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### Approximation Schemes for Packing Splittable Items with Cardinality Constraints

Leah Epstein, Asaf Levin, Rob van Stee
2010 Algorithmica
We continue the study of bin packing with splittable items and cardinality constraints. In this problem, a set of items must be packed into as few bins as possible.  ...  We first present a scheme for the case k = 2 and then for the general case of constant k. Additionally, we present dual approximation schemes for k = 2 and constant k.  ...  Conclusions In this paper, we provided approximation schemes for bin packing of splittable items with cardinality constraints for all values of k. We also provided dual approximation schemes.  ...

### Approximation Schemes for Packing Splittable Items with Cardinality Constraints [chapter]

Leah Epstein, Rob van Stee
2008 Lecture Notes in Computer Science
We continue the study of bin packing with splittable items and cardinality constraints. In this problem, a set of items must be packed into as few bins as possible.  ...  We first present a scheme for the case k = 2 and then for the general case of constant k. Additionally, we present dual approximation schemes for k = 2 and constant k.  ...  Conclusions In this paper, we provided approximation schemes for bin packing of splittable items with cardinality constraints for all values of k. We also provided dual approximation schemes.  ...

### EPTAS for the dual of splittable bin packing with cardinality constraint [article]

G. Jaykrishnan, Asaf Levin
2022 arXiv   pre-print
The problem considered is the splittable bin packing with cardinality constraint.  ...  Two versions of the splittable bin packing with cardinality constraint have been studied in the literature.  ...  Many variations of the splittable bin packing with cardinality constraint have been studied.  ...

### On Packing Splittable Items with Cardinality Constraints [chapter]

2010 IFIP Advances in Information and Communication Technology
We show that NEXT FIT gives a ( 1 + 1 k ) -approximation asymptotically, for k ≥ 2.  ...  This problem can be modeled as a variation of bin packing where each item corresponds to a different type and the normalized weight of each item can be greater than 1, which is the size of a bin.  ...  Packing splittale items with a cardinality constraint of k parts of items per bin is NP-hard in the strong sense for any fixed k ≥ 3. Proof.  ...

### Approximation Algorithms for Scheduling with Class Constraints [article]

Klaus Jansen and Alexandra Lassota and Marten Maack
2019 arXiv   pre-print
Even though this problem is closely related to the Class Constraint Bin Packing, the Class Constraint Knapsack and the Cardinality Constraint variants, CCS lacks results regarding approximation algorithms  ...  Further we developed the first simple approximation algorithms with a constant approximation ratio running in strongly polynomial time.  ...  For CCS approximation schemes are known for two special cases.  ...

### Approximation of the k-batch consolidation problem

Sung-Pil Hong, Myoung-Ju Park, Soo Y. Chang
2009 Theoretical Computer Science
In particular, this implies an upper bound on the approximation factor, 2H k − 1, where H k = 1 + 1 2 + · · · + 1 k .  ...  We will show that the k-batch consolidation problem admits an approximation whose factor is twice that of the k-set cover problem.  ...  Another related model is packing splittable items with cardinality constraints, or PSIC  .  ...

### Empowering the Configuration-IP - New PTAS Results for Scheduling with Setups Times

Klaus Jansen, Kim-Manuel Klein, Marten Maack, Malin Rau, Michael Wagner
2018 Innovations in Theoretical Computer Science
For both of the variants that jobs can be executed in parallel or not, we obtain an efficient polynomial time approximation scheme (EPTAS) of running time f (1/ε) × poly(|I|) with a single exponential  ...  Integer linear programs of configurations, or configuration IPs, are a classical tool in the design of algorithms for scheduling and packing problems, where a set of items has to be placed in multiple  ...  Acknowledgements We thank Syamantak Das for helpful discussions on the problem. 1 German Research Foundation (DFG) project JA 612/20-1  ...

### Improved results for a memory allocation problem [article]

Leah Epstein, Rob van Stee
2006 arXiv   pre-print
We consider a memory allocation problem that can be modeled as a version of bin packing where items may be split, but each bin may contain at most two (parts of) items.  ...  A 3/2-approximation algorithm and an NP-hardness proof for this problem was given by Chung et al. We give a simpler 3/2-approximation algorithm for it which is in fact an online algorithm.  ...  Given such an instance of the 3-Partition problem we define an instance of the splittable item packing with cardinality constraints as follows.  ...

### Improved Results for a Memory Allocation Problem

Leah Epstein, Rob van Stee
2009 Theory of Computing Systems
This problem can be modeled as a version of bin packing where items may be split, but each bin may contain at most two (parts of) items. This problem was recently introduced by Chung et al.  .  ...  Additionally, we design an efficient approximation algorithm, for which the approximation ratio can be made arbitrarily close to 7 5 .  ...  NP-hardness of the problem (in the strong sense) Theorem 1 Packing splittable items with a cardinality constraint of k parts of items per bin is NP-hard in the strong sense for any fixed k ≥ 3.  ...

### Improved Results for a Memory Allocation Problem [chapter]

Leah Epstein, Rob van Stee
2007 Lecture Notes in Computer Science
This problem can be modeled as a version of bin packing where items may be split, but each bin may contain at most two (parts of) items. This problem was recently introduced by Chung et al.  .  ...  Additionally, we design an efficient approximation algorithm, for which the approximation ratio can be made arbitrarily close to 7 5 .  ...  NP-hardness of the problem (in the strong sense) Theorem 1 Packing splittable items with a cardinality constraint of k parts of items per bin is NP-hard in the strong sense for any fixed k ≥ 3.  ...

### Empowering the Configuration-IP - New PTAS Results for Scheduling with Setups Times [article]

Klaus Jansen, Kim-Manuel Klein, Marten Maack, Malin Rau
2018 arXiv   pre-print
For both of the variants that jobs can be executed in parallel or not, we obtain an efficient polynomial time approximation scheme (EPTAS) of running time f(1/ε)×poly(|I|) with a single exponential term  ...  Integer linear programs of configurations, or configuration IPs, are a classical tool in the design of algorithms for scheduling and packing problems, where a set of items has to be placed in multiple  ...  schemes for scheduling problems with setup times.  ...

### The Train Delivery Problem - Vehicle Routing Meets Bin Packing [chapter]

Aparna Das, Claire Mathieu, Shay Mozes
2011 Lecture Notes in Computer Science
Alternatively, we give a polynomial time approximation scheme for the case where W , an input parameter that corresponds to the bin size or the vehicle capacity, is polynomial in the number of items or  ...  The train delivery problem is strongly NP-Hard and does not admit an approximation ratio better than 3/2. We design the first approximation schemes for the problem.  ...  Such algorithms are called asymptotic polynomial time approximation schemes (APTAS). Our Results. We give the first approximation schemes for the train delivery problem.  ...

### Heterogeneous Resource Allocation under Degree Constraints

Olivier Beaumont, Lionel Eyraud-Dubois, Christopher Thraves Caro, Hejer Rejeb
2013 IEEE Transactions on Parallel and Distributed Systems
In this paper, we consider the problem of assigning a set of clients with demands to a set of servers with capacities and degree constraints.  ...  We first show that the degree constraint on the maximal number of clients that a server can handle is realistic in many contexts.  ...  Related Works A closely related problem is Bin Packing with Splittable Items and Cardinality Constraints, where the goal is to pack a given set of items in as few bins as possible.  ...

### Reliable Service Allocation in Clouds with Memory and Capacity Constraints [chapter]

Olivier Beaumont, Lionel Eyraud-Dubois, Pierre Pesneau, Paul Renaud-Goud
2014 Lecture Notes in Computer Science
In this context, the goal for the Cloud provider is to find an allocation of VMs onto PMs so as to satisfy, at minimal cost, both capacity and reliability constraints for each service.  ...  In this paper, we propose a simple model for reliability constraints and we prove that it is possible to derive efficient heuristics.  ...  Under this assumption, the packing problem was equivalent to a mono-dimensional splittable item bin packing problem, that can be solved in polynomial time without cardinality constraints.  ...

### The Vertex Coloring Problem and its generalizations

Enrico Malaguti
2008 4OR
Jansen and Oehring  and Jansen  have shown that, for general graph, no polynomial time approximation schemes can be derived, while they derive polynomial time approximation schemes for special  ...  The corresponding problem is known as Bounded Vertex Coloring Problem (BVCP) or Bin Packing Problem with Conflicts (where the Bin Packing Problem requires to assign a set of items, each one with a positive  ...  The problem of routing packets through the network should consider both the efficiency of the routing and its fairness, i.e. the contribution that each node gives to the network with respect to the benefit  ...
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