We consider some poorly studied clustering problems. The paper purpose is to present a short survey on some new results on the computational complexity of these problems, and on efficient algorithms with performance g...
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We consider optimal route planning when the objective function is a general nonlinear and non-monotonic function. Such an objective models user behavior more accurately, for example, when a user is risk-averse, or the...
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ISBN:
(纸本)9781577357605
We consider optimal route planning when the objective function is a general nonlinear and non-monotonic function. Such an objective models user behavior more accurately, for example, when a user is risk-averse, or the utility function needs to capture a penalty for early arrival. It is known that as nonlinearity arises, the problem becomes NP-hard and little is known about computing optimal solutions when in addition there is no monotonicity guarantee. We show that an approximately optimal non-simple path can be efficiently computed under some natural constraints. In particular, we provide a fully polynomial approximation scheme under hop constraints. Our approximation algorithm can extend to run in pseudo-polynomial time under a more general linear constraint that sometimes is useful. As a by-product, we show that our algorithm can be applied to the problem of finding a path that is most likely to be on time for a given deadline.
In this paper we give randomized approximation algorithms for stochastic cumulative VRPs for split and unsplit deliveries. The approximation ratios are 2(1 + alpha) and 7 respectively, where a is the approximation rat...
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ISBN:
(纸本)9783319292212;9783319292205
In this paper we give randomized approximation algorithms for stochastic cumulative VRPs for split and unsplit deliveries. The approximation ratios are 2(1 + alpha) and 7 respectively, where a is the approximation ratio for the metric TSP. The approximation factor is further reduced for trees and paths. These results extend the results in [Technical note - approximation algorithms for VRP with stochastic demands. Operations Research, 2012] and [Routing vehicles to minimize fuel consumption. Operations Research Letters, 2013].
The most basic form of the max-sum dispersion problem ( MSD ) is as follows: given n points in R q and an integer k, select a set of k points such that the sum of the pairwise distances within the set is maximal. This...
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The most basic form of the max-sum dispersion problem ( MSD ) is as follows: given n points in R q and an integer k, select a set of k points such that the sum of the pairwise distances within the set is maximal. This is a prominent diversity problem, with wide applications in web search and information retrieval, where one needs to find a small and diverse representa- tive subset of a large dataset. The problem has recently received a great deal of attention in the computational geometry and operations research communities; and since it is NP-hard, research has focused on efficient heuristics and approximation algorithms. Several classes of distance functions have been considered in the literature. Many of the most common distances used in applications are induced by a norm in a real vector space. The focus of this thesis is on MSD over these geometric instances. We provide for it simple and fast polynomial-time approximation schemes (PTASs), as well as improved constant-factor approximation algorithms. We pay special attention to the class of negative-type distances, a class that includes Euclidean and Manhattan distances, among many others. In order to exploit the properties of this class, we apply several techniques and results from the theory of isometric embeddings. We explore the following variations of the MSD problem: matroid and matroid-intersection constraints, knapsack constraints, and the mixed-objective problem that maximizes a combi- nation of the sum of pairwise distances with a submodular monotone function. In addition to approximation algorithms, we present a core-set for geometric instances of low dimension, and we discuss the efficient implementation of some of our algorithms for massive datasets, using the streaming and distributed models of computation.
We study the prize-collecting version of the node-weighted Steiner tree problem (NWPCST) restricted to planar graphs. We give a new primal-dual Lagrangian-multiplier-preserving (LMP) 3-approximation algorithm for plan...
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We consider the problem of scheduling with renewable speed-up resources. Given m identical machines, n jobs and c different discrete resources, the task is to schedule each job non-preemptively onto one of the machine...
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ISBN:
(纸本)9783959770187
We consider the problem of scheduling with renewable speed-up resources. Given m identical machines, n jobs and c different discrete resources, the task is to schedule each job non-preemptively onto one of the machines so as to minimize the makespan. In our problem, a job has its original processing time, which could be reduced by utilizing one of the resources. As resources are different, the amount of the time reduced for each job is different depending on the resource it uses. Once a resource is being used by one job, it can not be used simultaneously by any other job until this job is finished, hence the scheduler should take into account the job-To-machine assignment together with the resource-To-job assignment. We observe that, the classical unrelated machine scheduling problem is actually a special case of our problem when m = c, i.e., the number of resources equals the number of machines. Extending the techniques for the unrelated machine scheduling, we give a 2-approximation algorithm when both m and c are part of the input. We then consider two special cases for the problem, with m or c being a constant, and derive PTASes (Polynomial Time approximation Schemes) respectively. We also establish the relationship between the two parameters m and c, through which we are able to transform the PTAS for the case when m is constant to the case when c is a constant. The relationship between the two parameters reveals the structure within the problem, and may be of independent interest.
We study variants of the capacitated vehicle routing problem. In the multiple depot capacitated k-travelling repairmen problem (MD-CkTRP), we have a collection of clients to be served by one vehicle in a fleet of k id...
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We consider clustering problems with non-uniform lower bounds and outliers, and obtain the first approximation guarantees for these problems. We have a set F of facilities with lower bounds {Li}i∈F and a set D of cli...
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ISBN:
(纸本)9783959770132
We consider clustering problems with non-uniform lower bounds and outliers, and obtain the first approximation guarantees for these problems. We have a set F of facilities with lower bounds {Li}i∈F and a set D of clients located in a common metric space {c(i, j)}i,j∈F∪D, and bounds k, m. A feasible solution is a pair (S ⊆ F, σ : D → S ∪ {out}), where σ specifies the client assignments, such that |S| ≤ k, |σ-1(i)| ≥ Li for all i ∈ S, and |σ-1(out)| ≤ m. In the lPower-bounded min-sum-of-radii with outliers (LBkSRO) problem, the objective is to minimize Σi∈S maxj∈σ-1(i) c(i, j), and in the lower-bounded k-supplier with outliers (LBkSupO) problem, the objective is to minimize maxi∈S maxj∈σ-1(i) c(i, j). We obtain an approximation factor of 12.365 for LBkSRO, which improves to 3.83 for the non-outlier version (i.e., m = 0). These also constitute the first approximation bounds for the min-sum-of-radii objective when we consider lower bounds and outliers separately. We apply the primal-dual method to the relaxation where we Lagrangify the |S| ≤ k constraint. The chief technical contribution and novelty of our algorithm is that, departing from the standard paradigm used for such constrained problems, we obtain an O(1)-approximation despite the fact that we do not obtain a Lagrangian-multiplier-preserving algorithm for the Lagrangian relaxation. We believe that our ideas have broader applicability to other clustering problems with outliers as well. We obtain approximation factors of 5 and 3 respectively for LBkSupO and its non-outlier version. These are the first approximation results for k-supplier with non-uniform lower bounds.
A tree cover is a collection of subtrees of a graph such that each vertex is a part of at least one subtree. The bounded tree cover problem (BTC) requires to find a tree cover with minimum number of subtrees of bounde...
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The Generalized Traveling Salesman Problem (GTSP) is a generalization of the well known Traveling Salesman Problem (TSP), where along with a weighted graph G = (V, E, w) we are given by a partition of its node set V =...
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