1,239 research outputs found
Constant Factor Approximation for Capacitated k-Center with Outliers
The -center problem is a classic facility location problem, where given an
edge-weighted graph one is to find a subset of vertices ,
such that each vertex in is "close" to some vertex in . The
approximation status of this basic problem is well understood, as a simple
2-approximation algorithm is known to be tight. Consequently different
extensions were studied.
In the capacitated version of the problem each vertex is assigned a capacity,
which is a strict upper bound on the number of clients a facility can serve,
when located at this vertex. A constant factor approximation for the
capacitated -center was obtained last year by Cygan, Hajiaghayi and Khuller
[FOCS'12], which was recently improved to a 9-approximation by An, Bhaskara and
Svensson [arXiv'13].
In a different generalization of the problem some clients (denoted as
outliers) may be disregarded. Here we are additionally given an integer and
the goal is to serve exactly clients, which the algorithm is free to
choose. In 2001 Charikar et al. [SODA'01] presented a 3-approximation for the
-center problem with outliers.
In this paper we consider a common generalization of the two extensions
previously studied separately, i.e. we work with the capacitated -center
with outliers. We present the first constant factor approximation algorithm
with approximation ratio of 25 even for the case of non-uniform hard
capacities.Comment: 15 pages, 3 figures, accepted to STACS 201
Robust Branch-Cut-and-Price for the Capacitated Minimum Spanning Tree Problem over a Large Extended Formulation
This paper presents a robust branch-cut-and-price algorithm for the Capacitated Minimum Spanning Tree Problem (CMST). The variables are associated to q-arbs, a structure that arises from a relaxation of the capacitated prize-collecting arbores- cence problem in order to make it solvable in pseudo-polynomial time. Traditional inequalities over the arc formulation, like Capacity Cuts, are also used. Moreover, a novel feature is introduced in such kind of algorithms. Powerful new cuts expressed over a very large set of variables could be added, without increasing the complexity of the pricing subproblem or the size of the LPs that are actually solved. Computational results on benchmark instances from the OR-Library show very signi¯cant improvements over previous algorithms. Several open instances could be solved to optimalityNo keywords;
Centrality of Trees for Capacitated k-Center
There is a large discrepancy in our understanding of uncapacitated and
capacitated versions of network location problems. This is perhaps best
illustrated by the classical k-center problem: there is a simple tight
2-approximation algorithm for the uncapacitated version whereas the first
constant factor approximation algorithm for the general version with capacities
was only recently obtained by using an intricate rounding algorithm that
achieves an approximation guarantee in the hundreds.
Our paper aims to bridge this discrepancy. For the capacitated k-center
problem, we give a simple algorithm with a clean analysis that allows us to
prove an approximation guarantee of 9. It uses the standard LP relaxation and
comes close to settling the integrality gap (after necessary preprocessing),
which is narrowed down to either 7, 8 or 9. The algorithm proceeds by first
reducing to special tree instances, and then solves such instances optimally.
Our concept of tree instances is quite versatile, and applies to natural
variants of the capacitated k-center problem for which we also obtain improved
algorithms. Finally, we give evidence to show that more powerful preprocessing
could lead to better algorithms, by giving an approximation algorithm that
beats the integrality gap for instances where all non-zero capacities are
uniform.Comment: 21 pages, 2 figure
Capacitated Center Problems with Two-Sided Bounds and Outliers
In recent years, the capacitated center problems have attracted a lot of
research interest. Given a set of vertices , we want to find a subset of
vertices , called centers, such that the maximum cluster radius is
minimized. Moreover, each center in should satisfy some capacity
constraint, which could be an upper or lower bound on the number of vertices it
can serve. Capacitated -center problems with one-sided bounds (upper or
lower) have been well studied in previous work, and a constant factor
approximation was obtained.
We are the first to study the capacitated center problem with both capacity
lower and upper bounds (with or without outliers). We assume each vertex has a
uniform lower bound and a non-uniform upper bound. For the case of opening
exactly centers, we note that a generalization of a recent LP approach can
achieve constant factor approximation algorithms for our problems. Our main
contribution is a simple combinatorial algorithm for the case where there is no
cardinality constraint on the number of open centers. Our combinatorial
algorithm is simpler and achieves better constant approximation factor compared
to the LP approach
Minimum Makespan Multi-vehicle Dial-a-Ride
Dial a ride problems consist of a metric space (denoting travel time between
vertices) and a set of m objects represented as source-destination pairs, where
each object requires to be moved from its source to destination vertex. We
consider the multi-vehicle Dial a ride problem, with each vehicle having
capacity k and its own depot-vertex, where the objective is to minimize the
maximum completion time (makespan) of the vehicles. We study the "preemptive"
version of the problem, where an object may be left at intermediate vertices
and transported by more than one vehicle, while being moved from source to
destination. Our main results are an O(log^3 n)-approximation algorithm for
preemptive multi-vehicle Dial a ride, and an improved O(log t)-approximation
for its special case when there is no capacity constraint. We also show that
the approximation ratios improve by a log-factor when the underlying metric is
induced by a fixed-minor-free graph.Comment: 22 pages, 1 figure. Preliminary version appeared in ESA 200
A review of the Tabu Search Literature on Traveling Salesman Problems
The Traveling Salesman Problem (TSP) is one of the most widely studied problems inrncombinatorial optimization. It has long been known to be NP-hard and hence research onrndeveloping algorithms for the TSP has focused on approximate methods in addition to exactrnmethods. Tabu search is one of the most widely applied metaheuristic for solving the TSP. Inrnthis paper, we review the tabu search literature on the TSP, point out trends in it, and bringrnout some interesting research gaps in this literature.
Capacitated Vehicle Routing with Non-Uniform Speeds
The capacitated vehicle routing problem (CVRP) involves distributing
(identical) items from a depot to a set of demand locations, using a single
capacitated vehicle. We study a generalization of this problem to the setting
of multiple vehicles having non-uniform speeds (that we call Heterogenous
CVRP), and present a constant-factor approximation algorithm.
The technical heart of our result lies in achieving a constant approximation
to the following TSP variant (called Heterogenous TSP). Given a metric denoting
distances between vertices, a depot r containing k vehicles with possibly
different speeds, the goal is to find a tour for each vehicle (starting and
ending at r), so that every vertex is covered in some tour and the maximum
completion time is minimized. This problem is precisely Heterogenous CVRP when
vehicles are uncapacitated.
The presence of non-uniform speeds introduces difficulties for employing
standard tour-splitting techniques. In order to get a better understanding of
this technique in our context, we appeal to ideas from the 2-approximation for
scheduling in parallel machine of Lenstra et al.. This motivates the
introduction of a new approximate MST construction called Level-Prim, which is
related to Light Approximate Shortest-path Trees. The last component of our
algorithm involves partitioning the Level-Prim tree and matching the resulting
parts to vehicles. This decomposition is more subtle than usual since now we
need to enforce correlation between the size of the parts and their distances
to the depot
A biased random-key genetic algorithm for the capacitated minimum spanning tree problem
This paper focuses on the capacitated minimum spanning tree(CMST)problem.Given a central
processor and a set of remote terminals with specified demands for traffic that must flow between the central processor and terminals,the goal is to design a minimum cost network to carry this demand.
Potential links exist between any pair of terminals and between the central processor and the terminals.
Each potential link can be included in the design at a given cost.The CMST problem is to design a
minimum-cost network connecting the terminals with the central processor so that the flow on any arc of the network is at most Q. A biased random-keygenetic algorithm(BRKGA)is a metaheuristic for combinatorial optimization which evolves a population of random vectors that encode solutions to the combinatorial optimization problem.This paper explores several solution encodings as well as different
strategies for some steps of the algorithm and finally proposes a BRKGA heuristic for the CMST problem.
Computational experiments are presented showing the effectivenes sof the approach:Seven newbest-
known solutions are presented for the set of benchmark instances used in the experiments.Peer ReviewedPostprint (author’s final draft
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