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A unified approach to approximating partial covering problems
 IN PROCEEDINGS OF THE 14TH ANNUAL EUROPEAN SYMPOSIUM ON ALGORITHMS
, 2006
"... An instance of the generalized partial cover problem consists of a ground set U and a family of subsets S ⊆ 2 U. Each element e ∈ U is associated with a profit p(e), whereas each subset S ∈ S has a cost c(S). The objective is to find a minimum cost subcollection S ′ ⊆ S such that the combined prof ..."
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An instance of the generalized partial cover problem consists of a ground set U and a family of subsets S ⊆ 2 U. Each element e ∈ U is associated with a profit p(e), whereas each subset S ∈ S has a cost c(S). The objective is to find a minimum cost subcollection S ′ ⊆ S such that the combined profit of the elements covered by S ′ is at least P, a specified profit bound. In the prizecollecting version of this problem, there is no strict requirement to cover any element; however, if the subsets we pick leave an element e ∈ U uncovered, we incur a penalty of π(e). The goal is to identify a subcollection S ′ ⊆ S that minimizes the cost of S ′ plus the penalties of uncovered elements. Although problemspecific connections between the partial cover and the prizecollecting variants of a given covering problem have been explored and exploited, a more general connection remained open. The main contribution of this paper is to establish a formal relationship between these two variants. As a result, we present a unified framework for approximating problems that can be formulated or interpreted as special cases of generalized partial cover. We demonstrate the applicability of our method on a diverse collection of covering problems, for some of which we obtain the first nontrivial approximability results.
Scheduling with Outliers
"... Abstract. In classical scheduling problems, we are given jobs and machines, and have to schedule all the jobs to minimize some objective function. What if each job has a specified profit, and we are no longer required to process all jobs? Instead, we can schedule any subset of jobs whose total profi ..."
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Abstract. In classical scheduling problems, we are given jobs and machines, and have to schedule all the jobs to minimize some objective function. What if each job has a specified profit, and we are no longer required to process all jobs? Instead, we can schedule any subset of jobs whose total profit is at least a (hard) target profit requirement, while still trying to approximately minimize the objective function. We refer to this class of problems as scheduling with outliers. This model was initiated by Charikar and Khuller (SODA ’06) for minimum maxresponse time in broadcast scheduling. In this paper, we consider three other wellstudied scheduling objectives: the generalized assignment problem, average weighted completion time, and average flow time, for which LPbased approximation algorithms are provided. Our main results are: – For the minimum average flow time problem on identical machines, we give an LPbased logarithmic approximation algorithm for the unit profits case, and complement this result by presenting a matching integrality gap. – For the average weighted completion time problem on unrelated machines, we give a constantfactor approximation. The algorithm is based on randomized rounding of the timeindexed LP relaxation strengthened by knapsackcover inequalities. – For the generalized assignment problem with outliers, we outline a simple reduction to GAP without outliers to obtain an algorithm whose makespan is within 3 times the optimum makespan, and whose cost is at most (1 + ɛ) times the optimal cost. 1
Path hitting in acyclic graphs
 In Proceedings of the 14th Annual European Symposium on Algorithms
, 2006
"... Abstract. An instance of the path hitting problem consists of two families of paths, D and H, in a common undirected graph, where each path in H is associated with a nonnegative cost. We refer to D and H as the sets of demand and hitting paths, respectively. When p ∈ H and q ∈ D share at least one ..."
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Abstract. An instance of the path hitting problem consists of two families of paths, D and H, in a common undirected graph, where each path in H is associated with a nonnegative cost. We refer to D and H as the sets of demand and hitting paths, respectively. When p ∈ H and q ∈ D share at least one mutual edge, we say that p hits q. The objective is to find a minimum cost subset of H whose members collectively hit those of D. In this paper we provide constant factor approximation algorithms for path hitting, confined to instances in which the underlying graph is a tree, a spider, or a star. Although such restricted settings may appear to be very simple, we demonstrate that they still capture some of the most basic covering problems in graphs. 1
Approximate kSteiner Forests via the Lagrangian Relaxation Technique with Internal Preprocessing
 In 14th Annual European Symposium on Algorithms
, 2006
"... An instance of the kSteiner forest problem consists of an undirected graph G = (V, E), the edges of which are associated with nonnegative costs, and a collection D = {(s1, t1),..., (sd, td)} of distinct pairs of vertices, interchangeably referred to as demands. We say that a forest F ⊆ G connects ..."
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An instance of the kSteiner forest problem consists of an undirected graph G = (V, E), the edges of which are associated with nonnegative costs, and a collection D = {(s1, t1),..., (sd, td)} of distinct pairs of vertices, interchangeably referred to as demands. We say that a forest F ⊆ G connects a demand (si, ti) when it contains an siti path. Given a requirement parameter k ≤ D, the goal is to find a minimum cost forest that connects at least k demands in D. This problem has recently been studied by Hajiaghayi and Jain [SODA ’06], whose main contribution in this context was to relate the inapproximability of kSteiner forest to that of the dense ksubgraph problem. However, Hajiaghayi and Jain did not provide any algorithmic result for the respective settings, and posed this objective as an important direction for future research. In this paper, we present the first nontrivial approximation algorithm for the kSteiner forest problem, which is based on a novel extension of the Lagrangian relaxation technique. Specifically, our algorithm constructs a feasible forest whose cost is within a factor of O(min{n 2/3, √ d} · log d) of optimal, where n is the number of vertices in the input graph and d is the number of demands. We believe that the approach illustrated in the current writing is of independent interest, and may be applicable in other settings as well.
LAGRANGIAN RELAXATION AND PARTIAL COVER (EXTENDED ABSTRACT)
, 2008
"... Lagrangian relaxation has been used extensively in the design of approximation algorithms. This paper studies its strengths and limitations when applied to Partial Cover. We show that for Partial Cover in general no algorithm that uses Lagrangian relaxation and a Lagrangian Multiplier Preserving ( ..."
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Lagrangian relaxation has been used extensively in the design of approximation algorithms. This paper studies its strengths and limitations when applied to Partial Cover. We show that for Partial Cover in general no algorithm that uses Lagrangian relaxation and a Lagrangian Multiplier Preserving (LMP) αapproximation as a black box can yield an approximation factor better than 4 3 α. This matches the upper bound given by Könemann et al. (ESA 2006, pages 468–479). Faced with this limitation we study a specific, yet broad class of covering problems: Partial Totally Balanced Cover. By carefully analyzing the inner workings of the LMP algorithm we are able to give an almost tight characterization of the integrality gap of the standard linear relaxation of the problem. As a consequence we obtain improved approximations for the Partial version of Multicut and Path Hitting on Trees, Rectangle Stabbing, and Set Cover with ρBlocks.
Primaldual algorithms for combinatorial optimization problems
, 2007
"... Combinatorial optimization problems such as routing, scheduling, covering and packing problems abound in everyday life. At a very high level, a combinatorial optimization problem amounts to finding a solution with minimum or maximum cost among a large number of feasible solutions. An algorithm for a ..."
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Combinatorial optimization problems such as routing, scheduling, covering and packing problems abound in everyday life. At a very high level, a combinatorial optimization problem amounts to finding a solution with minimum or maximum cost among a large number of feasible solutions. An algorithm for a given optimization problem is said to be exact if it always returns an optimal solution and is said to be efficient if it runs in time polynomial on the size of its input. The theory of NPcompleteness suggests that exact and efficient algorithms are unlikely to exist for the class of NPhard problems. Unfortunately, a large number of natural and interesting combinatorial optimization problems are NPhard. One way to cope with NPhardness is to relax the optimality requirement and instead look for solutions that are provably close to the optimum. This is the main idea behind approximation algorithms. An algorithm is said to be a ρapproximation if it always returns a solution whose cost is at most a ρ factor away from the optimal cost. Arguably, one of the most important techniques in the design of combinatorial algorithms is the primaldual schema in which the cost of the primal solution is compared to the cost of a dual solution. In this dissertation we study the primaldual schema in the design of approximation algorithms for a number of covering and scheduling problems.
LAGRANGIAN RELAXATION AND PARTIAL COVER
"... Lagrangian relaxation has been used extensively in the design of approximation algorithms. This paper studies its strengths and limitations when applied to Partial Cover. We show that for Partial Cover in general no algorithm that uses Lagrangian relaxation and a Lagrangian Multiplier Preserving (LM ..."
Abstract

Cited by 1 (0 self)
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Lagrangian relaxation has been used extensively in the design of approximation algorithms. This paper studies its strengths and limitations when applied to Partial Cover. We show that for Partial Cover in general no algorithm that uses Lagrangian relaxation and a Lagrangian Multiplier Preserving (LMP) αapproximation as a black box can yield an approximation factor better than 4/3 α. This matches the upper bound given by Könemann et al. (ESA 2006, pages 468–479). Faced with this limitation we study a specific, yet broad class of covering problems: Partial Totally Balanced Cover. By carefully analyzing the inner workings of the LMP algorithm we are able to give an almost tight characterization of the integrality gap of the standard linear relaxation of the problem. As a consequence we obtain improved approximations for the Partial version of Multicut and Path Hitting on Trees, Rectangle Stabbing, and Set Cover with ρBlocks.
Approximation Algorithms for kHurdle Problems
"... Abstract. The polynomialtime solvable khurdle problem is a natural generalization of the classical st minimum cut problem where we must select a minimumcost subset S of the edges of a graph such that ..."
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Abstract. The polynomialtime solvable khurdle problem is a natural generalization of the classical st minimum cut problem where we must select a minimumcost subset S of the edges of a graph such that