Practical Partitioning-Based Methods for the Steiner Problem

Polzin, Tobias; Daneshmand, Siavash Vahdati

doi:10.1007/11764298_22

Cited by 13 publications

(10 citation statements)

References 6 publications

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“…It seems promising to extend our data reduction scheme to practical solutions of other graph problems. A loosely related approach-also based on graph separation but without the gadgeteering-has been used for solving Steiner tree problems (Polzin and Vahdati Daneshmand, 2006). - Estivill-Castro et al (2006, Sect.…”

Section: Discussionmentioning

confidence: 99%

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Separator-based data reduction for signed graph balancing

2009

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Polynomial-time data reduction is a classical approach to hard graph problems. Typically, particular small subgraphs are replaced by smaller gadgets. We generalize this approach to handle any small subgraph that has a small separator connecting it to the rest of the graph. The problem we study is the NP-hard BALANCED SUBGRAPH problem, which asks for a 2-coloring of a graph that minimizes the inconsistencies with given edge labels. It has applications in social networks, systems biology, and integrated circuit design. The data reduction scheme unifies and generalizes a number of previously known data reductions, and can be applied to a large number of graph problems where a coloring or a subset of the vertices is sought. To solve the instances that remain after reduction, we use a fixed-parameter algorithm based on iterative compression with a very effective heuristic speedup. Our implementation can solve biological real-world instances exactly for which previously only approximations were known. In addition, we present experimental results for financial networks and random networks.

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Section: Discussionmentioning

confidence: 99%

“…A similar method has been suggested by Polzin and Vahdati Daneshmand (2006) for the STEINER TREE problem. However, they do not employ gadgets and have no formal characterization of reducible cases.…”

Section: Rule 1 Let G = (V E) Be a Vertex Bipartization Instance Andmentioning

confidence: 99%

Separator-based data reduction for signed graph balancing

2009

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“…Regarding G's treewidth tw, the oldest but yet strongest result is due to Korach and Solel [24]; yet this technical report has never been officially published and has been cited only rarely, e.g., by Polzin and Daneshmand [26], Betzler [9], and Gassner [22]. Their algorithm achieves a runtime of O(tw 4tw · |V |) but the paper's description is very sketchy and leaves many details unclear; it does not contain a formal proof of either the running time nor of its correctness.…”

Section: Previous Workmentioning

confidence: 98%

Improved Steiner tree algorithms for bounded treewidth

Chimani

Mutzel

Zey

2012

Journal of Discrete Algorithms

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We propose a new algorithm that solves the Steiner tree problem on graphs with vertex set V to optimality in O(B 2 tw+2 · tw · |V |) time, where tw is the graph's treewidth and the Bell number B k is the number of partitions of a k-element set. This is a lineartime algorithm for graphs with fixed treewidth and a polynomial algorithm for tw ∈ O(log |V |/ log log |V |).While being faster than the previously known algorithms, the coloring scheme used in our algorithm can be extended to give new, improved algorithms for the prize-collecting Steiner tree as well as the k-cardinality tree problems with similar runtime bounds.

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“…Hence, numerous implementations exist that are able to solve this problem, often very well, in practice. The most important exact algorithms are due to Zachariasen and Winter [20] for geometric instances, Koch and Martin [21] using integer linear programming techniques, and Polzin and Daneshmand [22,23,24] with the strongest results for general graphs. Also, many powerful heuristics exist, see, for example, [25,26,27,28].…”

Section: Introductionmentioning

confidence: 99%

Dealing with Large Hidden Constants: Engineering a Planar Steiner Tree PTAS

Tazari

Müller‐Hannemann

2009

2009 Proceedings of the Eleventh Workshop on Algorithm Engineering and Experiments (ALENEX)

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We present the first attempt on implementing a highly theoretical polynomial-time approximation scheme (PTAS) with huge hidden constants, namely, the PTAS for Steiner tree in planar graphs by Mathieu (SODA 2007, WADS 2007). Whereas this result, and several other PTAS results of the recent years, are of high theoretical importance, no practical applications or even implementation attempts have been known to date, due to the extremely large constants that are involved in them. We describe techniques on how to circumvent the challenges in implementing such a scheme. Our main contribution is the engineering of several details of the original algorithm to make it work in practice. With today's limitations on processing power and space, we still have to sacrifice approximation guarantees for improved running times by choosing some parameters empirically. But our experiments show that with our choice of parameters, we do get the desired approximation ratios, suggesting that a much tighter analysis might be possible. Hence, we show that it is possible to actually implement and run this algorithm, even on large instances, already today -but under some compromises. Further improvements, both in theory and practice, might make these great theoretical works finally bear practical fruits in the future.First computational experiments with benchmark instances from SteinLib and large artificial instances well exceeded our own expectations. We demonstrate that we are able to handle instances with up to a million nodes and several hundreds of terminals in 1.5 hours on a standard PC. On the rectilinear preprocessed instances from SteinLib, we observe a monotonous improvement for smaller values of ε, with an average gap below 1% for ε = 0.1. We compare our implementation against the well-known batched 1-Steiner heuristic and observe that on very large instances, we are able to produce comparable solutions much faster.

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Practical Partitioning-Based Methods for the Steiner Problem

Cited by 13 publications

References 6 publications

Separator-based data reduction for signed graph balancing

Separator-based data reduction for signed graph balancing

Improved Steiner tree algorithms for bounded treewidth

Dealing with Large Hidden Constants: Engineering a Planar Steiner Tree PTAS

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