123 research outputs found
Unsplittable Euclidean Capacitated Vehicle Routing: A (2+?)-Approximation Algorithm
In the unsplittable capacitated vehicle routing problem, we are given a metric space with a vertex called depot and a set of vertices called terminals. Each terminal is associated with a positive demand between 0 and 1. The goal is to find a minimum length collection of tours starting and ending at the depot such that the demand of each terminal is covered by a single tour (i.e., the demand cannot be split), and the total demand of the terminals in each tour does not exceed the capacity of 1.
Our main result is a polynomial-time (2+?)-approximation algorithm for this problem in the two-dimensional Euclidean plane, i.e., for the special case where the terminals and the depot are associated with points in the Euclidean plane and their distances are defined accordingly. This improves on recent work by Blauth, Traub, and Vygen [IPCO\u2721] and Friggstad, Mousavi, Rahgoshay, and Salavatipour [IPCO\u2722]
The Geometry of Scheduling
We consider the following general scheduling problem: The input consists of n
jobs, each with an arbitrary release time, size, and a monotone function
specifying the cost incurred when the job is completed at a particular time.
The objective is to find a preemptive schedule of minimum aggregate cost. This
problem formulation is general enough to include many natural scheduling
objectives, such as weighted flow, weighted tardiness, and sum of flow squared.
Our main result is a randomized polynomial-time algorithm with an approximation
ratio O(log log nP), where P is the maximum job size. We also give an O(1)
approximation in the special case when all jobs have identical release times.
The main idea is to reduce this scheduling problem to a particular geometric
set-cover problem which is then solved using the local ratio technique and
Varadarajan's quasi-uniform sampling technique. This general algorithmic
approach improves the best known approximation ratios by at least an
exponential factor (and much more in some cases) for essentially all of the
nontrivial common special cases of this problem. Our geometric interpretation
of scheduling may be of independent interest.Comment: Conference version in FOCS 201
A Tight 4/3 Approximation for Capacitated Vehicle Routing in Trees
Given a set of clients with demands, the Capacitated Vehicle Routing problem is to find a set of tours that collectively cover all client demand, such that the capacity of each vehicle is not exceeded and such that the sum of the tour lengths is minimized. In this paper, we provide a 4/3-approximation algorithm for Capacitated Vehicle Routing on trees, improving over the previous best-known approximation ratio of (sqrt{41}-1)/4 by Asano et al.[Asano et al., 2001], while using the same lower bound. Asano et al. show that there exist instances whose optimal cost is 4/3 times this lower bound. Notably, our 4/3 approximation ratio is therefore tight for this lower bound, achieving the best-possible performance
A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms
Parameterization and approximation are two popular ways of coping with
NP-hard problems. More recently, the two have also been combined to derive many
interesting results. We survey developments in the area both from the
algorithmic and hardness perspectives, with emphasis on new techniques and
potential future research directions
Online Admission Control and Embedding of Service Chains
The virtualization and softwarization of modern computer networks enables the
definition and fast deployment of novel network services called service chains:
sequences of virtualized network functions (e.g., firewalls, caches, traffic
optimizers) through which traffic is routed between source and destination.
This paper attends to the problem of admitting and embedding a maximum number
of service chains, i.e., a maximum number of source-destination pairs which are
routed via a sequence of to-be-allocated, capacitated network functions. We
consider an Online variant of this maximum Service Chain Embedding Problem,
short OSCEP, where requests arrive over time, in a worst-case manner. Our main
contribution is a deterministic O(log L)-competitive online algorithm, under
the assumption that capacities are at least logarithmic in L. We show that this
is asymptotically optimal within the class of deterministic and randomized
online algorithms. We also explore lower bounds for offline approximation
algorithms, and prove that the offline problem is APX-hard for unit capacities
and small L > 2, and even Poly-APX-hard in general, when there is no bound on
L. These approximation lower bounds may be of independent interest, as they
also extend to other problems such as Virtual Circuit Routing. Finally, we
present an exact algorithm based on 0-1 programming, implying that the general
offline SCEP is in NP and by the above hardness results it is NP-complete for
constant L.Comment: early version of SIROCCO 2015 pape
A Unified Framework of FPT Approximation Algorithms for Clustering Problems
In this paper, we present a framework for designing FPT approximation algorithms for many k-clustering problems. Our results are based on a new technique for reducing search spaces. A reduced search space is a small subset of the input data that has the guarantee of containing k clients close to the facilities opened in an optimal solution for any clustering problem we consider. We show, somewhat surprisingly, that greedily sampling O(k) clients yields the desired reduced search space, based on which we obtain FPT(k)-time algorithms with improved approximation guarantees for problems such as capacitated clustering, lower-bounded clustering, clustering with service installation costs, fault tolerant clustering, and priority clustering
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