the underlying combinatorial structures, and highlights these properties as the basis for concentration bounds. Matroids and matroid intersection provide a unifying framework for several known applications [19], [23], [7], [22], [2] as well as new ones, and their generality allows a richer set of constraints to be incorporated easily. We give some illustrative examples, with a more comprehensive discussion deferred to a later version of the paper.
New Constructive Aspects of the Lovász Local Lemma
"... The Lovász Local Lemma (LLL) is a powerful tool that gives sufficient conditions for avoiding all of a given set of “bad ” events, with positive probability. A series of results have provided algorithms to efficiently construct structures whose existence is nonconstructively guaranteed by the LLL, ..."
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Cited by 23 (4 self)
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The Lovász Local Lemma (LLL) is a powerful tool that gives sufficient conditions for avoiding all of a given set of “bad ” events, with positive probability. A series of results have provided algorithms to efficiently construct structures whose existence is nonconstructively guaranteed by the LLL, culminating in the recent breakthrough of Moser & Tardos. We show that the output distribution of the MoserTardos algorithm wellapproximates the conditional LLLdistribution – the distribution obtained by conditioning on all bad events being avoided. We show how a known bound on the probabilities of events in this distribution can be used for further probabilistic analysis and give new constructive and nonconstructive results. We also show that when an LLL application provides a small amount of slack, the number of resamplings of the
Multibudgeted Matchings and Matroid Intersection via Dependent Rounding
"... Motivated by multibudgeted optimization and other applications, we consider the problem of randomly rounding a fractional solution x in the (nonbipartite graph) matching and matroid intersection polytopes. We show that for any fixed δ> 0, a given point x can be rounded to a random solution R su ..."
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Cited by 16 (1 self)
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Motivated by multibudgeted optimization and other applications, we consider the problem of randomly rounding a fractional solution x in the (nonbipartite graph) matching and matroid intersection polytopes. We show that for any fixed δ> 0, a given point x can be rounded to a random solution R such that E[1R] = (1 − δ)x and any linear function of x satisfies dimensionfree ChernoffHoeffding concentration bounds (the bounds depend on δ and the expectation µ). We build on and adapt the swap rounding scheme in our recent work [9] to achieve this result. Our main contribution is a nontrivial martingale based analysis framework to prove the desired concentration bounds. In this paper we describe two applications. We give a randomized PTAS for matroid intersection and matchings with any fixed number of budget constraints. We also give a deterministic PTAS for the case of matchings. The concentration bounds also yield related results when the number of budget constraints is not fixed. As a second application we obtain an algorithm to compute in polynomial time an εapproximate Paretooptimal set for the multiobjective variants of these problems, when the number of objectives is a fixed constant. We rely on a result of Papadimitriou and Yannakakis [26].
Energy Efficient Scheduling via Partial Shutdown
, 2010
"... Motivated by issues of saving energy in data centers we define a collection of new problems referred to as “machine activation ” problems. The central framework we introduce considers a collection of m machines (unrelated or related) with each machine i having an activation cost of ai. There is also ..."
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Cited by 9 (2 self)
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Motivated by issues of saving energy in data centers we define a collection of new problems referred to as “machine activation ” problems. The central framework we introduce considers a collection of m machines (unrelated or related) with each machine i having an activation cost of ai. There is also a collection of n jobs that need to be performed, and pi,j is the processing time of job j on machine i. Standard scheduling models assume that the set of machines is fixed and all machines are available. However, in our setting, we assume that there is an activation cost budget of A – we would like to select a subset S of the machines to activate with total cost a(S) ≤ A and find a schedule for the n jobs on the machines in S minimizing the makespan (or any other metric). We consider
AdCell: Ad Allocation in Cellular Networks
"... Abstract. With more than four billion usage of cellular phones worldwide, mobile advertising has become an attractive alternative to online advertisements. In this paper, we propose a new targeted advertising policy for Wireless Service Providers (WSPs) via SMS or MMS namely AdCell. In our model, a ..."
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Abstract. With more than four billion usage of cellular phones worldwide, mobile advertising has become an attractive alternative to online advertisements. In this paper, we propose a new targeted advertising policy for Wireless Service Providers (WSPs) via SMS or MMS namely AdCell. In our model, a WSP charges the advertisers for showing their ads. Each advertiser has a valuation for specific types of customers in various times and locations and has a limit on the maximum available budget. Each query is in the form of time and location and is associated with one individual customer. In order to achieve a nonintrusive delivery, only a limited number of ads can be sent to each customer. Recently, new services have been introduced that offer locationbased advertising over cellular network that fit in our model (e.g., ShopAlerts by AT&T). We consider both online and offline version of the AdCell problem and develop approximation algorithms with constant competitive ratio. For the online version, we assume that the appearances of the queries follow a stochastic distribution and thus consider a Bayesian setting. Furthermore, queries may come from different
Combinatorial Algorithm for Restricted MaxMin Fair Allocation
, 2014
"... We study the basic allocation problem of assigning resources to players so as to maximize fairness. This is one of the few natural problems that enjoys the intriguing status of having a better estimation algorithm than approximation algorithm. Indeed, a certain configurationLP can be used to estima ..."
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We study the basic allocation problem of assigning resources to players so as to maximize fairness. This is one of the few natural problems that enjoys the intriguing status of having a better estimation algorithm than approximation algorithm. Indeed, a certain configurationLP can be used to estimate the value of the optimal allocation to within a factor of 4 + ε. In contrast, however, the best known approximation algorithm for the problem has an unspecified large constant guarantee. In this paper we significantly narrow this gap by giving a 13approximation algorithm for the problem. Our approach develops a local search technique introduced by Haxell [Hax95] for hypergraph matchings, and later used in this context by Asadpour, Feige, and Saberi [AFS12]. For our local search procedure to terminate in polynomial time, we introduce several new ideas such as lazy updates and greedy players. Besides the improved approximation guarantee, the highlight of our approach is that it is purely combinatorial and uses the configurationLP only in the analysis.
Generalized Machine Activation Problems
, 2011
"... In this paper we consider a generalization of the machine activation problem introduced recently [“Energy efficient scheduling via partial shutdown ” by Khuller, Li and Saha (ACMSIAM 2010 Symp. on Discrete Algorithms)] where the unrelated parallel machine scheduling problem is studied with machine ..."
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In this paper we consider a generalization of the machine activation problem introduced recently [“Energy efficient scheduling via partial shutdown ” by Khuller, Li and Saha (ACMSIAM 2010 Symp. on Discrete Algorithms)] where the unrelated parallel machine scheduling problem is studied with machine activation cost. This is the standard unrelated parallel machine scheduling problem with a machine dependent activation cost that is incurred, if any job is assigned to the machine. The problem asks for a choice of machines to activate, and a schedule of all jobs on the active machines subject to the makespan constraint. The goal is to minimize the total activation cost. Our main generalization consists of a general activation cost model, where the activation cost for a machine is a nondecreasing function of its load. We develop a greedy algorithm that yields a fractional assignment of jobs, such that at least n − ɛ jobs are assigned fractionally and the total cost is at most 1 + ln(n/ɛ) times the optimum. Combining with standard rounding methods yields improved bounds for several machine activation problems. In addition, we study the machine activation problem with d linear constraints (these could model makespan constraints, as well as other types of constraints). Our method yields a schedule with machine activation cost of O ( 1 ɛ log n) times the optimum and a constraint violation by a factor of 2d + ɛ. This result matches our previous bound for the case d = 1. As a byproduct, our method also yields a ln n + 1 approximation factor for the nonmetric universal facility location problem for which the cost of opening a facility is an arbitrary nondecreasing function of the number of clients assigned to it. This gives an affirmative answer to the open question posed in earlier work on universal facility location.
New Approximation Results for Resource Replication Problems
"... Abstract. We consider several variants of a basic resource replication problem in this paper, and propose new approximation results for them. These problems are of fundamental interest in the areas of P2P networks, sensor networks and ad hoc networks, where optimal placement of replicas is the main ..."
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Abstract. We consider several variants of a basic resource replication problem in this paper, and propose new approximation results for them. These problems are of fundamental interest in the areas of P2P networks, sensor networks and ad hoc networks, where optimal placement of replicas is the main bottleneck on performance. We observe that the threshold graph technique, which has been applied to several kcenter type problems, yields simple and efficient approximation algorithms for resource replication problems. Our results range from positive (efficient, small constant factor, approximation algorithms) to extremely negative (impossibility of existence of any algorithm with nontrivial approximation guarantee, i.e., with positive approximation ratio) for different versions of the problem. 1
PTAS for Ordered Instances of Resource Allocation Problems
"... Abstract We consider the problem of fair allocation of indivisible goods where we are given a set I of m indivisible resources (items) and a set P of n customers (players) competing for the resources. Each resource j ∈ I has a same value v j > 0 for a subset of customers interested in j and it h ..."
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Abstract We consider the problem of fair allocation of indivisible goods where we are given a set I of m indivisible resources (items) and a set P of n customers (players) competing for the resources. Each resource j ∈ I has a same value v j > 0 for a subset of customers interested in j and it has no value for other customers. The goal is to find a feasible allocation of the resources to the interested customers such that in the MaxMin scenario (also known as Santa Claus problem) the minimum utility (sum of the resources) received by each of the customers is as high as possible and in the MinMax case (also known as RC max problem), the maximum utility is as low as possible. In this paper we are interested in instances of the problem that admit a PTAS. These instances are not only of theoretical interest but also have practical applications. For the MaxMin allocation problem, we start with instances of the problem that can be viewed as a convex bipartite graph; there exists an ordering of the resources such that each customer is interested (has positive evaluation) in a set of consecutive resources and we demonstrate a PTAS. For the MinMax allocation problem, we obtain a PTAS for instances in which there is an ordering of the customers (machines) and each resource (job) is adjacent to a consecutive set of customers (machines). Next we show that our method for the MaxMin scenario, can be extended to a broader class of bipartite graphs where the resources can be viewed as a tree and each customer is interested in a subtree of a bounded number of leaves of this tree (e.g. a subpath).