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Maximizing a Submodular Set Function subject to a Matroid Constraint (Extended Abstract)
 PROC. OF 12 TH IPCO
, 2007
"... Let f: 2 N → R + be a nondecreasing submodular set function, and let (N, I) be a matroid. We consider the problem maxS∈I f(S). It is known that the greedy algorithm yields a 1/2approximation [9] for this problem. It is also known, via a reduction from the maxkcover problem, that there is no (1 ..."
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Cited by 112 (14 self)
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Let f: 2 N → R + be a nondecreasing submodular set function, and let (N, I) be a matroid. We consider the problem maxS∈I f(S). It is known that the greedy algorithm yields a 1/2approximation [9] for this problem. It is also known, via a reduction from the maxkcover problem, that there is no (1 − 1/e + ɛ)approximation for any constant ɛ> 0, unless P = NP [6]. In this paper, we improve the 1/2approximation to a (1−1/e)approximation, when f is a sum of weighted rank functions of matroids. This class of functions captures a number of interesting problems including set coverage type problems. Our main tools are the pipage rounding technique of Ageev and Sviridenko [1] and a probabilistic lemma on monotone submodular functions that might be of independent interest. We show that the generalized assignment problem (GAP) is a special case of our problem; although the reduction requires N  to be exponential in the original problem size, we are able to interpret the recent (1 − 1/e)approximation for GAP by Fleischer et al. [10] in our framework. This enables us to obtain a (1 − 1/e)approximation for variants of GAP with more complex constraints.
Maximizing a Monotone Submodular Function subject to a Matroid Constraint
, 2008
"... Let f: 2 X → R+ be a monotone submodular set function, and let (X, I) be a matroid. We consider the problem maxS∈I f(S). It is known that the greedy algorithm yields a 1/2 approximation [14] for this problem. For certain special cases, e.g. max S≤k f(S), the greedy algorithm yields a (1 − 1/e)app ..."
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Cited by 62 (0 self)
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Let f: 2 X → R+ be a monotone submodular set function, and let (X, I) be a matroid. We consider the problem maxS∈I f(S). It is known that the greedy algorithm yields a 1/2 approximation [14] for this problem. For certain special cases, e.g. max S≤k f(S), the greedy algorithm yields a (1 − 1/e)approximation. It is known that this is optimal both in the value oracle model (where the only access to f is through a black box returning f(S) for a given set S) [28], and also for explicitly posed instances assuming P � = NP [10]. In this paper, we provide a randomized (1 − 1/e)approximation for any monotone submodular function and an arbitrary matroid. The algorithm works in the value oracle model. Our main tools are a variant of the pipage rounding technique of Ageev and Sviridenko [1], and a continuous greedy process that might be of independent interest. As a special case, our algorithm implies an optimal approximation for the Submodular Welfare Problem in the value oracle model [32]. As a second application, we show that the Generalized Assignment Problem (GAP) is also a special case; although the reduction requires X  to be exponential in the original problem size, we are able to achieve a (1 − 1/e − o(1))approximation for GAP, simplifying previously known algorithms. Additionally, the reduction enables us to obtain approximation algorithms for variants of GAP with more general constraints.
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.
Broadcast scheduling: algorithms and complexity
, 2007
"... Broadcast Scheduling is a popular method for disseminating information in response to client requests. There are n pages of information, and clients request pages at different times. However, multiple clients can have their requests satisfied by a single broadcast of the requested page. In this pape ..."
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Cited by 24 (2 self)
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Broadcast Scheduling is a popular method for disseminating information in response to client requests. There are n pages of information, and clients request pages at different times. However, multiple clients can have their requests satisfied by a single broadcast of the requested page. In this paper we consider several related broadcast scheduling problems. One central problem we study simply asks to minimize the maximum response time (over all requests). Another related problem we consider is the version in which every request has a release time and a deadline, and the goal is to maximize the number of requests that meet their deadlines. While approximation algorithms for both these problems were proposed several years back, it was not known if they were NPcomplete. One of our main results is that both these problems are NPcomplete. In addition, we use the same unified approach to give a simple NPcompleteness proof for minimizing the sum of response times. A very complicated proof was known for this version. Furthermore, we give a proof that FIFO is a 2competitive online algorithm for minimizing the maximum response time (this result had been claimed earlier with no proof) and that there is no better deterministic online algorithm (this result was claimed earlier as well, but with an incorrect proof). A preliminary version of this paper was presented at the ACMSIAM Symposium on Discrete
When LP is the Cure for Your Matching Woes: Improved Bounds for Stochastic Matchings (Extended Abstract)
"... Abstract. Consider a random graph model where each possible edge e is present independently with some probability pe. We are given these numbers pe, and want to build a large/heavy matching in the randomly generated graph. However, the only way we can find out whether an edge is present or not is to ..."
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Cited by 23 (5 self)
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Abstract. Consider a random graph model where each possible edge e is present independently with some probability pe. We are given these numbers pe, and want to build a large/heavy matching in the randomly generated graph. However, the only way we can find out whether an edge is present or not is to query it, and if the edge is indeed present in the graph, we are forced to add it to our matching. Further, each vertex i is allowed to be queried at most ti times. How should we adaptively query the edges to maximize the expected weight of the matching? We consider several matching problems in this general framework (some of which arise in kidney exchanges and online dating, and others arise in modeling online advertisements); we give LProunding based constantfactor approximation algorithms for these problems. Our main results are: • We give a 5.75approximation for weighted stochastic matching on general graphs, and a 5approximation on bipartite graphs. This answers an open question from [Chen et al. ICALP 09]. • Combining our LProunding algorithm with the natural greedy algorithm, we give an improved 3.88approximation for unweighted stochastic matching on general graphs and 3.51approximation on bipartite graphs. • We introduce a generalization of the stochastic online matching problem [Feldman et al. FOCS 09] that also models preferenceuncertainty and timeouts of buyers, and give a constant factor approximation algorithm. 1
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].
An Online Scalable Algorithm for Average Flowtime in Broadcast Scheduling
 In SODA 10: Proceedings of the twentyfirst annual ACMSIAM symposium on Discrete algorithms
, 2010
"... In this paper the online pullbased broadcast model is considered. In this model, there are n pages of data stored at a server and requests arrive for pages online. When the server broadcasts page p, all outstanding requests for the same page p are simultaneously satisfied. We consider the problem o ..."
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Cited by 16 (12 self)
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In this paper the online pullbased broadcast model is considered. In this model, there are n pages of data stored at a server and requests arrive for pages online. When the server broadcasts page p, all outstanding requests for the same page p are simultaneously satisfied. We consider the problem of minimizing average (total) flow time online where all pages are unitsized. For this problem, there has been a decadelong search for an online algorithm which is scalable, i.e. (1 + ɛ)speed O(1)competitive for any fixed ɛ> 0. In this paper, we give the first analysis of an online scalable algorithm. 1
A primaldual approximation algorithm for partial vertex cover: Making educated guesses
 In 8th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, volume 3624 of LNCS
, 2005
"... We study the partial vertex cover problem. Given a graph G = (V,E), a weight function w: V → R +, and an integer s, our goal is to cover all but s edges, by picking a set of vertices with minimum weight. The problem is clearly NPhard as it generalizes the wellknown vertex cover problem. We provide ..."
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Cited by 13 (2 self)
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We study the partial vertex cover problem. Given a graph G = (V,E), a weight function w: V → R +, and an integer s, our goal is to cover all but s edges, by picking a set of vertices with minimum weight. The problem is clearly NPhard as it generalizes the wellknown vertex cover problem. We provide a primaldual 2approximation algorithm which runs in O(nlog n+m) time. This represents an improvement in running time from the previously known fastest algorithm. Our technique can also be used to get a 2approximation for a more general version of the problem. In the partial capacitated vertex cover problem each vertex u comes with a capacity ku. A solution consists of a function x: V → N0 and an orientation of all but s edges, such that the number of edges oriented toward vertex u is at most xuku. Our objective is to find a cover that minimizes � v∈V xvwv. This is the first 2approximation for the problem and also runs in O(nlog n + m) time.
Online scheduling to minimize the maximum delay factor
 IN SODA 09: PROCEEDINGS OF THE TWENTIETH ANNUAL ACMSIAM SYMPOSIUM ON DISCRETE ALGORITHMS
, 2009
"... In this paper two scheduling models are addressed. First is the standard model (unicast) where requests (or jobs) are independent. The other is the broadcast model where broadcasting a page can satisfy multiple outstanding requests for that page. We consider online scheduling of requests when they h ..."
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Cited by 12 (4 self)
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In this paper two scheduling models are addressed. First is the standard model (unicast) where requests (or jobs) are independent. The other is the broadcast model where broadcasting a page can satisfy multiple outstanding requests for that page. We consider online scheduling of requests when they have deadlines. Unlike previous models, which mainly consider the objective of maximizing throughput while respecting deadlines, here we focus on scheduling all the given requests with the goal of minimizing the maximum delay factor. The delay factor of a schedule is defined to be the minimum α ≥ 1 such that each request i is completed by time ai + α(di − ai) where ai is the arrival time of request i and di is its deadline. Delay factor generalizes the previously defined measure of maximum stretch which is based only the processing times of requests [9, 11]. We prove strong lower bounds on the achievable competitive ratios for delay factor scheduling even with unittime requests. Motivated by this, we consider resource augmentation analysis [24] and prove the following positive results. For the unicast model we give algorithms that are (1 + ɛ)speed O ( 1 ɛ)competitive in both the single machine and multiple machine settings. In the broadcast model we give an algorithm for samesized pages that is (2 + ɛ)speed O ( 1 ɛ 2)competitive. For arbitrary page sizes we give an algorithm that is (4 + ɛ)speed O ( 1 ɛ 2)competitive.