Obtaining Optimal<i>k</i>-Cardinality Trees Fast

Chimani, Markus; Kandyba, Maria; Ljubić, Ivana; Mutzel, Petra

doi:10.1137/1.9781611972887.3

Cited by 10 publications

(14 citation statements)

References 19 publications

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“…Goemans and Myung [7] or Chimani et al [2], for example. Observe that by construction of the graph H and by indegree inequalities (3), x j 1 j 2 = y j 2 and x i 1 i 2 = y i 2 , and thus inequality (8) follows immediately.…”

Section: The Equivalence Of the Gsec Model And The Directed Cut Modelmentioning

confidence: 99%

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The regenerator location problem

2009

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In this paper we introduce the regenerator location problem (RLP), which deals with a constraint on the geographical extent of transmission in optical networks. Specifically, an optical signal can only travel a maximum distance of d max before its quality deteriorates to the point that it must be regenerated by installing regenerators at nodes of the network. As the cost of a regenerator is high we wish to deploy as few regenerators as possible in the network, while ensuring all nodes can communicate with each other. We show that the RLP is NP-Complete. We then devise three heuristics for the RLP. We show how to represent the RLP as a max leaf spanning tree problem (MLSTP) on a transformed graph. Using this fact we model the RLP as a Steiner arborescence problem (SAP) with a unit degree constraint on the root node. We also devise a branch-and-cut procedure to the directed cut formulation for the SAP problem. In our computational results over 740 test instances, the heuristic procedures obtained the optimal solution in 454 instances, while the branch-and-cut procedure obtained the optimal solution in 536 instances. These results indicate the quality of the heuristic solutions are quite good, and the branch-and-cut approach is viable for the optimal solution of problems with up to 100 nodes. Our approaches are also directly applicable to the MLSTP indicating that both the heuristics and branch-and-cut approach are viable options for the MLSTP.

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Section: The Equivalence Of the Gsec Model And The Directed Cut Modelmentioning

confidence: 99%

“…Consequently, in Set2, we focused our attention only on creating instances on randomly generated networks. In fact, we use the very same instances from Set1 and modify them by assigning an integer weight uniformly in [2,4]. 3 Thus like Set1, there are a total of 200 WRLP instances.…”

Section: Data Setsmentioning

confidence: 99%

The regenerator location problem

2009

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“…i,j ∈δ − j y ij ≥ 1, ∀j ∈ V \ {0}, 3.14 to directed cut-based formulations, see 20,21 . Inequalities 3.13 avoid short cycles corresponding to a single edge, Inequalities 3.14 assure that each node has one incoming arc.…”

Section: 12mentioning

confidence: 99%

Solving the Minimum Label Spanning Tree Problem by Mathematical Programming Techniques

Chwatal

Raidl

2011

Advances in Operations Research

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We present exact mixed integer programming approaches including branch-and-cut and branchand-cut-and-price for the minimum label spanning tree problem as well as a variant of it having multiple labels assigned to each edge. We compare formulations based on network flows and directed connectivity cuts. Further, we show how to use odd-hole inequalities and additional inequalities to strengthen the formulation. Label variables can be added dynamically to the model in the pricing step. Primal heuristics are incorporated into the framework to speed up the overall solution process. After a polyhedral comparison of the involved formulations, comprehensive computational experiments are presented in order to compare and evaluate the underlying formulations and the particular algorithmic building blocks of the overall branch-and-cut-andprice framework.

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“…These problems consist of finding one or more trees (or paths) on a given graph satisfying some given constraints while minimizing or maximizing an objective function. Some COT problems have been considered and solved in the literature, e.g., Degree Constrained Minimum Spanning Tree (DCMST) [45,7], Bounded Diameter Minimum Spanning Tree (BDMST) [35], Capacitated Minimum Spanning Tree problem (CMST) [56,3], Minimum Diameter Spanning Tree (MDST) [50], EdgeWeighted k-Cardinality Tree (KCT), [20,25], Steiner Minimal Tree (SMT) [66,28], Optimum Communication Spanning Tree problems (OCST) [32], etc. We also see many COP problems which have been studied and solved in the literature.…”

Section: Introductionmentioning

confidence: 99%

“…Most of these COT/COP problems are NP-hard. They are often approached by dedicated algorithms including exact methods, such as the Lagrangian-based heuristic [7], the ILP-based algorithm using directed cuts [25], the Lagrangian-based branch and bound in [15], and the vertex labeling algorithm from [30]; there are also meta-heuristic algorithms such as a hybrid evolutionary algorithm [19], ant colony optimization [21], and local search [20]. These techniques exploit the structure of the constraints and the objective functions but are often difficult to extend or reuse.…”

Section: Introductionmentioning

confidence: 99%

LS(Graph): a constraint-based local search for constraint optimization on trees and paths

2012

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Constrained optimum tree (COT) and constrained optimum path (COP) problems arise in many real-life applications and are ubiquitous in communication networks. They have been traditionally approached by dedicated algorithms, which are often hard to extend with side constraints and to apply widely. This paper proposes a constraint-based local search framework for COT/COP applications, bringing the compositionality, reuse, and extensibility at the core of constraint-based local search and constraint programming systems. The modeling contribution is the ability to express compositional models for various COT/COP applications at a high level of abstraction, while cleanly separating the model and the search procedure. The main technical contribution is a connected neighborhood based on rooted spanning trees to find high-quality solutions to COP problems. This framework is applied to some COT/COP problems, e.g., the quorumcast routing problem, the edge-disjoint paths problem, and the routing and wavelength assignment with delay side constraints problem. Computational results show the potential importance of the approach.

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Obtaining Optimalk-Cardinality Trees Fast

Cited by 10 publications

References 19 publications

The regenerator location problem

The regenerator location problem

Solving the Minimum Label Spanning Tree Problem by Mathematical Programming Techniques

LS(Graph): a constraint-based local search for constraint optimization on trees and paths

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