53 research outputs found
An optimal bifactor approximation algorithm for the metric uncapacitated facility location problem
We obtain a 1.5-approximation algorithm for the metric uncapacitated facility
location problem (UFL), which improves on the previously best known
1.52-approximation algorithm by Mahdian, Ye and Zhang. Note, that the
approximability lower bound by Guha and Khuller is 1.463.
An algorithm is a {\em (,)-approximation algorithm} if
the solution it produces has total cost at most , where and are the facility and the connection
cost of an optimal solution. Our new algorithm, which is a modification of the
-approximation algorithm of Chudak and Shmoys, is a
(1.6774,1.3738)-approximation algorithm for the UFL problem and is the first
one that touches the approximability limit curve
established by Jain, Mahdian and Saberi. As a consequence, we obtain the first
optimal approximation algorithm for instances dominated by connection costs.
When combined with a (1.11,1.7764)-approximation algorithm proposed by Jain et
al., and later analyzed by Mahdian et al., we obtain the overall approximation
guarantee of 1.5 for the metric UFL problem. We also describe how to use our
algorithm to improve the approximation ratio for the 3-level version of UFL.Comment: A journal versio
The interval constrained 3-coloring problem
In this paper, we settle the open complexity status of interval constrained
coloring with a fixed number of colors. We prove that the problem is already
NP-complete if the number of different colors is 3. Previously, it has only
been known that it is NP-complete, if the number of colors is part of the input
and that the problem is solvable in polynomial time, if the number of colors is
at most 2. We also show that it is hard to satisfy almost all of the
constraints for a feasible instance.Comment: minor revisio
An Optimal Bifactor Approximation Algorithm for the Metric Uncapacitated Facility Location Problem
We consider the metric uncapacitated facility location problem(UFL). In this paper we modify the (1 + 2/e)-approximation algorithm of Chudak and Shmoys to obtain a new (1.6774,1.3738)- approximation algorithm for the UFL problem. Our linear programing rounding algorithm is the first one that touches the approximability limit curve established by Jain et al. As a consequence, we obtain the first optimal approximation algorithm for instances dominated by connection costs. Our new algorithm - when combined with a (1.11,1.7764)-approxima- tion algorithm proposed by Jain, Mahdian and Saberi, and later analyzed by Mahdian, Ye and Zhang - gives a 1.5-approximation algorithm for the metric UFL problem. This algorithm improves over the previously best known 1.52-approximation algorithm by Mahdian, Ye and Zhang, and it cuts the gap with the approximability lower bound by 1/3. The algorithm is also used to improve the approximation ratio for the 3-level version of the problem
Better Approximation Bounds for the Joint Replenishment Problem
The Joint Replenishment Problem (JRP) deals with optimizing shipments of
goods from a supplier to retailers through a shared warehouse. Each shipment
involves transporting goods from the supplier to the warehouse, at a fixed cost
C, followed by a redistribution of these goods from the warehouse to the
retailers that ordered them, where transporting goods to a retailer has
a fixed cost . In addition, retailers incur waiting costs for each
order. The objective is to minimize the overall cost of satisfying all orders,
namely the sum of all shipping and waiting costs.
JRP has been well studied in Operations Research and, more recently, in the
area of approximation algorithms. For arbitrary waiting cost functions, the
best known approximation ratio is 1.8. This ratio can be reduced to 1.574 for
the JRP-D model, where there is no cost for waiting but orders have deadlines.
As for hardness results, it is known that the problem is APX-hard and that the
natural linear program for JRP has integrality gap at least 1.245. Both results
hold even for JRP-D. In the online scenario, the best lower and upper bounds on
the competitive ratio are 2.64 and 3, respectively. The lower bound of 2.64
applies even to the restricted version of JRP, denoted JRP-L, where the waiting
cost function is linear.
We provide several new approximation results for JRP. In the offline case, we
give an algorithm with ratio 1.791, breaking the barrier of 1.8. In the online
case, we show a lower bound of 2.754 on the competitive ratio for JRP-L (and
thus JRP as well), improving the previous bound of 2.64. We also study the
online version of JRP-D, for which we prove that the optimal competitive ratio
is 2
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