In this paper, we consider the problem of choosing a minimum cost set of resources for executing a specified set of jobs. Each input job is an interval, determined by its start-time and end-time. Each resource is also an interval determined by its start-time and end-time; moreover, every resource has a capacity and a cost associated with it. We consider two versions of this problem. In the partial covering version, we are also given as input a number k, specifying the number of jobs that must be performed. The goal is to choose k jobs and find a minimum cost set of resources to perform the chosen k jobs (at any point of time the capacity of the chosen set of resources should be sufficient to execute the jobs active at that time). We present an O(log n)-factor approximation algorithm for this problem. We also consider the prize collecting version, wherein every job also has a penalty associated with it. The feasible solution consists of a subset of the jobs, and a set of resources, to perform the chosen subset of jobs. The goal is to find a feasible solution that minimizes the sum of the costs of the selected resources and the penalties of the jobs that are not selected. We present a constant factor approximation algorithm for this problem
We also show a constant factor approximation algorithm for the problem, by reducing it to the zero-one version of the problem. Jain and Vazirani @cite_2 provide a general framework for achieving approximation algorithms for partial covering problems, wherein the prize collecting version is considered. In this framework, under suitable conditions, a constant factor approximation for the prize collecting version implies a constant factor approximation for the partial version as well. However, their result applies only when the prize collecting algorithm has a certain strong property, called the Lagrangian Multiplier Preserving (LMP) property. While we are able to achieve a constant factor approximation for the problem, our algorithm does not have the LMP property. Thus, the Jain-Vazirani framework does not apply to our scenario.
@cite_2
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In this paper we study the prize-collecting version of the Generalized Steiner Tree problem. To the best of our knowledge, there is no general combinatorial technique in approximation algorithms developed to study the prize-collecting versions of various problems. These problems are studied on a case by case basis by [5] by applying an LP-rounding technique which is not a combinatorial approach. The main contribution of this paper is to introduce a general combinatorial approach towards solving these problems through novel primal-dual schema (without any need to solve an LP). We fuse the primal-dual schema with Farkas lemma to obtain a combinatorial 3-approximation algorithm for the Prize-Collecting Generalized Steiner Tree problem. Our work also inspires a combinatorial algorithm [19] for solving a special case of Kelly's problem [22] of pricing edges.We also consider the k-forest problem, a generalization of k-MST and k-Steiner tree, and we show that in spite of these problems for which there are constant factor approximation algorithms, the k-forest problem is much harder to approximate. In particular, obtaining an approximation factor better than O(n1 6-e) for k-forest requires substantially new ideas including improving the approximation factor O(n1 3-e) for the notorious densest k-subgraph problem. We note that k-forest and prize-collecting version of Generalized Steiner Tree are closely related to each other, since the latter is the Lagrangian relaxation of the former. We show that adaptively sampled O (k ) centers give a constant factor bi-criteria approximation for the k -means problem, with a constant probability. Moreover, these O (k ) centers contain a subset of k centers which give a constant factor approximation, and can be found using LP-based techniques of Jain and Vazirani [JV01] and [CGTS02]. Both these algorithms run in effectively O (nkd ) time and extend the O (logk )-approximation achieved by the k -means++ algorithm of Arthur and Vassilvitskii [AV07]. In a Content Distribution Network application, we have a set of servers and a set of clients to be connected to the servers. Often there are a few server types and a hard budget constraint on the number of deployed servers of each type. The simplest goal here is to deploy a set of servers subject to these budget constraints in order to minimize the sum of client connection costs. These connection costs often satisfy metricity, since they are typically proportional to the distance between a client and a server within a single autonomous system. A special case of the problem where there is only one server type is the well-studied k-median problem. In this paper, we consider the problem with two server types and call it the budgeted red-blue median problem. We show, somewhat surprisingly, that running a single-swap local search for each server type simultaneously, yields a constant factor approximation for this case. Its analysis is however quite non-trivial compared to that of the k-median problem (, 2004; Gupta and Tangwongsan, 2008). Later we show that the same algorithm yields a constant approximation for the prize-collecting version of the budgeted red-blue median problem where each client can potentially be served with an alternative cost via a different vendor. In the process, we also improve the approximation factor for the prize-collecting k-median problem from 4 (, 2001) to 3+e, which matches the current best approximation factor for the k-median problem. We consider a class of optimization problems where the input is an undirected graph with two weight functions defined for each node, namely the node's profit and its cost. The goal is to find a connected set of nodes of low cost and high profit. We present approximation algorithms for three natural optimization criteria that arise in this context, all of which are NP-hard. The budget problem asks for maximizing the profit of the set subject to a budget constraint on its cost. The quota problem requires minimizing the cost of the set subject to a quota constraint on its profit. Finally, the prize collecting problem calls for minimizing the cost of the set plus the profit (here interpreted as a penalty) of the complement set. For all three problems, our algorithms give an approximation guarantee of @math , where @math is the number of nodes. To the best of our knowledge, these are the first approximation results for the quota problem and for the prize collecting problem, both of which are at least as hard to approximate as the set cover. For the budget problem, our results improve on a previous @math result of Our methods involve new theorems relating tree packings to (node) cut conditions. We also show similar theorems (with better bounds) using edge cut conditions. These imply bounds for the analogous budget and quota problems with edge costs which are comparable to known (constant factor) bounds.