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 give the first constant-factor approximation algorithm for the rooted Orienteering problem, as well as a new problem that we call the Discounted-Reward traveling salesman problem (TSP), motivated by robot navigation. In both problems, we are given a graph with lengths on edges and rewards on nodes, and a start node @math . In the Orienteering problem, the goal is to find a path starting at @math that maximizes the reward collected, subject to a hard limit on the total length of the path. In the Discounted-Reward TSP, instead of a length limit we are given a discount factor @math , and the goal is to maximize the total discounted reward collected, where the reward for a node reached at time @math is discounted by @math . This problem is motivated by an approximation to a planning problem in the Markov decision process (MDP) framework under the commonly employed infinite horizon discounted reward optimality criterion. The approximation arises from a need to deal with exponentially large state spaces that emerge when trying to model one-time events and nonrepeatable rewards (such as for package deliveries). We also consider tree and multiple-path variants of these problems and provide approximations for those as well. Although the unrooted Orienteering problem, where there is no fixed start node @math , has been known to be approximable using algorithms for related problems such as @math -TSP (in which the amount of reward to be collected is fixed and the total length is approximately minimized), ours is the first to approximate the rooted question, solving an open problem in [E. M. Arkin, J. S. B. Mitchell, and G. Narasimhan, Proceedings of the @math th ACM Symposium on Computational Geometry, 1998, pp. 307-316] and [B. Awerbuch, Y. Azar, A. Blum, and S. Vempala, SIAM J. Comput., 28 (1998), pp. 254-262]. We complement our approximation result for Orienteering by showing that the problem is APX-hard. Reducing hidden bias in the data and ensuring fairness in algorithmic data analysis has recently received significant attention. We complement several recent papers in this line of research by introducing a general method to reduce bias in the data through random projections in a fair'' subspace. We apply this method to densest subgraph and @math -means. For densest subgraph, our approach based on fair projections allows to recover both theoretically and empirically an almost optimal, fair, dense subgraph hidden in the input data. We also show that, under the small set expansion hypothesis, approximating this problem beyond a factor of @math is NP-hard and we show a polynomial time algorithm with a matching approximation bound. We further apply our method to @math -means. In a previous paper, [NIPS 2017] showed that problems such as @math -means can be approximated up to a constant factor while ensuring that none of two protected class (e.g., gender, ethnicity) is disparately impacted. We show that fair projections generalize the concept of fairlet introduced by to any number of protected attributes and improve empirically the quality of the resulting clustering. We also present the first constant-factor approximation for an arbitrary number of protected attributes thus settling an open problem recently addressed in several works. We consider the problem of computing the permanent of a 0,1n by n matrix. For a class of matrices corresponding to constant degree expanders we construct a deterministic polynomial time approximation algorithm to within a multiplicative factor ([email protected])^n, for arbitrary @e>0. This is an improvement over the best known approximation factor e^n obtained in Linial, Samorodnitsky and Wigderson (2000) [9], though the latter result was established for arbitrary non-negative matrices. Our results use a recently developed deterministic approximation algorithm for counting partial matchings of a graph (Bayati, Gamarnik, Katz, Nair and Tet ali (2007) [2]) and Jerrum-Vazirani method (Jerrum and Vazirani (1996) [8]) of approximating permanent by near perfect matchings. We present a general framework for solving resource allocation and scheduling problems. Given a resource of fixed size, we present algorithms that approximate the maximum throughput or the minimum loss by a constant factor. Our approximation factors apply to many problems, among which are: (i) real-time scheduling of jobs on parallel machines, (ii) bandwidth allocation for sessions between two endpoints, (iii) general caching, (iv) dynamic storage allocation, and (v) bandwidth allocation on optical line and ring topologies. For some of these problems we provide the first constant factor approximation algorithm. Our algorithms are simple and efficient and are based on the local-ratio technique. We note that they can equivalently be interpreted within the primal-dual schema.