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Pricing on Paths: A PTAS for the Highway Problem
"... In the highway problem, we are given an nedge line graph (the highway), and a set of paths (the drivers), each one with its own budget. For a given assignment of edge weights (the tolls), the highway owner collects from each driver the weight of the associated path, when it does not exceed the budg ..."
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In the highway problem, we are given an nedge line graph (the highway), and a set of paths (the drivers), each one with its own budget. For a given assignment of edge weights (the tolls), the highway owner collects from each driver the weight of the associated path, when it does not exceed the budget of the driver, and zero otherwise. The goal is choosing weights so as to maximize the profit. A lot of research has been devoted to this apparently simple problem. The highway problem was shown to be strongly NPhard only recently [Elbassioni,Raman,Ray,Sitters’09]. The bestknown approximation is O(log n / log log n) [Gamzu,Segev’10], which improves on the previousbest O(log n) approximation [Balcan,Blum’06]. Better approximations are known for a number of special cases. Finding a constant (or better!) approximation algorithm for the general case is a challenging open problem. In this paper we present a PTAS for the highway problem, hence closing the complexity status of the problem. Our result is based on a novel randomized dissection approach, which has some points in common with Arora’s quadtree dissection for Euclidean network design [Arora’98]. The basic idea is enclosing the highway in a bounding path, such that both the size of the bounding path and the position of the highway in it are random variables. Then we consider a recursive O(1)ary dissection of the bounding path, in subpaths of uniform optimal weight. Since the optimal weights are unknown, we construct the dissection in a bottomup fashion via dynamic programming, while computing the approximate solution at the same time. Our algorithm can be easily derandomized. The same basic approach provides PTASs also for two generalizations of the problem: the tollbooth problem with a constant number of leaves and the maximumfeasibility subsystem problem on interval matrices. In both cases the previous best approximation factors are polylogarithmic [Gamzu,Segev’10,Elbassioni,Raman,Ray,Sitters’09].
How UnsplittableFlowCovering Helps Scheduling with JobDependent Cost Functions,
"... Abstract. Generalizing many wellknown and natural scheduling problems, scheduling with jobspecific cost functions has gained a lot of attention recently. In this setting, each job incurs a cost depending on its completion time, given by a private cost function, and one seeks to schedule the jobs ..."
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Abstract. Generalizing many wellknown and natural scheduling problems, scheduling with jobspecific cost functions has gained a lot of attention recently. In this setting, each job incurs a cost depending on its completion time, given by a private cost function, and one seeks to schedule the jobs to minimize the total sum of these costs. The framework captures many important scheduling objectives such as weighted flow time or weighted tardiness. Still, the general case as well as the mentioned special cases are far from being very well understood yet, even for only one machine. Aiming for better general understanding of this problem, in this paper we focus on the case of uniform job release dates on one machine for which the state of the art is a 4approximation algorithm. This is true even for a special case that is equivalent to the covering version of the wellstudied and prominent unsplittable flow on a path problem, which is interesting in its own right. For that covering problem, we present a quasipolynomial time (1 + ε)approximation algorithm that yields an (e + ε)approximation for the above scheduling problem. Moreover, for the latter we devise the best possible resource augmentation result regarding speed: a polynomial time algorithm which computes a solution with optimal cost at 1 + ε speedup. Finally, we present an elegant QPTAS for the special case where the cost functions of the jobs fall into at most log n many classes. This algorithm allows the jobs even to have up to log n many distinct release dates. All proposed quasipolynomial time algorithms require the input data to be quasipolynomially bounded. 1
Improved Approximation Algorithms for Unsplittable Flow on a Path with Time Windows
"... In the wellstudied Unsplittable Flow on a Path problem (UFP), we are given a path graph with edge capacities. Furthermore, we are given a collection of n tasks, each one characterized by a subpath, a weight, and a demand. Our goal is to select a maximum weight subset of tasks so that the total de ..."
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In the wellstudied Unsplittable Flow on a Path problem (UFP), we are given a path graph with edge capacities. Furthermore, we are given a collection of n tasks, each one characterized by a subpath, a weight, and a demand. Our goal is to select a maximum weight subset of tasks so that the total demand of selected tasks using each edge is upper bounded by the corresponding capacity. Chakaravarthy et al. [ESA’14] studied a generalization of UFP, bagUFP, where tasks are partitioned into bags, and we can select at most one task per bag. Intuitively, bags model jobs that can be executed at different times (with different duration, weight, and demand). They gave a O(logn) approximation for bagUFP. This is also the best known ratio in the case of uniform weights. In this paper we achieve the following main results: •We present an LPbased O(logn / log logn) approximation for bagUFP. We remark that, prior to our work, the best known integrality gap (for a nonextended formulation) was O(logn) even in the special case of UFP [Chekuri et al., APPROX’09]. •We present an LPbasedO(1) approximation for uniformweight bagUFP. This also generalizes the integrality gap bound for uniformweight UFP by Anagnostopoulos et al. [IPCO’13]. •We consider a relevant special case of bagUFP, twUFP, where tasks in a bag model the possible ways in which we can schedule a job with a given processing time within a given time window. We present a QPTAS for twUFP with quasipolynomial demands and under the Bounded TimeWindow Assumption, i.e. assuming that the time window size of each job is within a constant factor from its processing time. This generalizes the QPTAS for UFP by Bansal et al. [STOC’06].