The paper studies set covering in fuel-considered vehicle routing problems (FVRP). Firstly, we study the FVRP with distance constraint and time windows (FVRP-TW) whose objective is to find a set covering with the mini...
详细信息
The paper studies set covering in fuel-considered vehicle routing problems (FVRP). Firstly, we study the FVRP with distance constraint and time windows (FVRP-TW) whose objective is to find a set covering with the minimum cardinality, which means the number of used vehicles is minimized and hence the fuel consumption is minimized in the real logistics. We give a bicriteriaapproximation algorithm for this problem. Secondly, we study the set covering in the FVRP with distance constraint and constant time windows (FVRP-CTW), which has a constant number of the time windows provided by logistics companies. We give a bicriteriaapproximation algorithm with (2+epsilon, O(log1/epsilon)) for this problem, in which the first term is the approximation ratio on the distance constraint and the second term is the approximation ratio on the cardinality of the covering set. Thirdly, we study the set covering in general FVRP and propose a lower bound for this problem which is based on total unimodularity. Finally, we design an algorithm framework based on the lower bound for solving the set covering in general FVRP. Simulation results demonstrate the effectiveness of the algorithms for solving the set covering in FVRP-TW and general FVRP. (C) 2015 Elsevier B.V. All rights reserved.
Given a graph H = (V, F) with edge weights {w(e) : e is an element of F}, the weighted degree of a node v in H is Sigma{w(vu): vu is an element of F}. We give bicriteria approximation algorithms for problems that seek...
详细信息
Given a graph H = (V, F) with edge weights {w(e) : e is an element of F}, the weighted degree of a node v in H is Sigma{w(vu): vu is an element of F}. We give bicriteria approximation algorithms for problems that seek to find a minimum cost directed graph that satisfies both intersecting supermodular connectivity requirements and weighted degree constraints. The input to such problems is a directed graph G = (V, E) with edge-costs Ice : e E E) and edge-weights {we : e E E), an intersecting supermodular set-function f on V. and degree bounds {b(v) : v B c V). The goal is to find a minimum cost f-connected subgraph H = (V, F) (namely, at least f (S) edges in F enter every S C V) of G with weighted degrees <= b(v). Our algorithm computes a solution of cost <= 2. opt, so that the weighted degree of every v E V is at most: 7b(v) for arbitrary f and 5b(v) for a 0, 1-valued f;2b(v) + 4 for arbitrary f and 2b(v) 2 for a 0, 1-valued f in the case of unit weights. Another algorithm computes a solution of cost <= 3. opt and weighted degrees <= 6b(v). We obtain similar results when there are both indegree and outdegree constraints, and better results when there are indegree constraints only: a (1, 4b(v))-approximation algorithm for arbitrary weights and a polynomial time algorithm for unit weights. Similar results are shown for crossing supermodular f. We also consider the problem of packing maximum number k of pairwise edge-disjoint arborescences so that their union satisfies weighted degree constraints, and give an algorithm that computes a solution of value at least left perpendiculark/36right perpendicular. Finally, for unit weights and without trying to bound the cost, we give an algorithm that computes a subgraph so that the degree of every v is an element of V is at most b(v) + 3, improving over the approximation b(v) + 4 of Bansal et al. (2008) [2]. (C) 2010 Elsevier B.V. All rights reserved.
We study the problem of efficiently optimizing submodular functions under cardinality constraints in distributed setting. Recently, several distributed algorithms for this problem have been introduced which either ach...
详细信息
ISBN:
(纸本)9781450345934
We study the problem of efficiently optimizing submodular functions under cardinality constraints in distributed setting. Recently, several distributed algorithms for this problem have been introduced which either achieve a sub-optimal solution or they run in super-constant number of rounds of computation. Unlike previous work, we aim to design distributed algorithms in multiple rounds with almost optimal approximation guarantees at the cost of outputting a larger number of elements. Toward this goal, we present a distributed algorithm that, for any epsilon > 0 and any constant r, outputs a set S of O(rk/epsilon(1/r)) items in r rounds, and achieves a (1 - epsilon)-approximation of the value of the optimum set with k items. This is the first distributed algorithm that achieves an approximation factor of (1 - epsilon) running in less than log 1/epsilon number of rounds. We also prove a hardness result showing that the output of any 1 - epsilon approximation distributed algorithm limited to one distributed round should have at least Omega(k/epsilon) items. In light of this hardness result, our distributed algorithm in one round, r = 1, is asymptotically tight in terms of the output size. We support the theoretical guarantees with an extensive empirical study of our algorithm showing that achieving almost optimum solutions is indeed possible in a few rounds for large-scale real datasets.
Traditional network functions such as firewalls and Intrusion Detection Systems (IDS) are implemented in costly dedicated hardware, making the networks expensive to manage and inflexible to changes. Network function v...
详细信息
Traditional network functions such as firewalls and Intrusion Detection Systems (IDS) are implemented in costly dedicated hardware, making the networks expensive to manage and inflexible to changes. Network function virtualization enables flexible and inexpensive operation of network functions, by implementing virtual network functions (VNFs) as software in virtual machines (VMs) that run in commodity servers. However, VNFs are vulnerable to various faults such as software and hardware failures. Without efficient and effective fault tolerant mechanisms, the benefits of deploying VNFs in networks can be traded-off. In this paper, we investigate the problem of fault tolerant VNF placement in cloud networks, by proactively deploying VNFs in stand-by VM instances when necessary. It is challenging because VNFs are usually stateful. This means that stand-by instances require continuous state updates from active instances during their operation, and the fault tolerant methods need to carefully handle such states. Specifically, the placement of active/stand-by VNF instances, the request routing paths to active instances, and state transfer paths to stand-by instances need to be jointly considered. To tackle this challenge, we devise an efficient heuristic algorithm for the fault tolerant VNF placement. We also propose two bicriteria approximation algorithms with provable approximation ratios for the problem without compute or bandwidth constraints. We then consider the dynamic fault recovery problem given that some placed active instances of VNFs may go faulty, for which we propose an approximation algorithm that dynamically switches traffic processing from faulty VNFs to stand-by instances. Simulations with realistic settings show that our algorithms can significantly improve the request admission rate compared to conventional approaches. We finally evaluate the performance of the proposed algorithm for the dynamic fault recovery problem in a real test-bed consisting of bo
暂无评论